tag:blogger.com,1999:blog-75655689080213984842024-02-21T22:26:08.051-08:00A General Relativistic Stationary UniverseAlex Mittelmann, alexmittelmann@yahoo.comhttp://www.blogger.com/profile/03457606761033752726noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-7565568908021398484.post-15621216235229153212020-09-05T04:11:00.000-07:002016-03-17T05:06:59.261-07:00<div style="text-align: center;">
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<span class="Apple-style-span" style="color: #222222; font-size: 18px;"><span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;">Is the universe expanding or not? Judging from the redshift seen in the spectrum of distant celestial objects it would appear so, unless there exists a viable alternative to this interpretation of the spectral shift.</span></span><br />
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<span class="Apple-style-span" style="color: #222222; font-size: 18px;"><span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;">There are two possible interpretations for cosmological redshift z that show wavelength independence over 19 octaves of the spectrum: (1) A change in the scale factor to the metric, implying the expansion of space and the recession of objects in it (i.e., the radius of the universe or scale-factor changes with time </span><i><span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;">t</span></i><span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;">. (2) The general relativistic curved spacetime interpretation (implying a static metric in a stationary universe). In addition to illuminating how redshift z is caused in a globally curved four-dimensional spacetime manifold, it will be shown how objects (such as galaxy clusters and superclusters) remain stable against gravitational collapse without the requirement of a cosmological constant (or vacuum pressure). It is emphasized that global curvature plays an essential role in cosmology and provides a natural explanation for various empirical observations. Too, it is exemplified this point of view by considering a novel version of Einstein's 1916-1917 world-model, where cosmological redshift z is directly related to the large-scale structure of the universe.</span></span></div>
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<b>Key words</b>: theory, cosmology, general relativity, non-Euclidean geometry, hyperbolic spacetime, spherical spacetime, geodesic, isotropy, homogeneity, static universe, stationary universe, local instability, global stability, manifolds, Gaussian surface, pseudo-Riemannian, topology, SNe Ia, observation. </div>
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<b>Introduction:</b></div>
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A great deal of misunderstanding is involved in current interpretation of both general relativity and non-Euclidean geometry when considering a stationary (non-expanding) universe. The purpose of the present work is to provide a short qualitatively and conceptual account of those misinterpretations and to present a new formulation of the problem. Several novel features result form this investigation: (1) redshift z observed in the spectra of astronomical objects may be due to a curved spacetime phenomenon. Moreover it precludes expansion (or the expansion of space) from being a notion consistent with the physical universe in which we live. (2) The stability of the universe follows directly from general relativity when gravity is considered a geometric property of spacetime. The interpretation that the universe is unstable against gravitational collapse, or dispersion, is found to be untenable.</div>
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The stability, or rather, the <i>instability</i> associated with the Einstein field equations was the single most important problem that lead to big bang cosmology, before 1929. That was why Einstein added the cosmological term. The interpretation of redshift as a Doppler effect followed, in a sense, from the apparent instability. The two factors combined to form the foundation of modern cosmology. So pointing out how stability is maintained in a general relativistic universe is just as important and elucidating the origin of redshift z (if it is not a relativistic Doppler effect, that is).</div>
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The standard approach to the problem, to some extent, suppresses the distinction between local and global geometry, while the inclusion of an <i>ad hoc</i> vacuum pressure completely differentiates the local from the global field dynamics. The assumption in this text is that local gravitational fields (surrounding massive bodies) influence the way objects move, while globally the field (which defines the geometric shape of the universe, or the global deviation from linearity) exerts little influence on massive bodies. The result here is that stability is maintained against gravitational collapse without the need of supplemental pressures or forces (e.g., the cosmological constant) to counter gravity. The conclusion is that redshift z is caused as electromagnetic radiation propagates along geodesic paths, while matter remains unaffected by the globally curved non-Euclidean field (the topological space with time <i>t</i> that underlies the definition of the metric tensor). This is an <span style="font: normal normal normal 18px/normal 'Times New Roman';">attempt to make the geometric attributes of the spacetime manifold itself (the metric) account for the observed redshift z.</span></div>
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The implications are numerous and will be discussed, but one of the most obvious ones is that the universe is non-expanding, and there is no big bang event in the past. In contrast to the standard big bang cosmological model, this is a <i>stationary</i>, evolving and dynamic universe that is infinite spatiotemporally, in both the past and future directions of time.</div>
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The following discussion stems from a thread at <a href="http://scienceforums.com/"><span style="color: #3b2e7b; text-decoration: underline;">scienceforums.com</span></a> entitled <a href="http://scienceforums.com/topic/3031-redshift-z/"><span style="color: #3b2e7b; text-decoration: underline;">Redshift z</span></a> which began 07/05/2005.</div>
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<b>A brief note regarding terminology:</b></div>
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<i>Global curvature</i> here refers to the geometry of spacetime (in particular the topology) of the visible universe (and beyond?). This is the curvature induced on the manifold as a result of the presence of all the gravitating mass-energy in the cosmos. We assume the continuum to be, to a good approximation, homogeneous (the same at all locations in space) and isotropic (the same in all spatial directions) at this time, with a nonzero value for gravitational potential (i.e., gravity is everywhere present). For empirical reasons we'll be discussing mostly the visible universe, but there is no need to assume the global curvature will stop at the visible horizon (the cosmic light horizon), since an observer located near the horizon should see the universe much as we do (as if centered in the celestial sphere). So 'global' is not a limiting term in this respect. Observations are limiting of course.</div>
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<i>Local curvature</i> refers here to the geometry of spacetime surrounding massive bodies (in accord with general relativity), where ambient matter acts as the local source of intrinsic curvature, much as electric charge acts as the source of electric fields governed by Gauss' law (whereby local divergence of electric fields relates to to the charge density). There would be deviations from homogeneity locally (humps and bumps or hills and valleys throughout the continuum). Local geometry does not determine the global geometry completely, but it does limit the possibilities, particularly a geometry of a smooth constant curvature. For this reason, the universe is taken to be a <a href="http://en.wikipedia.org/wiki/Geodesic_manifold"><span style="color: #0536cd; text-decoration: underline;">geodesic manifold</span></a>, free of topological defects, or gravitational perturbations. Relaxing either of these geometries would complicate the analysis considerably, but would not invalidate the general conclusion; just as mountains and valleys on the surface of the earth do not change the global topology of the earth (which remains more or less spherical). Often, the term <i>local</i> refers to the entire visible universe (<a href="http://en.wikipedia.org/wiki/Shape_of_the_Universe"><span style="color: #0536cd; text-decoration: underline;">see here for example)</span></a>. That will not be the case in what follows. </div>
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Let's recap by showing schematic diagrams of three global geometries and asking a question: Which of the following spacetime manifolds would appear to be consistent with observations, as viewed from an observers reference frame, and taking into consideration distance measurements, surface brightness tests, light curves or rise times and redshifts of distant Type Ia supernovae (SNe Ia), currently interpreted as an accelerating expansion?<br />
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<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibZ-GlZMUgqkdnGTSDSTVeOtCjIwWU7C0-1j6W4_0Ug39-BYQ5ZIfEcGfuttNj6o-2xHlbjuNWpeS6_S0RlXTf7yHb4hY3KBVnv0HbDH28B0b1f59utdw6vi7bvgacrKXsnMawJpUhIfg/s1600/Three+Geometries+for+the+Visible+Universe+14cm150dpi+ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.09375) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></div>
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This diagram <span style="font: normal normal normal 18px/normal 'Times New Roman';">illustrates schematically</span> three different spacetime manifolds in reduced dimension (cross-section equatorial slices of the visible universe). These are three different geometric models of the universe. The scale ranges from the observer to the visible horizon. The 'concentric' circles represent spherical shells centered around the observer. The central illustration (B) represents a Euclidean manifold. Theses are three <span style="font: normal normal normal 18px/normal 'Times New Roman';">maps of the universe shown as grids composed of time slices and spatial distances on those slices (circles), with the observer at the center, and with incoming geodesic light-rays from all directions.</span></div>
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Note: the observer is located at the origin (at the center of either 'celestial sphere' A, B, or C). The outer edge of each illustration represents the visible horizon (the edge of the visible universe). So each circle surrounding the origin represents about 1.3 billion light years extending outwards about 13 billion light years (to the outer edge of each illustration). </div>
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Which geometric manifold is consistent with observations in the look-back time?</div>
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Manifestly, the difference between each manifold is that distances appear larger with look-back time in manifold A than they do compared to the Euclidean universe B. And distances appear smaller with look-back time in manifold C than they do compared to the Euclidean universe B. The result is that the apparent distance to the visible horizon is different in the three diagrams. (Recall, look-back time is the time required for light to travel from the emitting source to the observer.)</div>
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To begin, let's turn away from the standard expanding model and turn the clock back a hundred years, or so. We assume the universe to be globally homogeneous and isotropic, while locally inhomogeneous. We also assume the global geometric structure of spacetime to be non-Euclidean, since gravity is everywhere present. If the structure of spacetime is actually non-Euclidean, as postulated by general relativity (GR), then several very important physical features will manifest themselves globally. Several assumptions and conclusions follow.</div>
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Simply put, a homogeneous and isotropic gravitational field, has the same physical gradient or 'slope' at all points in the manifold. Equivalently, the metric tensor has everywhere the same value. All points are equal. All points on this manifold have the same value or magnitude of gravitational curvature (geometrically speaking), and it is nonzero. The fact that the potential would be nonzero implies that light would be affected relative to a stationary observer. As light is radiated outwards from the source it is traveling at c. However, from the observers frame that will 'appear' <span style="text-decoration: underline;">not</span> to be the case. The photon would progressively lose energy as it propagates through the homogeneous gravitational field. So diagram B would be ruled out by observation, since it represent a flat Euclidean space with time, or Minkowski space-time.</div>
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Locally, uniform gravity fields surrounding massive objects have a gravitational potential or magnitude of curvature at different points proportional to the altitude from the surface of the body. </div>
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The global field we are discussing has a potential or magnitude of curvature relative to the total mass-energy density (the mass-energy determines the curvature of the homogeneous field) of the universe at all points, since it permeates all of spacetime. In a homogeneous and isotropic universe the nonzero value of the gravitational potential is virtually the same at all points, even within intergalactic ('empty') space.</div>
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If indeed this global spacetime curvature exists, it is easy to see how objects themselves would not be affected by it, albeit intuitively that may not be so easy. The more pressing issue seems to be why, then, photons would be affected by the field. I've given much thought to this, but unfortunately there are few links or papers to which I can refer regarding this hypothesis. Not surprisingly so. It is not an area of active research. I think it should be. </div>
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What shows up most often on search engines are solutions such as <a href="http://scienceworld.wolfram.com/physics/StaticIsotropicMetric.html"><span style="color: #0536cd; text-decoration: underline;">static isotropic metric</span></a>, also called a <i>standard isotropic metric</i> which relates to the Schwarzschild solution (or the Schwarzschild vacuum).</div>
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There was an interesting and related work that popped up. It's entitled <a href="http://th-www.if.uj.edu.pl/acta/vol15/pdf/v15p0193.pdf%5D"><span style="color: #0536cd;">On the Physical Interpretation of a Solution of a Nonsymmetric Unified Field Theory</span></a>, dated 1983, by E.J. Vlachynsky (Department of Applied Mathematics, University of Sydney). This work examines a spherically symmetric static solution of the Einstein-Straus-Klotz non-symmetric field theory, in relation to a background pseudo-Riemannian spacetime, and proposes a new physical interpretation of spacetime. The paper mentions G.F.R, Ellis, whom we discussed earlier in the Redshift z thread (recall, Ellis, in 1978, wrote that redshift may be seen in terms of cosmological gravitational redshifts). Vlachynsky writes:</div>
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Conclusion: We have shown that the background space-time corresponding to GFT [nonsymmetric unified field theory] solution is equivalent to the SE [Einstein's universe] solution of General Relativity. Thus we must reject the interpretation of ([equation] 1) which asserts that (1) represents an expanding universe. Clearly (1) should be interpreted as representing the exterior geometry of a static black hole (as opposed to a primeval atom) in the background of a static universe. [...]</div>
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What is shown is that the line element (in equation 1) is equivalent to the general relativistic line element which describes a Schwarzschild black hole in the background of Einstein's world model. This solution is identified with a static, spherically symmetric, electric charge.</div>
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And I'm guessing that the manifold would look something like this:<br />
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<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXwm_0Nh2uj89uBlQaoKqS7TrKne7OKlhJioMjYOnG_cZEnrF0dITYAG77tjKbV4D-5UOn7GdYhAlbA6oiMS34s0rxCJmK88j8jmpOk_tAnz1bHTcVq7y7ieeRShLEO4AzzAe78fMI5Ek/s1600/Gaussian+Spacetime+Manifold+inverted+curvature+10cm+150dpi+ok+.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></div>
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure 2D</b></span></div>
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A Gaussian Spacetime Manifold with an attitude. This manifold represents the negative square root solution to the Schwarzchild-like metric, where the horizon is located at the outside circle, and the stationary observer at O. The manifold is homogeneous, isotropic and static.</div>
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Or perhaps this:<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure 3D</b></span><br />
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A reduced dimension schematic diagram representing a constant time equatorial slice through the Schwarzschild solution for a static (non-expanding) spherically symmetric Einstein world-model.</div>
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Figure 3D represents curvature of the Schwarzschild solution with a Flamm-like paraboloid. This diagram differs from diagram A, B and C above in that it is not a look-back time representation; it represents a constant time equatorial slice through the Schwarzschild solution. This manifold has the property that distances measured will match distances in the Schwarzschild metric. This is a cross-section at one moment of time (cosmic time), so all particles moving across it must have infinite velocity. See <a href="http://en.wikipedia.org/wiki/Schwarzschild_metric"><span style="color: #0536cd; text-decoration: underline;">Schwarzschild metric; Flamm's paraboloid</span></a>. <i>"Even a tachyon would not move along the path that one might naively expect from a "rubber sheet" analogy: in particular, if the dimple is drawn pointing upward rather than downward, the tachyon's path still curves toward the central mass, not away."</i> </div>
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The mainstream view leads to instabilities inherent both in Newtonian mechanics and Einsteinian dynamics. But this view may not be justified. I will try to demonstrate the idea that <i>global curvature is different from local curvature in that objects are not affected by geodesics, but light will be</i>, is entirely consistent with GR, the equivalence principle, the concept of curved spacetime in general, and the fundamentals of spherical geometry (below). Further into the discussion it will be shown, too, that this same idea works for hyperbolic geometry.</div>
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In other words, from the inertial rest-frame of an observer, the apparent 'action' may be considered spurious (nonexistent), at least in the context here. So where an 'action' or radial motion is expected, there is none. The distinction is an important one, since the type of action expected according to the standard model determines the structure and evolution of the universe, or the evolution and fate of the universe (i.e., the 'action' determines whether the universe collapses or expands with time).</div>
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Here is an interpretation of the standard concordance model of big bang cosmology, Lambda-Cold Dark Matter (ΛCDM, LCDM, or Lambda-CDM), with an arbitrary HUDF superimposition. The outer circle represents the universe now.<br />
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This model assumes the cosmological principle. The LCDM universe is homogeneous and isotropic. Time dilation and redshift z are attributed to a Doppler-like shift in electromagnetic radiation as it travels across expanding space. This model assumes a nearly "flat" spatial geometry. Light traveling in this expanding model moves along null geodesics. Light waves are 'stretched' by the expansion of space as a function of time. The expansion is accelerating due to a vacuum energy or dark energy inherent in empty space. Approximately 73% of the energy density of the present universe is estimated to be dark energy. In addition, a dark matter component is currently estimated to constitute about 23% of the mass-energy density of the universe. The 5% remainder comprises all the matter and energy observed as subatomic particles, chemical elements and electromagnetic radiation; the material of which gas, dust, rocks, planets, stars, galaxies, etc., are made. This model includes a single originating big bang event, or initial singularity, which constitutes an abrupt appearance of expanding space containing radiation. This event was immediately followed by an exponential expansion of space (inflation). </div>
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<span style="font: normal normal normal 18px/normal Times;">Despite the visual shape of the manifold above, this is actually a </span>non-geometric interpretation with respect to the observed shift of spectral lines (redshift z). It merely represents the rate of radial expansion, i.e., it smoothly mimics the effects of curved spacetime by means of a universal expansion that changes non-linearly with time. Distortions in both rulers and clocks are due to relative motion. Placing this together with some form of the equivalence principle obviously tends to suggest the accelerated expansion interpretation. <span style="font: normal normal normal 18px/normal Times;">We'll come back to the geometric structure observed in this schematic diagram.</span></div>
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<b>Part I</b></div>
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Local versus global curvature in a homogeneous isotropic general relativistic spacetime manifold</div>
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When forced to choose between two competing cosmologies, one should not automatically gravitate towards the stationary dynamically evolving general relativistic spacetime with Gaussian curvature and a pseudo-Riemannian manifold approach, as opposed to the standard pseudo-Newtonian neo-Euclidean expansion of a quasi-Einsteinian Minkowski space. Empirical evidence should determine the decision making process. One of the problems is that these two cosmologies are <i>a priori</i> indistinguishable one from the other observationally, in accord with a wide interpretation of the Einstein equivalence principle. The other problem, historically, has been the idea that a static universe is unstable (like a pencil balancing on its point).</div>
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Tantalizingly, perhaps not for the first time, it is shown how a general relativistic universe (a spherically symmetric Einsteinian manifold) remains free for-all-time of gravitational instabilities that would otherwise cause catastrophic collapse or wholesale propulsive expansion, <span style="text-decoration: underline;">without</span> the use of an <i>ad hoc</i> cosmological constant-like vacuum energy term to counter the attractive force of gravity.</div>
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The idea of finite space is of no particular interest to this investigation (since finiteness does not alter the final result). We consider the universe philosophically to be infinite and without bounds: meaning that even an Einstein universe would be infinite in spatiotemporal extent. The observer (from her view-point) is centrally located inside a sphere. The universe appears on average homogeneous and isotropic. She can only observe objects out to the visible horizon. But the 2-dimensional sphere has no boundary since we (philosophically) push what would be the surface of a sphere, in 4-dimensions, to infinity. Whether the curvature extends that far, or whether the manifold takes on a Minkowski-type metric at exceeding great distances beyond the horizon is debatable (but not here). Note: this is similar to the infamous Einstein universe. Recall that Einstein's 1917 cosmological considerations regarding the field equations lead him to the conclusion that even spherical solutions would not require any boundary condition at infinity.</div>
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A key point will be to show that there is no preferred direction or center of gravity to a sphere of constant curvature, the manifold is not a <a href="http://en.wikipedia.org/wiki/Compact_space"><span style="color: #0536cd; text-decoration: underline;">compact space</span></a> (bounded and geodesically complete), as could otherwise be imagined in a finite spherically symmetric space, unless we’re forced by observation to some other conclusion. In the absence of such, and when gravity is considered a 4-dimensional geometric phenomenon, we assume (for now) that global curvature would not induce the displacement of massive bodies towards one another, but that electromagnetic radiation (EMR) would be affected by such a curvature (as seen by redshift z and time dilation), when viewed in the look-back time (only) by comparing distances (with the techniques mentioned previously). </div>
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Note too, even without a center of gravity the same would hold true: all the objects in the universe would not be forced or pulled together, free-fall towards each other, or move toward one another gravitationally, since in a homogeneous and isotropic universe there would be no gradient or difference of potential from one location to the other. So a natural equilibrium without an <i>ad hoc</i> repulsive force is obtained. </div>
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In the language of curved spacetime the picture is equivalent, for both spherical and hyperbolic geometries: In a spacetime that is homogeneous and isotropic the value, gradient, potential or magnitude of curvature is everywhere the same. Nobody (or no body) is located in any "special" position in the manifold.</div>
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Let's look at real-world observations. Though there are no privileged reference frames in a globally curved homogeneous universe, according to observations there are "special" spatial positions; there would appear to exist privileged reference frames: this is based on the fact that every observer views the universe from her particular position, her unique inertial reference frame, as she finds herself centered in the celestial sphere at the present time. But all coordinated systems are special in this sense. That is because of the limited velocity of light. So even though the universe "looks the same" when viewed from any spatiotemporal location, all spatiotemporal locations do not "look the same" when viewed from the rest-frame of the observer immersed within the manifold. Both spatial and temporal increments and intervals "appear" different at other locations far removed than those measured locally. Spatial increments appear to change with distance and temporal intervals appear to be dilated with distance, from the rest-frame of the observer. (Note, saying that all reference frames are privileged is the same as saying that none are). In sum, there is no immediate reason to prefer certain systems of coordinates over others, i.e., we arrive at the requirement of <a href="http://en.wikipedia.org/wiki/General_covariance"><span style="color: #5588b1;">general covariance</span></a>.</div>
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It is well known, generally accepted, and confirmed empirically that phenomena such as <a href="http://en.wikipedia.org/wiki/Gravitational_redshift"><span style="color: #0536cd;">gravitational redshift</span></a> and <a href="http://en.wikipedia.org/wiki/Time_dilation"><span style="color: #0536cd;">time dilation</span></a> occur locally. It is less well known, generally accepted, or confirmed empirically that these same (or similar) phenomena would occur globally, or cosmologically, in a nonexpanding universe. It is here argued, and demonstrated below, that similar phenomena are observed globally, when the observer scrutinizes the universe from her reference frame, due to the general curvature of spacetime, and the limited velocity of light. </div>
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Notice here I neglect the special relativistic time dilation associated with the relative velocity of objects in an expanding universe (where the requirement is that space itself has to expand, leading to what is called a <a href="http://en.wikipedia.org/wiki/Relativistic_Doppler_shift"><span style="color: #5588b1;">relativistic Doppler shift</span></a>), except as an analogy to the gravitational redshift-like effect proposed here. This redshift (with its associated relativistic time dilation) is different from the classical gravitational redshift that occurs locally, since it describes the total difference in observed frequencies and possess the required <a href="http://en.wikipedia.org/wiki/Lorentz_symmetry"><span style="color: #264b9f;">Lorentz symmetry</span></a>. This gravitational redshift-like effect (or cosmological redshift z) is produced as light propagates through the global Gaussian curvature of the field, not as light escapes the local field of a massive object. </div>
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What concerns us now is the general relativistic notion of cosmological redshift z and time dilation with the look-back time in a static homogeneous and isotropic manifold associated with a pseudo-Riemannian curvature of spacetime. In this case, both redshift z and time dilation are observable effects relative to the rest-frame of an observer as she looks out into a universe that exhibits nonzero Gaussian curvature. </div>
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The hunch here is that by setting the total "force" of the gravitational field curvature equal to zero at every point, the differential equation for a straight line in Euclidean space, or a geodesic in a non-Euclidean space can be derived. If the classical requirement that physical space be Euclidean is vacated, and a non-Euclidean Riemannian space is introduced, the apparent motion of bodies in the gravitational field may be described by an equation of motion without recourse to any global gravitational "force" (<a href="http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/field_equations.htm"><span style="color: #0536cd; text-decoration: underline;">source</span></a>). In other words, one need not change any equations of GR for what is to follow.</div>
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It is well known that Einstein investigated cosmological modeling based on his general theory but found it would not satisfy the conditions of homogeneity, isotropy, and staticity unless an additional term was added to the equations: the cosmological constant. I argue that this was an unfortunate maneuver. GR can indeed satisfy the conditions of homogeneity, isotropy, and staticity without lambda (see below).</div>
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<b>Geodesics in a homogeneous spacetime of constant curvature</b></div>
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A brief note on geodesics in a homogeneous spacetime of constant positive or negative curvature: It is well known that there are different types of geodesics. A geodesic is a generalization of the notion of a straight line to a curved space, in differential geometry. The shortest path between points in the space defines the geodesic (locally). On the surface of a sphere, for example, a geodesic is a segment of a great circle. The velocity (and corresponding motion) of a test particle (or point particle) traveling between two points on a geodesic are usually parametrized as constant.</div>
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General relativity describes the geodesic motion of a point particle under the influence of gravity (a curved spacetime). Typically, geodesics in a curved spacetime are the paths (or trajectories) taken by a freely falling particle or an object, such as the earth, orbiting another, say, the sun.</div>
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Generally, objects can travel a path where their movement is constrained in various ways (they are not free). This is a topic of Sub-Riemannian geometry.</div>
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The velocity of a test particle (or point particle, whose size and gravitational field are ignored) traveling between two points on a geodesic path (or world-line) in a four-dimensional spacetime manifold of constant curvature need not be constant, when viewed from the reference frame of an observer.</div>
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Massive gravitating objects can affect the background manifold of constant Gaussian curvature. This case leads to a problem of determining to what extent the situation approximates 'true' geodesic motion. Qualitatively, the smaller the gravitational field produced by an object, when compared to the globally homogeneous gravitational field in which it resides, the more this object's motion will approach a geodesic. In sum, the greater the mass of an objects, the less the it will be affected by the global curvature The smaller the mass, the more the object (or particle) will be affected by the global field curvature.</div>
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[A related side note: The above considerations imply that the acceleration induced by a globally curved spacetime on an object located within the manifold is proportional to its mass.]</div>
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In other words, a photon will travel the shortest distance between two points (a 'true' geodesic) on the globally homogeneous spacetime manifold of constant curvature, while a massive gravitating object will virtually not be affected. The deviation from this 'true' geodesic will be greater with increasing mass of the object.</div>
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This situation is comparable, locally, to the geodesic motion of a small planet relative to a companion star. The smaller the mass of the planet, the closer it will follow the geodesic of the local gravity field produced by the star.</div>
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What this implies, then, is that electromagnetic radiation, propagating through a four-dimensional (pseudo-)Riemannian spacetime continuum of constant (positive or negative) Gaussian curvature, will travel a geodesic, while massive bodies will not be confined to propagate along the same geodesic. Objects do not have to follow the shortest path between two points of a great circle (for example). And due to the distortion along the geodesic path of the photon, spectral lines will be redshifted when viewed from the rest-frame of an observer (the greater the distance, the higher the redshift).</div>
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It follows too that massive bodies remain virtually unaccelerated by the global curvature, the greater the mass of the object, but are, in every practical sense, only affected (accelerated) by local intrinsic gravitational fields (and interactions thereof), i.e., they will follow local geodesics of the submanifold of which they form a part.</div>
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Conclusion: Redshift z occurs in a non-expanding universe, and increases with distance from an observer. Global stability is maintained against gravitational collapse. The universe does not expand or collapse.<br />
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Figure PRMCC represents a homogeneous pseudo-Riemannian manifold of constant positive Gaussian curvature. Placed on the manifold is a galaxy cluster, Cl 0024+17 (ZwCl 0024+1652) from Hubble's Advanced Camera for Surveys.</div>
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It should be easy to see, in looking at Figure PRMCC, why a light ray would follow a great circle arc towards any observer (the shortest distance between two points), while the galaxy cluster will follow local geodesics (not great circle arcs) that depend on its location relative to other cluster (not shown). Yet, at the same time the cluster is embedded in the Riemannian (or pseudo-Riemannian) manifold.</div>
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<b>Homogeneous gravitational field</b></div>
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In classical Newton’s mechanics, a homogeneous gravitational field describes the same gradient of the gravitational potential at every point. The gravitational field does not fall off with distance. That is, its intensity is of constant magnitude (its piecewise magnitude is constant). This field is produced by an infinite material surface with constant mass density. It is generated by a system of masses uniformly distributed in space. A test particle released at rest into a homogeneous field would 'feel' a acceleration from all directions, since the gravitational potential, or magnitude of the proper acceleration, is everywhere the same. And since it was introduced at rest, it remains at rest, since there is no gravitational 'pull' at all, or preferred direction in which to move.</div>
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A comparison of homogeneous gravitational fields in classical Newtonian mechanics and in the general theory of relativity reveal a fundamental difference. The concept of a homogeneous gravitational field in general relativity appears to be in violation of the causality principle, the correspondence principle, and in part the equivalence principle. In addition, there arises in the latter an unccountable singularity that has no physical explanation or meaning. Thus, the homogeneous field in general relativity cannot be acceptable as a real gravitational field (as opposed to one that is entirely due to inertial force), i.e., it has no physical meaning. See <a href="http://arxiv.org/pdf/gr-qc/0202058v1"><span style="color: #5588b1;">Can the notion of a homogeneous gravitational field be transferred from classical mechanics to the Relativistic Theory of Gravity? </span></a>"Unfortunately, the solution obtained according to the Relativistic Theory of Gravitation [RTG] can’t be accepted because it doesn’t fulfill the Causality Principle in this theory. So, it remains open in RTG the problem of finding a generalization of the classical concept of homogeneous gravitational field." There are indeed difficulties inherent in finding anything in general relativity that represents a uniform gravitational field.</div>
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In general relativistic terms, a necessary condition for any spacetime that would represent a globally homogeneous uniform field is that the scalar curvature should be constant, and four translation symmetries would be expected. The problem of finding a homogeneous gravitational field in general relativity has been, to some extent, considered in the relevant literature, usually in terms of the coordinates of an underlying Minkowski space-time. As it stands now, general relativity does not admit any spacetime with all the global properties we would like for a uniform gravitational field. Said differently, there is no global solution to the Einstein field equations that uniquely and satisfactorily embodies Newtonian ideas about a uniform field, i.e, the desired properties for a uniform gravitational field in GR cannot all be satisfied at once (without a metric that becomes degenerate). (See <a href="http://www.lightandmatter.com/html_books/genrel/"><span style="color: #5588b1;">Crowell </span></a>). The problem exists when considering the field equations with a curved pseudo-Riemannian manifold of Lorentz signature (3,1) or equivalently (1,3). It appears that every point in the field does not have the same gradient of curvature (or potential), i.e., all point are not equal. In essence, a generalization of the homogeneous gravitational field does not exist in GR. The global field that emerges from GR can only be considered a special case. A consequence of this type of field is that material objects are forced to move (accelerated) toward on another gravitationally, leading to the well-known gravitational instability associated with the field equations (and coupled with the cosmic time component, the varying scale factor with time, and the observed spectral shifts of distant galaxies) that ultimately pointed towards big bang cosmology.</div>
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In order to resolve these problems, it seems inevitable that a solution must be found that describes a homogeneous gravitational field as a global manifold of continuous curvature in a manner consistent with general relativity (which describes gravity as a curved spacetime continuum), and at the same time, consistent with classical mechanics whereby the causality principle is unviolated. The beauty of this solution (if it does not exist already), amongst other things perhaps, would be that the singularity would be avoided (i.e., physical laws would not break-down), and the need to supplement cosmological considerations of general relativity with inflation, dark energy and cold-dark matter would be unnecessary.<br />
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Figure ESU represents an equatorial slice of the visible universe (on an oblique angle) with the observer centered at the origin of the past light cone. A slice through any point of origin, and at any angle will be the same. All distances are in look-back time from the observes view point. Each concentric circle around O represents about 2 Gly. This universe has a spherical topology, where light is redshifted due to the propagation of photons along great circle arcs in reduced dimension (shown here schematically as the lines converging towards the origin). Stability is maintained because massive objects move along local geodesic paths determined by spacetime curvature in the local vicinity of the objects under consideration, just as the earth remains stable against gravitational collapse into the sun. (Local curvature is not shown in this diagram).<span style="color: #999999; font: normal normal normal 22px / normal "arial";"><b> </b></span></div>
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Notice that light propagates in what appears to be straight lines in the Euclidean sense. Curvature no longer appears as it does on the surface of a sphere, where light follows the curve of the surface along great circle arcs. Now, in a four-dimensional relativistic spacetime, the photon paths are essentially straight geodesic lines (excluding local gravitational effects, such as lensing). The curvature, or distortion, occurs along the path itself. Though the path is straight from the viewpoint of the observer, there is a distortion, plainly visible in the schematic diagram above. And the distortion becomes increasingly apparent the further the distance considered. This is exemplified by the cross-sections of the spherical shells centered on the observer, which appear to become closer together with distance. The volume of this positively curved universe appears smaller than those of its Euclidean or hyperbolic counterparts.</div>
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Recall that in reduced dimension, i.e., on the surface of a sphere, light propagates along the very same lines towards the origin (centered on the any observer, always located at what would look like a North or South pole: compare with Figure PRMCC). The distortion in the path is the cause of cosmological redshift z in a static Einstein universe. There is a loss of energy associated with increasing distance of propagation from the observer in the non-Euclidean manifold: the result is redshift z and time dilation. This is exactly what would be observed from the rest-frame of any observer located anywhere in the Einstein static four-dimensional manifold of constant positive intrinsic Gaussian curvature.</div>
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<b>The Einstein equivalence principle and cosmological considerations</b></div>
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Lets compare two coordinate systems. A coordinate system S<span style="font: normal normal normal 15px/normal 'Times New Roman';">1</span> is accelerating in "empty" space at a rate <span style="font: normal normal normal 18px/normal Times;">g</span> in the x direction. The second, S<span style="font: normal normal normal 15px/normal 'Times New Roman';">2</span>, is at rest in a homogeneous gravitational field that imparts to every material object in the field an acceleration of –<span style="font: normal normal normal 18px/normal Times;">g</span> in the x direction. Einstein observed that: […] <i>as far as we know, the physical laws with respect to the S</i><span style="font: normal normal normal 15px/normal 'Times New Roman';"><i>1</i></span><i> system do not differ from those with respect to the S</i><span style="font: normal normal normal 15px/normal 'Times New Roman';"><i>2</i></span><i> system </i>[…] w<i>e shall therefore assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.</i></div>
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Extrapolating now from the equivalence principle to cosmology it would seem difficult (though perhaps not impossible) to determine empirically if the universe is expanding along with everything in it, or if the universe is stationary and everything in it are immersed in a gravitational field. I<span style="font: normal normal normal 18px/normal 'Times New Roman';">nertial acceleration and gravitation appear intrinsically identical.</span></div>
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Judging from spectral shifts of EMR from distant sources and other measurements, the apparent radial motion of an object in a special relativistic Minkowski manifold will be <i>indistinguishable</i> from an object at rest in a gravitational field (from the rest frame of an observer). Radial motion would be apparent (from redshift), and could be interpreted as such. Of course, an object at rest in a gravitational field would look as if radially moving from the observer. The observation of motion would be either apparent or real but not both (for our purpose here, both could be operational locally since all objects have intrinsic motion, but not globally). Both situations would exhibit a redshift independent of wavelength across the entire electromagnetic spectrum, along with an associated time dilation factor, from an inertial frame of reference.</div>
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In another way (and still extrapolating to cosmological scales), though the law of motion describes the paths followed by bodies in a gravitational field as geodesics in a non-Euclidean spacetime, there is an observational equivalence in the static case; where bodies at rest relative to one another and immersed in a homogeneous gravitational field (a non-Euclidean spacetime) will exhibit an 'apparent' or spurious motion when viewed from the rest-frame of the observer. This means that even though the objects are at rest relative to one another, there is still a natural geodesic path in the gravitational field upon which the photon must propagate.</div>
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In both situations redshift and time dilation are associated with the loss of energy and photon travel time through the continuum; thus in both cases observations look the same, and in both cases the objects are located in a gravitational field. But in one case the object follows a geodesic path, and in the other the object may not. It will be shown that objects are not accelerated is any particular direction on such a pseudo-Riemannian manifold because the manifold is of constant curvature and homogeneous at all points, while locally it is flat. No direction is preferred over any other, relative to the global topology.</div>
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Locally, observed accelerations of objects in the field, apparent or real, have the characteristic feature that they will all experience the <span style="text-decoration: underline;">same</span> inertial acceleration. That is, the inertial force on the various objects will be proportional to their masses, with the acceleration being a constant. Gravitational acceleration exhibits identical behavior in this respect. In other words, in classical mechanics the gravitational force on a body is proportional to its mass, while the acceleration is a constant at every point in the field. These observations lead to the identity of an equivalence between gravitational and inertial mass (found originally by Newton), used as a motivation toward general relativity by Herr Einstein. </div>
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As far as 'forces' acting on bodies, the predictions of general relativity locally would remain similar to Newton's anywhere in the cosmos, but when great distances are concerned, with increasing look-back time, and mass-densities compatible with those of the universe (whatever that may be) the predictions of general relativity would significantly diverge from those of Newton, and would be verifiable by astronomical observations. So GR and Newtonian mechanics diverge not just for large mass-densities or large intrinsic velocities, but for observations of distant astronomical objects viewed from the observer rest-frame as she detects the photon arrival. She notices that redshift and time dilation become large, and do so with increasing distance. But she also notices that what 'appears' to be a radial motion, which she may interpret, alternatively, to be a gravitational effect, perfectly in accord with her knowledge of GR, the field equations, the equivalence principle, and the fundamental principles of non-Euclidean geometry.</div>
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<span style="font: normal normal normal 18px/normal Times;">Local physics is global physics, meaning that the laws of physics remain valid throughout the universe, everywhere and at all times, but appearance and mechanics differs both geometrically and mechanically the larger the scale considered and the further back in time we probe (i.e., the further we look), and the same properties are observed for all frames of reference (i.e., independent of velocity). However, both t</span>he redshift z and stability discussed here is not a consequence of local effects; these are global effects. At any point along geodesic paths the local physics is identical. But the paths are embedded differently within the global spacetime manifold, and it is the different embedding within the manifold that accounts for the differences in proper distance and proper time: not radial motion away from all observers. This extrapolation of general relativity to cosmology requires us to abandon the notion that physical phenomena are governed exclusively by locally sensible influences (which compels us to assign like physical causes to like physical effects). Similarly, the identification of gravity with local spacetime curvature alone is reduced to a limiting case. The point is that a homogeneous arrangement of gravitating masses can produce a globally extended region of curved spacetime in which the metrical field approaches flatness locally (i.e., the field exerts no acceleration on material objects), yet globally light propagates along geodesics in the curved spacetime continuum (the background metric): producing the observed cosmological redshift z.<span style="font: normal normal normal 18px/normal Times;"> </span></div>
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In the general theory of relativity, spacetime is not simply the totality of all the relations between material objects. The spacetime metric field is endowed with its own ontological existence, as is clear from the fact that gravity itself is a source of gravity. In a sense, the non-linearity of general relativity is an expression of the ontological existence of spacetime itself. In this context it's not possible to draw the classical distinction between relational and absolute entities, because spatio-temporal relations themselves are active elements of the theory. (<a href="http://www.mathpages.com/rr/rrtoc.htm"><span style="color: #5588b1;">Brown, Reflections of General Relativity</span></a>)</div>
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It is straightforward to see why the definition of the equivalence principle has evolved to include other principles (e.g., the cosmological principle). Originally it expressed the idea that gravitational effects are physically equivalent to the effects of acceleration of a reference frame: though without explicitly making clear what these effects might be, or by what physical mechanism they may operate, or on whether these things may differ at different scales, times or places. There is no simple, unique, quantitatively well defined statement that embodies the equivalence principle.</div>
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The idea that a distinction between inertial and noninertial reference frames was suspect occurred to Einstein by 1915. The problem is that when looking out into the universe the observer is bootstrapped to a frame of reference; one that looks surprisingly Euclidean. If our observer takes this frame as a standard reference, then how does she verify whether galaxies are accelerating for "no reason", or that redshift z is due to a gravitational affect (global curvature)? By extrapolation of the equivalence principle beyond the local, there is a priori no way to determine whether objects are radially accelerating for "no reason" or whether the effect is caused as light propagates through a gravitational field.</div>
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In general relativity Lorentz frames occupy a privileged position locally because objects move along rectilinear paths (world-lines) when there is no nongravitational forces acting upon them. But there is no single Lorentz frame that covers the entire universe. Only in a particular neighborhood is it possible to define a Lorentz frame. The locality of Lorentz frames can be exemplified by stretching a string across the surface of the earth, from say, Paris to Shanghai. The curvature of the earths surface will not be noticeable to anyone located on the string because the radius of curvature extends thousands of kilometers. On a map of Paris we do not detect any curvature. Nor do we detect any curvature on a map of Shanghai. Yet it is impossible to draw a map that includes both Paris and Shanghai without observing extreme distortion (even though the string has the same length whether it is curved or flat). This situation also implies that clocks will run at a different rate when compared from either local frame (as we will elaborate on below) due to the geodesic path. The length of the string (or distance) will appear to change depending on the sign of curvature.</div>
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Now lets apply the same principle to an astronomical object. In this thought experiment we assume the universe to be globally curved and nonexpanding. We attach a sting to a high-z Type Ia Supernova (located near the visual horizon). Or better yet, we let a beam of light extending from the SN Ia play the role of the string. We don't notice global curvature in the neighborhood of the Local Group (or even the Local Supercluster) because the radius of curvature extends billions of light years. But it is not possible to plot a flat Euclidean map that includes the Local Group and the SN Ia without seeing severe distortion (even though the string, or light beam, has the same length or travels the same distance whether the path is distorted or flat). </div>
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Extrapolating globally, again, there is no way to tell the difference between observational data (by redshift z alone) obtained in an expanding universe or data obtained in a static universe where everything is immersed in a gravitational field. In both cases light is redshifted. In one case it is a Doppler redshift-like effect (which depends on the relative motion of the source and the observer), and in the other a gravitational redshift-like effect (the cosmological version only depends on the relative location of the source and the observer). Locally we know from empirical evidence that the flow of time changes rates with height (or altitude) in a gravitational field; a fact that is required by the equivalence principle. Globally, we find that there should be a gravitational effect on the energy of a photon emitted from a distant source. The fractional loss of energy should be equal to the (Newtonian) loss of energy associated with a Doppler-like effect in an expanding universe due to the change in velocity during the photon's flight-time (with its time dilation factor).</div>
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There is little doubt that the equivalence principle, combined with the universality of local Lorentz covariance, makes the curvature interpretation for redshift z eminently viable, and it is likely the 'strongest' interpretation of Einstein's general relativity. But the fact remains, it isn't the only possible interpretation.</div>
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In the case of an expanding universe that appears to be accelerating with time (in accord with the standard model) it follows that the geodesic paths are "curved" and that objects which follow those paths are being influenced by some "force field." And since gravity tends to bring objects together (rather than accelerate objects away from one another) a "repulsive force" had to be evoked that would not only counter gravity, but would overcome it on large scales. This is exactly analogous, observationally, to the non-linearity of the geodesics that would be measured with respect to distance in a curved spacetime manifold. So a deviation in linearity (of the type exemplified by the spectra of distant supernovae Type Ia) should not be regarded as exclusively indicative of the presence of cosmological constant-like vacuum energy (or dark energy). </div>
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If the intrinsic curvature is nonzero, in the case of a stationary universe, then the non-linearity observed in the spectra of SNe Ia simply represents evidence of Gaussian curvature (K = -1, or K = +1). The conceptual necessity revolves not around <i>identifying</i> the purely geometrical effects of non-inertial coordinates with the physical phenomenon of gravitation, but to interpret redshift z and a purely geometrical effect.</div>
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The classical interpretation requires that electromagnetic waves have inertia and a resulting force in the direction or the wave-front propagation. In this sense, light waves act as if they had momentum. Whatever has inertia must also participate in gravitational interactions, in accord with the equivalence principle (gravitational and inertial mass are always proportional to one another). Therefore, light waves act as if they had weight, and lose energy as they propagate along geodesic paths (sections of great circles on a curved sphere in reduced dimensions). The gravitational mass of a light beam with energy <i>E</i> is<i>E/c^2</i> (<a href="http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html"><span style="color: #0536cd; text-decoration: underline;">Crowell, 2010</span></a>). Since electromagnetic waves have both gravitational mass and inertial mass, it seems evident that the equivalence principle must hold in universe where there is a general Newtonian radial motion and/or where light propagates through the globally static gravitational field of a homogeneous nonexpanding universe. In both cases the electromagnetic field spectrum becomes distorted; redshift z and time dilation are the result. </div>
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The disambiguation between the two types of curvature (local and global) will be made below in order to show that local curvature induces massive bodies to move in certain ways (geodesically), while global curvature does not. There is a problem with the standard interpretation of general relativity (GR) related to the <a href="http://en.wikipedia.org/wiki/Einstein_equivalence_principle"><span style="color: #0536cd; text-decoration: underline;">Einstein equivalence principle</span></a>, in the absence of a gravitational force (<a href="http://resources.metapress.com/pdf-preview.axd?code=m15269232107267t&size=largest"><span style="color: #0536cd; text-decoration: underline;">source)</span></a>. It will be shown that the standard concept expressed by Misner <i>et al</i> (1973, p. 5): <i>"Space tells matter how to move. Matter tells space how to curve"</i> is justifiable locally where the submanifolds have a Lorentzian signature, but untenable when global curvature is considered. It can be interpreted from cosmological considerations of the equivalence principle two general ideas (that are incompatible with each the other): there is an equivalence observationally between radial velocity and dilation of the metric. Just as there is an equality of gravitational and inertial masses (in the Newtonian sense), there is an equivalence of acceleration and space curvature (in the Einsteinian sense).</div>
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It will be found too that the general theory of relativity <i>does</i> indeed satisfy the conditions of homogeneity, isotropy, and staticity without an additional cosmological term (lambda).</div>
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<i>En passant</i>, by equating physics of bodies undergoing a free-fall in a gravitational field with the physics in free-fall in the absence of a gravitational field, we are claiming that local physics is not exactly global physics. Likewise, when we claim that a "flat" or quasi-Euclidean <span style="text-decoration: underline;">space itself is expanding</span> along with all the matter and energy in it, we are essentially claiming that physics is not exactly the same for all times and places in the universe; locally and globally. Indeed there is no corollary or analogue on scales compatible with the <a href="http://en.wikipedia.org/wiki/Virgo_Supercluster"><span style="color: #0536cd; text-decoration: underline;">Local Supercluster</span></a> on down. Quite the contrary, all the evidence points to the fact that space does not expand. Things expand <span style="text-decoration: underline;">into</span> space (e.g., isothermal and adiabatic expansion). General relativity says nothing about expanding space, but everything about curved spacetime!</div>
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One more side-note before moving on to geometrical argument:</div>
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Local dynamics and global dynamics according to the standard model (Lambda-CDM) are two different animals. Simply put, locally, gravitating systems operate by an evolutionary process not dissimilar to the Darwinian approach, where massive objects are formed and evolve into quasi-stable self-gravitating systems only if they are located or move to orbital positions that prevent gravitational collapse or dispersion. The orbital velocities of massive objects have to be adequate for such systems to subsist (a kind of natural selection, or <i>environmental selection</i> process). In other words, the collapse of material tends to slow down and stop at some threshold because of conservation of angular momentum. (If all the matter of a system under consideration ended up at the same point, angular momentum would be zero). Our solar system likely formed this way, and it seems this is a ubiquitous process that extends to superclusters, and perhaps beyond.</div>
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Locally, there is no repulsive force or vacuum energy required, whereas a repulsive-like force and a large dark matter component are required in order to to agree with observations globally.</div>
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Just to say that local dynamics hardly resembles global dynamics, where on the one hand quasi-stable equilibrium configurations are formed and maintained gravitationally, while on the other, the universe is blowing apart at the seams. </div>
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Certainly it should hardly be expected that physical phenomena should act the same at all scales. But I would hope the law of physics need not be modified to make theory fit observation. <i>Au contraire</i>. To end this side-note I'll post a link to further discrepancies between that which is observed locally and that which is believed to occur globally: <a href="http://arxiv.org/PS_cache/arxiv/pdf/1006/1006.1647v1.pdf"><span style="color: #0536cd; text-decoration: underline;">Local-Group tests of dark-matter Concordance Cosmology</span></a>.</div>
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To be continued</div>
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Coldcreation</div>
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Barcelona, June 14-15, 2010</div>
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<b>Part II</b></div>
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Disambiguation and the Stability of the Cosmos:</div>
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Global curvature of a homogeneous isotropic general relativistic spacetime</div>
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<b>Geometrical Arguments</b></div>
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General relativity is very dynamic and rich in that a very wide variety of physical interpretations can be drawn from it when extended to cosmology. Two of these interpretations are discussed in this thread. One of these interpretations postulates that spacetime globally is geometrically curved, resulting in redshift z as light propagates through the continuum, yet superclusters, clusters, galaxies, stars, planets and people are not affected by this globally homogeneous and isotropic gravitational field, i.e., light follows a geodesic path, while material objects do not. This would appear to show that physics is not the same locally as globally, since it is well known empirically that both light throughout the electromagnetic spectrum and objects follow geodesics locally.</div>
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The answer to this dilemma is not at all obvious. It would indeed seem as if gravity—whether treated as an attractive force or a curved spacetime phenomenon—would cause instabilities of the type that would affect the fate of the entire universe; leading either to collapse (a big crunch) or expansion. Even a static universe, at first glance, would not be safe against such instability.</div>
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I've argued that what 'appears' to be transpiring may not be what is actually happening. Part of the reason for this assumption is due to the geometric factor encoded in Einstein's general principle of relativity, and its relation to a quasi-trivial generalization of <b>Gauss'</b> <a href="http://en.wikipedia.org/wiki/Theorema_Egregium"><span style="color: #0536cd;"><b>Theorema Egregium</b></span><span style="color: #0536cd; text-decoration: underline;"> (meaning "Remarkable Theorem" in Latin)</span></a>. Gauss introduced this <i>remarkable theorem</i> in his famous 1827 paper "Disquisitiones generales circa superficies curvas" (General Investigations of Curved Surfaces).</div>
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In what follows, I will attempt to explains how local gravitational fields (surrounding astronomical objects) and the globally homogeneous and isotropic field (the curvature or geometry of the universe) result in two different interpretations of dynamics. In other words, local curvature surrounding massive bodies contributes to local dynamics but global curvature does not contribute dynamically, yet both local and global curvature induce a wavelength independent changes in the spectra of EMR towards the red end (or blue end, locally) of the spectrum, along with time dilation. One is dependent on the position of the observer in the field and the other not. </div>
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The topic of local dynamics is well known, and so it will not be elaborated upon in present outline (no need to repeat the basics). What is important for us is that global curvature must be considered an <span style="text-decoration: underline;">intrinsic curvature</span> of spacetime, the origin and the magnitude of which depend on the total mass-energy density of the cosmos. This is to be considered the most general class of the curvature tensor.</div>
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Let's quickly run through some of the basic terminology and concepts we'll be using below with regard to geometry. The idea that the global geometry of the universe is an <i>intrinsic</i> curvature is essential to the concept presented here.</div>
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<span style="color: #4388b6;"><a href="http://mathworld.wolfram.com/IntrinsicCurvature.html">Intrinsic curvature</a></span> such as Gaussian curvature is detectable to the "inhabitants" of a surface. Surfaces can have intrinsic curvature, independent of an embedding.</div>
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In contrast, <a href="http://mathworld.wolfram.com/ExtrinsicCurvature.html"><span style="color: #3b2e7b; text-decoration: underline;">extrinsic curvature</span></a> is defined at each point in a Riemannian manifold. An <a href="http://en.wikipedia.org/wiki/Extrinsic_curvature"><span style="color: #3b2e7b; text-decoration: underline;">extrinsic curvature</span></a> is not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides (they only have a curvature given an embedding). The curvature of a submanifold depends on its particular embedding. Examples of extrinsic curvature include the curvature and torsion of curves in three-space, or the mean curvature of surfaces in three-space. E<span style="font: normal normal normal 18px/normal 'Times New Roman';">xtrinsic properties of surfaces measure the rate of deviation between one surface and another. However, curvature from an intrinsic standpoint, describes relations between points within the surface itself. The results of measurements of intrinsic distances on a manifold can be encapsulated in the form of a <i>metric tensor</i> relative to any system of coordinates on the manifold. </span></div>
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Gaussian curvature is, in fact, an <i>intrinsic</i> property of the surface, meaning it does not depend on the particular embedding of the surface. Because curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.<br />
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Stereographic projection of a complex number <i>A</i> onto a point αof the Riemann sphere.</div>
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<a href="http://en.wikipedia.org/wiki/Riemann_sphere">See this article for a text regarding the details of this illustration.</a></div>
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Space of three or more dimensions can be intrinsically curved; the full mathematical description is described at <a href="http://en.wikipedia.org/wiki/Curvature_of_Riemannian_manifolds"><span style="color: #3b2e7b; text-decoration: underline;">Curvature of Riemannian manifolds</span></a>. T<span style="font: normal normal normal 18px/normal 'Times New Roman';">he basic idea of intrinsic curvature remains essentially the same in four dimensions.</span></div>
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After the discovery of the intrinsic definition of curvature (closely connected with non-Euclidean geometry) many mathematicians and scientists (natural philosophers) questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In general relativity the idea is generalized to the "curvature of space-time," where spacetime is a pseudo-Riemannian manifold. When a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the choice of time coordinates is largely arbitrary, it is the underlying space-time curvature that is physically significant.</div>
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Although an arbitrarily-curved space is very complex to describe, the curvature of a space which is globally isotropic and homogeneous is described by a single Gaussian curvature, as for a surface; mathematically these are strong conditions, but they correspond to reasonable physical assumptions (<span style="text-decoration: underline;">all points and all directions are indistinguishable</span>). (<a href="http://en.wikipedia.org/wiki/Extrinsic_curvature"><span style="color: #3b2e7b; text-decoration: underline;">Same source as above.</span></a>)</div>
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Since mechanics has its foundations in geometry, let's run through some more of the basics.</div>
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"The intrinsic geometry and the extrinsic curvature of a three-dimensional hypersurface embedded in a four-dimensional Riemannian spacetime have the same definition and same geometric significance as those of a two-dimensional surface in a three-dimensional flat space." <a href="http://books.google.es/books?id=FFeyTEU10BAC&pg=PA122&lpg=PA122&dq=consequences+of+Theorema+Egregium+in+spacetime&source=bl&ots=epIQCMpxzb&sig=9sWSjWKlxK_4KG4iBvPkyvmBZ4U&hl=en&ei=sEQWTNOrGon24Aapq728CQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CB8Q6AEwAw#v=onepage&q&f=false"><span style="color: #0536cd; text-decoration: underline;">Source: General Relativity and Gravitation, Canonical quantum gravity, Karel V Kuchai</span></a>. </div>
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Gauss discovered many remarkable theorems during his life time, but this one Gauss himself believed to be truly remarkable: he called the result <i>theorema egregium</i>. </div>
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The reason Gauss was so enthusiastic is that his formula proves the Gaussian curvature of a surface is indeed <span style="text-decoration: underline;">intrinsic</span>, that is, it is not dependent on the embedding of the surface in higher dimensional space.</div>
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Gauss' <i>theorema egregium</i> states that <a href="http://en.wikipedia.org/wiki/Gaussian_curvature"><span style="color: #0536cd; text-decoration: underline;">Gaussian curvature</span></a> of a surface can be determined from the measurements of length or distance on the surface itself. The idea Gauss had was that certain properties can be measured regardless of how a curve is positioned in space. This is important for our purposes, since it means we can determine the topology of the visible universe; something that plays a significant role in our discussion of curvature, redshift z and, as we will see, staticity.</div>
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The tricky part is arriving at the correct way of defining global curvature. It can be very difficult to distinguish intrinsic curvature (which is real) from extrinsic curvature (which does not produce directly observable effects). Gauss showed that spheres have intrinsic curvature, while cylinders do not. <i>"The manifestly intrinsic tensor notation protects us from being misled in this respect. If we can formulate a definition of curvature expressed using only tensors that are expressed without reference to any preordained coordinate system, then we know it is physically observable, and not just a superficial feature of a particular model."</i> (Crowell, 2010)</div>
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The astonishing feature is that the intrinsic properties of a surface—or the generalization of a manifold—are definable and measurable without regard to any external frame of reference. The Gaussian curvature is such a property, but the <a href="http://en.wikipedia.org/wiki/Principal_curvature"><span style="color: #0536cd; text-decoration: underline;">principal curvatures</span></a> are not.</div>
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Recall that to find the principal curvatures, one must take perpendicular slices, which requires that our surface sit in some higher-dimensional space. This is an extrinsic view. The fact that the Gaussian curvature of a surface, as computed by the principal curvatures, yields an intrinsic quantity is quite remarkable. It means that creatures confined to live on the two-dimensional surface of a sphere <span style="text-decoration: underline;">could</span> tell that the geometry of their space was different from the geometry of a flat piece of paper. This intrinsic difference is due to the intrinsic curvature of the sphere's surface. For practical purposes, it means that any observer could tell that the geometry of their local vicinity was different from the geometry of a flat Euclidean spacetime manifold, if indeed the global curvature was intrinsic. And it is relatively straightforward to show that redshift z could indeed be due to intrinsic curvature. This means that creatures embedded in a four-dimensional spacetime manifold <span style="text-decoration: underline;">could</span> tell that the geometry of their universe was different from the that of a flat Minkowski space (one that is expanding or not).</div>
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This is important in our discussion of spacetime in the presence of a global gravity field in a homogeneous and isotropic universe. It means that the curvature of the four-dimensional manifold in which we live can be measured and understood, without having to speculate on the existence of any other dimensions, i.e., physical reality could be described without extra dimensions (it has yet to be shown how this would affect other fields of physics, e.g., QM or string theory); <a href="http://stason.org/TULARC/education-books/startrek-relativity-FTL/5-4-Manifolds-Geodesics-Curvature-and-Local-Flatness-Gen.html"><span style="color: #0536cd; text-decoration: underline;">source</span></a>. But not only that, it means that light propagating in such a manifold would be redshifted as it follows the geodesic: Whereas, <span style="text-decoration: underline;">material objects would not</span> be affected (i.e., massive bodies react as if embedded in a flat spacetime).</div>
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Redshift would increase with distance, and do so without the steep requirements of large gravitational potentials, of the kind observed locally for gravitational redshift, though this has yet to be determined quantitatively. Results may very well indicate that criticisms regarding the missing-mass problem often associated with the gravitationally curved spacetime approach (in lieu of expansion) to generate the required redshift may not be pertinent: for example in the case of Ellis, G.F.R. (1977, Is the Universe Expanding?, General Relativity and Gravitation, Vol. 9, No. 2 (1978), pp. 87-94), Ellis writes that a <i>“spherically symmetric static general relativistic cosmological space-times can reproduce the same cosmological observations as the currently favored Friedmann-Robertson-Walker universes”</i> and adds (in a manuscript note) that it is difficult to fit the mass-redshift observations well within a static universe. Meaning that there does not appear to be enough mass m in the universe to attribute the redshift z to a gravitational effect, and so considers this as evidence <i>against</i> the stationary models.</div>
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Einstein's spherical model is simple in geometrical terms because of its symmetry: every point on the surface, in reduced dimensions, is equidistant from the center O (see Figure xyz). The geodesic paths between two points on a sphere are great circles (world-lines). A geodesic arc is simply the shortest path between two points along the surface. Of course, it would be shorter to go straight through the sphere between the two points, but that is not possible for the surface dweller. A great circle is just like every other circle with the additional constraint that its center lies at the center of the sphere. The curvature of world-lines, in this case, is measured to be the same for all observers regardless of a particular coordinate system. Thus an<i>intrinsic</i> measurement of curvature can be derived; and is so from the observer's local coordinate system (her rest-frame) as she measures distances. This will become clearer as we expand below.</div>
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If we want to measure <i>Gaussian curvature</i> locally the result will approach the Euclidean value as distance from the observer is small, since our spacetime appears locally flat (excluding the humps and bumps of the fields surrounding the planets, the sun, the Galaxy and so on); just as we see the earths surface to be "flat" locally. This is why Euclidean geometry, special relativity and Newtonian mechanics are a good approximation for small-scale measurements. It is interesting to note that according to general relativity, too, the character of spacetime locally (a <i>finite</i> region of the continuum with reference to which curvature, the Riemannian-Christoffel tensor, essentially vanishes) abides by the laws of special relativity (see <a href="http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf"><span style="color: #5588b1;">The Foundation of the General Theory of Relativity, Einstein, 1914-1917, pp. 176-179, pdf</span></a>). This serves as a guide to understanding how gravitating systems remain stable despite Gaussian curvature of the global field, i.e., how objects remain at rest in a static, homogeneous, spherically symmetric gravitational field, without need of a fictitious vacuum energy. In essence, relative to the Gaussian curvature of the manifold, all objects (e.g., clusters of galaxies) are situated in a region where the global field vanishes. Only local fields influence the motion of material objects. The latter is true because <span style="font: normal normal normal 18px/normal 'Times New Roman';">the gravitational field of an object (which clearly has non-zero intrinsic curvature) cannot simply be <i>transformed away</i> by a change of coordinates.</span></div>
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Curvature is thus a property of spacetime that increases with distance from the observer. Any observer will be entitled to consider her local area to be flat. As the range of observation increases, and the time-travel for photons increases, the departure from linearity as we see it in the look-back time becomes markedly large, and does so all the way to the visible horizon. Essentially, all photons emitted from outside the local region (and more so with increasing distance) will appear as if they had to 'climb' out of a gravitational well. The spectra will appear as if 'stretched' to longer wavelengths. The viewer can interpret this shift as due to motion or curvature: as a Doppler-<i>like</i> effect, or as a gravitational redshift-<i>like</i> effect. And because these shifts transpire throughout the entire electromagnetic spectrum, the two differing interpretations are empirically indistinguishable. To understand that which is observed beyond the local region (say, on scales beyond the Local Group or even the Local Supercluster) with respect to redshift z, we must turn towards general relativity.</div>
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Recall, here the assumption is that both the degradation in energy and the dilution in the rate of photon arrival results from the geodesic path the photon must travel (with distance and time) through a pseudo-Riemannian four-dimensional continuum of constant Gaussian curvature. T<span style="font: normal normal normal 18px/normal 'Times New Roman';">he paths of light delineate the structure of the manifold. The metric components of a pseudo-Riemannian manifold are continuous differentiable functions of relative position (not motion). </span>That is the meaning of redshift z as a gravitational phenomenon in a static, stationary yet dynamically evolving general relativistic spacetime manifold. (The dynamically evolving aspects of a stationary universe will be expanded upon in a subsequent post). So redshift is essentially a measure of global curvature. Redshift and time dilation give us an intrinsic measure of curvature.</div>
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A confusion, in my opinion, with this regard has arisen in part due to interpretations of the equivalence principle: since the metric for an accelerated observer, which arises only from the choice of the coordinates, is not a measure of intrinsic curvature (even though it does indicate the presence of a gravitational field). By the equivalence principle, the gravitational field experienced by an accelerating observer is indistinguishable from an acceleration arising from a gravitational field permeating all of space. The interpretation of gravity from the local point of view of an accelerated observer is certainly consistent with GR (just as the FLRW metric is consistent with GR), but it may very well be unrelated to Einstein's hope of finding solutions to the field equations that would accurately represent the essence of the physical universe and it's evolution in time.</div>
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The beauty of applying Gauss' <i>theorema egregium</i> (and the subsequent Riemannian geometric architecture) to general relativity is that it protects us from being misled in this respect, with reference to global curvature. If a definition of curvature expressed can be formulated using only tensors with reference to our local coordinate system (and by extrapolation to any other 'local' coordinate systems based on the assumption of homogeneity and isotropy), then we know the curvature is physically observable (and can be interpreted as curvature), not just a superficial feature of a particular model. </div>
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Again, this is a generic global field, the most general class of the curvature tensor in Einstein's general theory of relativity. This globally smooth field is an irreducible (not a local product of space) n-dimensional homogeneous pseudo-Riemannian manifold that contains an inversion symmetry about every point (i.e., a globally Riemannian symmetric space of constant sectional curvature where <a href="http://en.wikipedia.org/wiki/Riemannian_symmetric_space"><span style="color: #3b2e7b; text-decoration: underline;">geodesic symmetries</span></a> are defined on the entire manifold). However, when we extend the spherical model to 4-dimensions, straight lines (great circles on a sphere) do not come back to their starting point: This is a non-compact geometry that has open geodesics, i.e., the geometry of the universe is not compact, it is assumed here infinite in extent with infinite paths of constant direction and the space has no definable volume (not yet anyway).</div>
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There are three classes of Riemannian manifolds with constant sectional curvature over the entire surface: positive curvature (+1), zero curvature (0), and negative curvature (-1), corresponding to three geometries; a unit sphere, Euclidean space, and hyperbolic space. It will be shown below that when an observer measures distances (relative to her rest-frame) to astronomical objects in such spaces of geometry (+1) and (-1) she will find that the deviation from linearity (from zero curvature) increases with distance. In reduced dimension, this is equivalent to deviation in the sum of angels of a triangle from 180° in accord with Toponogov's theorem which characterizes sectional curvature in terms of how "fat" geodesic triangles (or how "thin" geodesic triangles) appear when compared to their Euclidean counterparts. </div>
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Locally, sufficiently small triangles will appear Euclidean to a close approximation, while with increasing distance non-linearity increases. The larger the triangle under consideration, the "fatter" or "thinner" it becomes. It follows that if an observer measures curvature via triangulation, where one of the points lies near or at the visual horizon, it would be found that curvature, or deviation from linearity, attains a maximum value (the magnitude of geodesic distortion is greatest).</div>
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Spacetime indeed plays a dual role in general relativity because it comprizes both the nonlinear dynamical object globally, and the context within which the nonlinear dynamics are defined locally. The metrical relations between objects determine the relative positions of those objects, and those positions in turn influence the spacetime metric. And because every form of stress-energy gravitates, including gravitation itself, an exact analytical solutions to the field equations with respect to global spacetime curvature will be very difficult to determine. The global field itself cannot be uniquely defined by the distribution of massive bodies since the field itself can serve as an "object" with its own <i>intrinsic</i> curvature.</div>
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<span style="color: #4388b6;"><a href="http://en.wikipedia.org/wiki/Differential_geometry">Differential geometry</a></span>, leading up to the mid-nineteenth century, was studied primarily from the <i>extrinsic</i> point of view. Curved surfaces were considered as residing in a Euclidean space of higher dimension (e.g., a the space surrounding a <a href="http://en.wikipedia.org/wiki/Mathematical_object"><span style="color: #0536cd; text-decoration: underline;">geometric object</span></a>). The intrinsic point of view, where there is no 'outside' of the object, was developed by Gauss and expanded upon by Riemann. The fundamental result is that of Gauss' <i>theorema egregium</i>. <a href="http://en.wikipedia.org/wiki/Gaussian_curvature"><span style="color: #0536cd; text-decoration: underline;">Gaussian curvature</span></a> is an intrinsic invariant.</div>
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Recall that <a href="http://en.wikipedia.org/wiki/General_theory_of_relativity"><span style="color: #0536cd;">Einstein's general theory of relativity</span></a> is expressed in the language of <i>differential geometry</i>. Accordingly, the universe is considered a smooth manifold endowed with <a href="http://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold"><span style="color: #0536cd;">pseudo-Riemannian metric</span></a>. This metric describes the curvature of spacetime. <span style="font: normal normal normal 18px/normal 'Times New Roman';">The spacetime metric <i>is</i> the field. </span>Understanding this curvature is essential, not solely for the positioning of satellites into orbit around the earth, or for understanding the dynamics of self-bounded gravitating systems, but for understanding the global geometric structure of the universe itself.</div>
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The intrinsic point of view is useful in general relativity when spacetime manifold cannot be taken as <i>extrinsic</i> (what is there beyond?). The global concept of curvature from the <i>intrinsic</i> point of view, however, is difficult to define in relation to submanifold structures. For example when mapping two local coordinate systems each its own Lorentz frame, though transitionally smooth, there is an entirely <i>extrinsic</i> difference between the two, i.e., intrinsic measurements available in general relativity are not capable of detecting an arbitrary change of coordinates (Crowell, 2010).</div>
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In order to reconcile the two different views, <i>extrinsic</i> curvature can be considered as a structural addition to <i>intrinsic</i>curvature. (<a href="http://en.wikipedia.org/wiki/Differential_geometry#Intrinsic_versus_extrinsic"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>). The point is, a local observer should be able to detect violations of the Pythagorean theorem globally, since measurements will lead to a metric that shows a non-Euclidean value for the ratio of the circumference of a circle to its radius.</div>
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Contrarily to the above arguments, the standard model interpretation of global curvature relates the expansion rate with the mass-energy content. Thus the universe can be either open, closed or flat (hyperbolic, spherical or Euclidean). This is not an intrinsic curvature. Models can be deceptive in this regard since they tend to impute physical reality to features or characteristics that are purely extrinsic (i.e., only present in that particular model). Intrinsic features, on the other hand, are logically implied by the axioms of the system itself. The existence of great circles (a geodesic by extension to 4-dimensions) is clearly an intrinsic feature of non-Euclidean geometries, because these spherical lines (or paths) can be defined independent of any model.</div>
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If indeed the global curvature of spacetime is intrinsic, would that not mean that is it is real and observable? It certainly would. Ultimately, if indeed gravity is concerned with redshift z, there is no unique inertial frame of reference which will correctly explain the large-scale geometric structure of the visible universe. All rest-frames are equivalent. Yet a spacetime diagram drawn for an inertial reference frame is unique to the observer located at the origin of the diagram. All observers are entitles to place themselves at the origin (and in fact have no other choice). That doesn't explain the way things truly are for all frames of reference. This coordinate-independence property is also called general covariance. And the global curvature falls precisely within this definition. It is truly an intrinsic curvature.</div>
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In sum, the curvature is intrinsic and the effects we observe are real, what is not real are the locations of objects, or distances between us and the objects; these are only 'apparent.' This problem is common in standard cosmology. Because of expansion and light travel time, objects are thought to be much further away than they appear to be. The difference here is that when expansion is replaced by curvature the "real" distance is not as far. I referred to this above, with a lack of rigor, as an <i>apparent curvature</i>, since the curvature distorts distances giving a false impression of distance, but too, because observers located elsewhere in the universe might see the Milky Way as immersed deep within a gravitational well (at high-z), which is obviously not the case. In that respect, curvature can be seen as illusory or apparent, but the best word to describe the phenomenon of global curvature is <i>relative</i>, as in depending on the observer's frame of reference (which is to a good approximation flat, and where time flows at a particular rate) compared to the rate and location of distant objects.</div>
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Yes, the curvature is intrinsic, yes photons are affected by it, and yes the magnitude of curvature can be determined. But the objects we observe are not really where they appear to be, i.e., objects such as distant galaxies are not at their proper distances. If you could fly out to one of those galaxies right NOW you would see that clocks run at the same rate as they do here on earth, and that the distance traveled would not coincide with the distance measured before departure, and that when looking back in the direction of our little blue planet, its neighborhood would no longer look Euclidean, time would run slower. </div>
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So even though the global curvature is intrinsic (real) the distortion caused by the curvature is entirely relative to how the observer sees the universe in time (in the look-back time) from her non-privileged rest frame. </div>
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We've seen above that the situation is slightly different locally with respect to gravity fields, where gravitational redshift is observed as the photon escapes the gravity of an objects such as the earth (as viewed by an observer at a higher altitude than the signal was emitted). The observer finds that the energy of the light decreases as it rises. In both cases though (in the global and local fields) redshift occurs as light travels a geodesic. And in both case the geometry of the field can be determined. As long as you know that the energy of light is related to its frequency (a wave with<span style="font: normal normal normal 18px/normal 'Lucida Grande';"> </span>crests and troughs), and if you could make note of the crests and troughs as they arrive, then you could calculate the frequency of the wave as 1/dt, where dt is the time between the point when one crest arrives and the point when the next crest arrives. So, if the energy of the light decreases, and thus its frequency decreases, then dt (the time between<span style="font: normal normal normal 18px/normal 'Lucida Grande';"> </span>crests) must increase. (<a href="http://stason.org/TULARC/education-books/startrek-relativity-FTL/5-1-Reasoning-for-its-Existence-General-Relativity.html"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>)</div>
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The geometry of a manifold can be completely determined if one knows the form of the invariant interval using a particular coordinate system on the manifold. Starting with the form of the invariant interval in some coordinate system on a manifold it can be determine the curvature of the manifold, the path of a geodesic in spacetime, and everything we need to know about the manifold's geometry. The mathematics used to describe these properties involves geometric constructs known as tensors. The invariant interval on a manifold is directly related to a tensor known as the metric tensor on the manifold. (<a href="http://stason.org/TULARC/education-books/startrek-relativity-FTL/5-5-The-Invariant-Interval-General-Relativity.html"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>)</div>
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If the intrinsic metric and extrinsic curvature of all possible sub-manifolds are connected to Gauss's remarkable theorem, it can be concluded with reasonable certainty that the space in which they are embedded is Euclidean (i.e., "flat"). As it turns out, even in a flat three-dimensional Lorentzian spacetime manifold Gauss' theorema egregium still holds (with a change of sign). What this means, in a practical sense, is that proper distances, if curvature were to vanish (if the distortion were no longer present) and if light was instantaneous, objects would be at their Euclidean distances. In another way, without gravitation (i.e., if curvature were to be removed) the manifold would be flat. We could thus conclude that curvature is embedded in a Euclidean universe. But that would not be an entirely accurate assessment of the real world.</div>
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Let's give an analogical example in reduced dimension to see how this might operate and to make our point: Consider a sphere, where perfectly round ball-bearings are allowed to roam freely with respect to one another. Lets assume the balls are confined to the surface of the sphere. These ball-bearings are not at all affected by the curvature of the sphere upon which they are confined, since all points on the sphere have the same gradient (essentially equal to zero). There is no slope or gradient that would impel the objects to move. Every object is located in a quasi-Euclidean submanifold (in its own rest-frame). All motions of these objects would be due to interactions between them, say, through local gravitational perturbations, just as planets, stars and galaxies interact: without all being compelled to coalesce in the same location. Likewise two, or more, ball-bearings placed on the manifold would not roll away from one another "down hill." Recall, our sphere has a smooth curvature, with no hills or valleys. Adding such to the discussion would simply correspond to local inhomogeneities (e.g., galaxies and their fields), not global ones. Extend this reduced dimension analogy to four-dimensions and the situation remains the same. A curved universe is free of large-scale instability. That's the short story.</div>
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<b>Geodesics and redshift</b></div>
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What about light? Why would light be redshifted yet objects not affected? Light emitted from distance objects (which by definition propagates, unlike massive bodies) would be impelled to travel a geodesic on the surface of such a sphere with intrinsic curvature. Because of the limited velocity of light c, an observer located on this surface (a very large sphere) will see EMR emitted from a distant source redshifted with the associated time dilation component, since light is traveling a geodesic path. The time it takes for a photon to travel from the source (far-removed) to the observer located anywhere on the surface, will be longer than if the line were straight (i.e., not geodesic, as on a flat Euclidean plane). </div>
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The world-line of photon on a spherical surface appears curved, but it is simply the analog of straight lines in the non-Euclidean geometry used to describe gravitational fields in general relativity. Due to the curved sphere, the further the distance that separates the points, the further the deviation from linearity will be manifest, from the observer's point of view. Say we add another luminous source further removed from the observer (any observer). The path traveled on the Gaussian manifold (the geodesic) will induce a larger deviation still, than the path of a Euclidean straight line. This difference is negligible as long as distance traveled by the photon is very small, but becomes larger with greater distance. Note, in passing, that a universe in which the spatial curvature is positive the circumference of a circle less than its Euclidean value. By consequence, distant objects will appear closer than in a Euclidean space. Redshift is due to the geodesic path of a photon with distance, not distance alone (since no matter how far removed is the source, redshift will not occur in a Euclidean, flat, universe). </div>
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Let's exemplify the situation with a schematic diagram:<br />
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<b>Figure G</b></div>
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Three <a href="http://mathworld.wolfram.com/SphericalTriangle.html"><span style="color: #0536cd;">spherical triangles</span></a> are drawn (albeit inaccurately here) on the surface of a sphere, which represents the visible universe (but which could represent the entire universe) in reduced dimensions on a spherical 'plane', consistent with Einstein's infamous static model, 1916-1917. This is a topological manifold without a boundary, and without a center.</div>
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Remark: <b>Figure G</b> differs from diagrams A, B and C above (Figure 1ABC), which are cross-section slices of the visible universe in the look-back time as seen by an observer at O. The geometry represented in Figure G is, however, compatible with <a href="http://i289.photobucket.com/albums/ll226/DVDjHex/GeneralRelativisticSpacetimeMani-1.jpg"><span style="color: #0536cd;"><b>Figure 1C</b></span></a> and 1Cb below. The poles in G can be disregarded since any point on the sphere can be considered a pole or a non-pole. All points are equivalent. (See <a href="http://www.google.es/imgres?imgurl=http://mathworld.wolfram.com/images/eps-gif/SphericalTriangle_700.gif&imgrefurl=http://mathworld.wolfram.com/SphericalTriangle.html&usg=__FDh2-SBO1_msEOcxMZa0p7xIsYw=&h=223&w=219&sz=12&hl=en&start=95&um=1&itbs=1&tbnid=BXadoi61H3-oHM:&tbnh=107&tbnw=105&prev=/images%3Fq%3Dtriangle%2Bspherical%2Bgeometry%2Bgeodesic%26start%3D90%26um%3D1%26hl%3Den%26sa%3DN%26ndsp%3D18%26tbs%3Disch:1"><span style="color: #0536cd; text-decoration: underline;">here, for example</span></a>).</div>
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The shortest path between two points on a Gaussian sphere (which here represents in reduced dimension a 4-dimensional spacetime manifold with intrinsic spherical curvature) is a segment of a <a href="http://en.wikipedia.org/wiki/Great_circle"><span style="color: #0536cd; text-decoration: underline;">great circle</span></a> (these are analogue to 'straight lines' in Euclidean geometry). Every photon is traveling a segment of a great circle (a geodesic). This is the path a non-accelerating photon would follow. The geodesics in spacetime depends on the Riemannian metric (or pseudo-Riemannian), which affects the notions of distance, and the notion of time. The geodesic affects the energy of every photon (i.e., they are seen as redshifted to an observer with look-back time), yet the geodesic <b>does not affect objects (</b>i.e., objects are not displaced by the global curvature<b>)</b>. Again, photons are confined to propagate along the shortest path between two points along the geodesic, but object are not, simply because there is no acceleration associated with a spherical plane in any direction. The intrinsic Gaussian curvature is equal everywhere, i.e, the geometric curvature is constant as opposed to varying from point to point. On a sphere of radius ρ, we have Gaussian curvature K=1/ρ2. (<a href="http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>)</div>
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"If the geodesics defined by an airplane and a radio wave differ from one another, then it is not possible to treat both problems exactly using the same geometrical theory. In general relativity, this would be analogous to a violation of the equivalence principle. General relativity's validity as a purely geometrical theory of gravity requires that the equivalence principle be exactly satisfied in all cases." (Benjamin Crowell, <a href="http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html"><span style="color: black;">A Geometrical theory of Spacetime, section 1.5.2</span></a>).</div>
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The angles of all planar triangles (or <a href="http://mathworld.wolfram.com/SphericalTriangle.html"><span style="color: #0536cd; text-decoration: underline;">spherical triangles</span></a>) on the sphere have the sum of the angles between π and 3π radians, i.e., between 180° for the smallest of triangles, and 540° (Zwillinger 1995, p. 469; see too Gauss's formulas p.471, or refer to <a href="http://mathworld.wolfram.com/GausssFormulas.html"><span style="color: #0536cd; text-decoration: underline;">this</span></a> or <a href="http://mathworld.wolfram.com/SphericalTrigonometry.html"><span style="color: #0536cd; text-decoration: underline;">this</span></a>). Though the task of measuring curvature via distances in a four-dimensional manifold (the universe) is daunting, it is nevertheless possible in principle, and emperically, to do so (e.g., using a variety of techniques, i.e., apparent brightness (or luminosity distance), rise times, of SNe Ia as standard candles, etc. See <a href="http://en.wikipedia.org/wiki/Cosmic_distance_ladder"><span style="color: #0536cd; text-decoration: underline;">cosmic distance ladder</span></a>).</div>
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Notice what happens in <b>Figure G</b>: when an observer studies regions locally (or nearby) she measures distances and angles that lead to a quasi-Euclidean geometry (see triangle <b>a</b>). The further she looks, the greater the distance, the greater the departure from linearity, the greater the curvature (see triangle <b>b</b> in <b>Figure G</b>). Notice the curvature with respect to the triangle formed by points <b>A</b>, <b>B</b> and <b>C</b>. The largest triangle here appears large and greatly curved, but it is small compared to the entire sphere (slightly larger than the surface of Africa compared to the earths surface). Much larger triangles can be drawn to exemplify the curvature which would manifest itself as redshift z at distances near the edge of the visible universe (where the distortion, or curvature, appears to attain a maximum value). </div>
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The amount by which curvature exceeds 180° is called the <a href="http://mathworld.wolfram.com/SphericalExcess.html"><span style="color: #0536cd; text-decoration: underline;">spherical excess</span></a>. One can easily see that the smaller the triangle, the smaller is the spherical excess (i.e., eventually reducing to the Euclidean plane law in the small area limit as a close approximation). As large triangles and greater distances are considered, the spherical excess increases. This surplus determines the surface area of any spherical triangle. The geodesic path on which all photons must propagate becomes progressively distorted (curved) with distance (as compared to what would be the case on a flat plane, where for obvious reasons curvature does not increase with distance, i.e. there is no spherical excess). Thus on a spherically curved 4-dimensional manifold redshift increases with distance up to a maximum at the horizon (as redshift z tends to infinity).</div>
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For hyperbolic triangles (discussed below) <i>excess</i> is replaced by <i>defect</i>. Interestingly, no matter how the surface of a sphere is bent, distorted or dented, due to local deviations in linearity or local inhomogeneities (gravitational wells of stars, galaxies, clusters, or even superclusters; no matter how deep the gravitational well) the total curvature of the sphere remains 4π (the Euler characteristic of a sphere being 2). See <a href="http://en.wikipedia.org/wiki/Gauss-Bonnet_theorem"><span style="color: #0536cd; text-decoration: underline;">Gauss-Bonnet theorem</span></a> and <a href="http://en.wikipedia.org/wiki/Euler_characteristic"><span style="color: #0536cd; text-decoration: underline;">Euler characteristics</span></a> for a further discussion and proofs. </div>
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Here is one more example of spherical triangles, only this time closer to home (mine that is):<br />
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<b>Figure G3</b></div>
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Illustration of spherical geometry where the angles of a triangle do not sum to 180° but do sum to 180° given small enough regions. The globe here is a projection centered on Spain at 40°28<span style="font: normal normal normal 15px/normal Symbol;">′</span>20<span style="font: normal normal normal 15px/normal Symbol;">″</span>N 3°33<span style="font: normal normal normal 15px/normal Symbol;">′</span>39<span style="font: normal normal normal 15px/normal Symbol;">″</span>W. The map is of Cannes, France at 43°33<span style="font: normal normal normal 15px/normal Symbol;">′</span>05<span style="font: normal normal normal 15px/normal Symbol;">″</span>N 7°00<span style="font: normal normal normal 15px/normal Symbol;">′</span>46<span style="font: normal normal normal 15px/normal Symbol;">″</span>E. GoogleEarth images with high-resolution aerial photography data. (The 60° figure is only a rough estimate).</div>
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This phenomenon, by which EMR loses energy (is redshifted) on the reduced dimension spherical manifold, can easily be extrapolated to four dimensions. This then becomes related to phenomena of <a href="http://en.wikipedia.org/wiki/General_relativity#Gravitational_time_dilation_and_frequency_shift"><span style="color: #0536cd; text-decoration: underline;">gravitational time dilation and frequency shift</span></a>that occurs locally. If indeed the equivalence principle holds globally, the topology of the universe influences the passage of time. Whereas locally light sent down into a gravity well is blueshifted, and light sent in the opposite direction (out of the gravity well) is redshifted, globally there is no gravitational well to fall in or climb out (all points are equal). Light is redshifted (not blueshifted) and distant clocks run slower than local clocks because EMR is propagating along geodesic paths, on a great circle, the opposite of free-fall, in a one-way direction towards the observer (any observer) and in accord with Einstein's relativity. The difference between this global inverse-free-fall and climbing out of a well locally, has a classical counter-part. Globally, the photon travels as if it where attached to a string that extends from the center of the sphere (as if it were affected by a fictitious force, a fictitious inertial force) from the fixed reference frame of the observer.</div>
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<span style="font: normal normal normal 18px/normal Times;">So, </span>the metrical <i>distance</i> between any two points in the Gaussian manifold is directly proportional to the time required by photons to travel from one point to the other. The paths of photons in this manifold correspond to the geodesic. These paths are straight lines in a four-dimensional spacetime from the perspective of an observer.</div>
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Interestingly, a local gravity field has the characteristic feature that lines of force converge towards the center of gravity (arrows point to the center). Light is gravitationally redshifted as it propagates away from the center of gravity, and blueshifted as light propagates towards the center of gravity (assuming we have observers immersed inside the field at different altitudes measuring the frequency shifts, of course), and spectral shifts increase with distance or field strength.</div>
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The global field has the opposite characteristic feature, that lines of force will appear to diverge away from the observer (arrows point towards the horizon, in all directions). That would be so since curvature increases with distance. Light is redshifted as it propagates towards the observer, and does so with distance, not with field 'strength'. Note: this property would be apparent in the case for both spherical and hyperbolic manifolds. [The corollary with expanding models is obviously: redshift occurs whether the expansion rate is decelerated, accelerated or constant]. Redshift increases with distance in both spherical and hyperbolic static manifolds. Here, however, there is no corollary with blueshift, since the notion of altitude in the global field has no physical meaning: All points are equal, so all observers are at the rest-frame origin of the coordinate system, as in Figure 1Cb below.</div>
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This concept finds application in <a href="http://en.wikipedia.org/wiki/Lagrangian_mechanics"><span style="color: #0536cd; text-decoration: underline;">Lagrangian mechanics</span></a>, where it plays a conjugate role to <a href="http://en.wikipedia.org/wiki/Generalized_coordinates"><span style="color: #0536cd; text-decoration: underline;">generalized coordinates</span></a>, by deriving <a href="http://en.wikipedia.org/wiki/Equations_of_motion"><span style="color: #0536cd; text-decoration: underline;">equations of motion</span></a> in terms of a general set of generalized coordinates, the results found will be valid for any coordinate system that is ultimately specified: Too, as seen in the <a href="http://en.wikipedia.org/wiki/D%27Alembert%27s_principle"><span style="color: #0536cd; text-decoration: underline;">Lagrange–d'Alembert principle</span></a> in accelerating systems and the <a href="http://en.wikipedia.org/wiki/Virtual_work#Principle_of_virtual_work_for_applied_forces"><span style="color: #0536cd; text-decoration: underline;">principle of virtual work for applied forces</span></a> in static equilibrium systems, where the force of constraint vanishes. Virtual work on a system is the work resulting from virtual forces acting through a real displacement, in this case the photon's propagation along a geodesic (as opposed to a real force acting through a virtual displacement). The virtual work on a photon must be zero since the forces are zero, yet due to the continuous displacement of the photon (until it arrives at the observer) it undergoes consistent and constant strain (i.e., its spectrum is deformed; redshifted) as it travels a geodesic straight line towards the observer. </div>
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<b>A Natural Fine-Tuning</b></div>
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The above is a classical example of an equivalence principle, where there is an equality between a virtual force acting through real displacement, and a real force acting through a virtual displacement. This is equivalent to the situation under review here regarding cosmological redshift z, where on one hand there is thought to be a real force (dark energy) acting on a virtual or spurious radial motion (expansion in accord with the standard model), and on the other we have a virtual force (attributed to global curvature) acting on real displacements (that of the photon). Redshift z results in both situations, and the two interpretation may be easily <a href="http://en.wikipedia.org/wiki/Conflated"><span style="color: #0536cd; text-decoration: underline;">conflated</span></a>. Indeed, in the general relativistic context there is no reason to assume a purely radial motion is necessary to explain observations. In fact, the classical assumption of radial motion is unnecessary.</div>
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This is also related to Newton's third law, where forces occur in pairs (action and reaction): On one hand, in the standard model, the cosmological constant is the opposite reaction the gravitational force (an anti-gravity-like force, or curvature with an opposite sign). Though in this case the equilibrium is unstable since lambda dominates gravity on large-scales (and <i>visa versa</i>) causing an accelerated expansion. Without lambda the universe ends in a big crunch. In both cases, with or without lambda, the universe is unstable as a pencil balancing on it's point. So the physical nature of the gravitational reaction (or force) within the standard model is NOT identical to that of the action itself.</div>
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And on the other hand, we have an action that is due to the combined gravitational force of all massive bodies (the mass-energy density, whatever the total may be) resulting in a continuously curved spacetime continuum, and a reaction, also due to gravity, which is equal and opposite in all directions. This result leads to intrinsic motion locally (since locally the universe is inhomogeneous, with differing values of the metric tensor at each point), but zero instability globally (since globally the universe is homogeneous, with the same value of the metric tensor at each point). This result is consistent with the notion of a natural fine-tuning that prevents wholesale collapse or gross expansion from occurring in the physical universe. Thus, local gravitational field curvature influences the motion of objects. Globally, that is not the case. The intrinsic curvature of the spacetime manifold has no affect on massive bodies because, in Newtonian terms, there is an equal force or acceleration emanating from all directions. And in relativistic terms, curvature has the same magnitude in all directions, where locally spacetime curvature is virtually flat. So local gravity fields dominate over the global field. And this is precisely why the cosmological term never should have been introduced into the field equations in the first place, i.e., there never was a need for a 'force,' vacuum pressure, or curvature with an opposite sign, that countered gravity. Indeed, when gravity is considered a curved spacetime phenomenon, in accord with general relativity, the notion of global instability vanishes. The physical nature of the global gravitational field reaction (or force) is identical to that of the action itself (i.e., it is equal to zero). Essentially, just as the observer is confined to a locally Euclidean field, so too are objects.</div>
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A couple of days ago I was driving through the beautiful rolling hills along the Spanish countryside of Guadalajara. The road on the way to Molina de Aragon had many curves and ups and downs. I couldn't help thinking of the situation here under review. The analogy could not have been clearer, with local humps, bumps and curves playing the role of massive bodies and their respective gravitational fields. At the speeds I was driving there was absolutely no effect due to the global curvature of the earth. Or if there was, it was entirely dwarfed by the local inhomogeneities. Obviously the situation would have been different if I were traveling at the velocity of light. The local humps and bumps would have been relatively smoothed out. The energy gained going into a 'well' would have virtually equalled the outbound energy-loss. Yet the global curvature would have induced a nonzero departure from linearity. Had I been riding on a photon a great distance from my destination I would have been redshifted as I traveled a geodesic along a great circle (the shortest distance between two points), from an observers rest-frame at Molina. Well, the analogy is not perfect, but fun nevertheless.</div>
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By the way, there is a corollary in Newtonian mechanics, where gravity is assumed to be a force which locally draws objects having mass towards the center of any massive body. Where locally we have instabilities to a certain threshold that conserves angular momentum, globally there is no such instability. If the universe were infinite, homogeneous and isotropic at any give time, there would be no center of gravity towards which all heavenly bodies would fall forming "one great spherical mass". In an infinite universe with a homogeneous distribution of matter (on average) there is no preferred direction in which gravity will pull. In the second edition of <i>Principia</i> Newton wrote, <i>"The fixed stars, being equally spread out in all points of the heavens, cancel out their mutual pulls by opposite attractions."</i> Each particle is pulled by gravitational forces in all directions and remains, thus, undisturbed in equilibrium. Yet locally gravity would cause matter to condense and form astronomical bodies and systems. (<a href="http://books.google.es/books?id=-8PJbcA2lLoC&pg=PA323&lpg=PA323&dq=infinite+newtonian+universe+gravity+is+the+same+in+all+directions&source=bl&ots=T3cSL4yTuY&sig=fSQz6Gb0JFNAb91TA2CwNUfR5aA&hl=en&ei=9TEjTP6JJtS7jAee96hB&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCEQ6AEwAw#v=onepage&q=infinite%20newtonian%20universe%20gravity%20is%20the%20same%20in%20all%20directions&f=false"><span style="color: #0536cd; text-decoration: underline;">E. R. Harrison, 2000, Cosmology: the science of the universe</span></a>). </div>
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It is though nevertheless often considered that the equilibrium described by a Newtonian universe is unstable. For example <a href="http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html"><span style="color: #0536cd; text-decoration: underline;">Cromwell writes</span></a>, "Any perturbation of the uniform density of matter breaks the symmetry, leading to the collapse of some pocket of the universe. If the radius of such a collapsing region is <i>r</i>, then its gravitational is proportional to <i>r^3</i> and its gravitational field is proportional to r^3/r^2=r. Since acceleration is proportional to its own size, the time it takes to collapse is independent of it size." </div>
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The error in thinking here revolves around the idea that the universe itself will have a self-similar structure, in that clumping on small scales behaves identically to clumping on large scales. But observational evidence suggest that clumping is not the same at all scales. On scales compatible with stars clumping forms a tightly compacted masses (stars). On galactic scales collapse forms dense nuclei, with a tight grouping of stars surrounding the core which becomes less dense with distance from the core. On scales of clusters and superclusters galaxies are often separated by vast regions of 'empty' space. The larger the structure the looser the clumping. Beyond that no one knows, but one could speculate that the trend continues, leading to looser and looser clumping, until collapse is no longer an option. At all scales, large and small, angular momentum would be conserved. Just as collapse tends to halt due to angular momentum conservation locally, collapse tends to halt due to angular momentum conservation globally. To assume that the universe in its entirety (even one that is spatiotemporally finite) would behave as a collapsing protoplanetary or protostellar cloud is not justified by our current knowledge. Empirical evidence (from clusters up) would not corroborate with such an assumption. So it is possible that a Newtonian universe, where gravity is a universal attractive force, would not collapse, lest a Newtonian universe that extends to infinity. Local perturbations would not affect the global symmetry, just as the formation of a new star out of gas and dust does not affect the dynamics of a supercluster.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure SB</b></span></div>
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Figure SB represents a two-dimensional Euclidean manifold, a static (nonexpanding, noncollapsing) spatiotemporally infinite Newtonian universe. This universe is globally homogeneous (uniform). The observed 'concentric' circles represent hierarchical structures (inhomogeneities) consisting of solar systems, galaxies, clusters, superclusters, and so on. In this view we are looking at a cross-section of the universe, a flat plane. There is no cosmological redshift z in this universe.</div>
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Newton's idea that all points would mutually (or collectively cancel) in a spatially infinite universe with no center of symmetry is equivalent to a homogeneous gravitationally curved global field (see Figure SC below). Newton's idea did not imply that gravity would be zero (as in not present), but that gravity would be present everywhere with the same acceleration forced upon particles, equal to zero (the forces cancel-out). In our case, the general curvature of the field has similar properties.</div>
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But that didn't stop Einstein from running into the same difficulties as Newton. In fact GR seemed to make the situation worse. Where before local instabilities were thought to be contagious, spreading to all scales, now the entire spacetime manifold can uniformly collapse, with or without a center of symmetry. Furthermore, relativity does not guarantee that angular momentum conservation will prevent wholesale collapse, as it does in the case of classical mechanics. Even if it did, no one knows how to extrapolate conservation of angular momentum to the universe in its entirety (least of all one that is infinite).</div>
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Enter lambda: The cosmological term was introduced by Einstein in order to prevent gravitational collapse. It acted as a repulsive force, or vacuum pressure that would counter the attractive force of gravity. It was found by physicists that Einstein's model, despite lambda, was unstable. There was a fine-tuned configuration between matter and the cosmological constant. For any small perturbation away from this configuration, Einstein's field equations showed that the universe would tend to amplify the perturbation. If the matter density was slightly greater, the universe would collapse. And so Einstein's static universe model was deemed unstable, and hence unphysical.</div>
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Such as collapse<span style="font: normal normal normal 13px/normal Arial;">—</span>in a homogenous and isotropic universe without a center towards which objects would accelerated<span style="font: normal normal normal 13px/normal Arial;">—</span>is consistent with the idea that all the matter in the universe, including spacetime itself, would end up in a Big Crunch. But this concept is different from the standard <i>gravitational collapse </i>(of the kind responsible for the formation of galaxies, stars and planets). Inversely to the concept of "expanding space" in the standard model, we would have "shrinking space" heading towards a big crunch. Ultimately, it could be imagined that the universe<span style="font: normal normal normal 13px/normal Arial;">—</span>including space and time<span style="font: normal normal normal 13px/normal Arial;">—</span>would disappear (ending as a black hole), similarly to the was it was created. Physics breaks down in both scenarios. So it seems that an unstable model is just as unphysical as a stable model, but in a different way.</div>
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But the impetus to add a cosmological term to the field equations in order to allow a static universe, may not have been justified on physical or philosophical grounds.</div>
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Let's see, in the next section, whether the introduction of lambda was justified on geometrical grounds, in the context of a manifold with constant positive curvature (and in the context of a universe that extends to infinity spatiotemporally, or not).</div>
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The question as to whether the sphere (the visible universe in four dimensions) is infinite spatiotemporally or not, can be solved to some extent by projecting stereographically the spherical topology onto a Euclidean plane. Neither plane or sphere has a boundary. Simply put, two parallel lines extending outwards in a spherically curved Gaussian manifold would converged as they tend toward infinity but they would never actually intersect (this is perhaps what instigated Einstein's notion that spacetime would be progressively Minkowskian as distance tended towards infinity). This is why Einstein's universe can (and should) be considered infinite spatiotemporally. As it turns out, parallel lines extending to infinity would also converge in a flat Minkowski space-time and a hyperbolic spacetime, from the point of view of any observer, making differentiation between three geometries virtually impossible in a four-dimensional universe. Only distances would vary according to geometry. (See the train tracks in Figure T below). </div>
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The error in thinking has been—by analogy with a sphere (which is the two-dimensional surface of constant positive curvature)—to expect that the total volume of a spherically curved universe is finite. That problem vanishes in 4-dimensions since one can always imaging additional spherical shells centered on an observer that extend to infinity. There is no reason why a boundary should exist, and there is no reason to assume a spherically curved 4-dimensional pseudo-Riemannian universe should have a finite volume. </div>
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Just as plane can be thought of as a sphere with infinite radius, a sphere can be thought of as a plane with infinite radius. Similarly, the concept of a geodesic path can (and should) be thought of as a straight line in the plane (albeit not without distortion of distances). Recall that a Gaussian sphere of constant positive mean <a href="http://en.wikipedia.org/wiki/Gaussian_curvature"><span style="color: #0536cd; text-decoration: underline;">Gaussian curvature</span></a> is the product of the two principle curvatures. It is an intrinsic property that can be determined by measuring length (or distance) and angles, and, it does not depend on the way the surface is <a href="http://en.wikipedia.org/wiki/Embedding"><span style="color: #0536cd; text-decoration: underline;">embedded</span></a> in space. Therefore, bending a surface locally (curvature associated with massive bodies or local inhomogeneities of the field) will not alter the Gaussian curvature. All these submanifolds would have 'boundaries,' but the sphere is a surface with constant positive Gaussian curvature <span style="text-decoration: underline;">without a boundary</span>. (<a href="http://en.wikipedia.org/wiki/Sphere"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>). Note: this would be in contrast to a compact <a href="http://en.wikipedia.org/wiki/Heine-Borel_theorem"><span style="color: #0536cd; text-decoration: underline;">Euclidean n-sphere</span></a>, where the sphere is the inverse image of a one-point set under a continuous function, which implies the sphere is closed (bounded). The former should not be mistake for the latter.</div>
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So, in contradistinction to a closed Friedmann universe with zero cosmological constant Λ the scale factor of which varies with time, a curvature index k that can be 0, 1 or −1 (corresponding to flat Euclidean geometry, positive or negative curvature) and oscillates between a big bang and a big crunch, the solution now has no scale factor. The universe can remain stable when the universe is homogeneous and isotropic with positive curvature (k=1) and has one precise value of density everywhere, as first postulated by Albert Einstein. As noted above in another way, local inhomogeneities do not affect the intrinsic curvature globally, since objects do not partake geodesically on the manifold (i.e., objects are not accelerated in any direction). The equilibrium is stable despite being inhomogeneous on smaller scales, where local spacetimes are Lorentz submanifolds, in according with general relativity. In accord, too, with the cosmological principle, the universe is homogeneous and isotropic globally; every point in space is like every other point, with the same value or magnitude of curvature, hence, the metric tensor must be the same everywhere. Note too that nowhere in the solution is the cosmological constant Λ required in order to maintain stability. </div>
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Evolution in the look-back time is somewhat beyond the scope of the present discussion, but suffice it say for now that no matter how fast or slow galaxies and clusters evolve over time <i>t</i> (thought the consensus for a stationary universe must be very slow evolution), the stability of the universe remains unaffected. (Evolution, CMBR, thermodynamic processes, along with formation of the light elements and their isotopes will be the topics of a subsequent post).</div>
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In another way, if we sprinkle test particles (all at rest) across the surface of a sphere at any arbitrarily chosen points, then they will neither accelerate or gravitationally collapse relative to one another, and the volume of the sphere will remain infinite. This is exactly what we would expect in a static general relativistic universe. The Gaussian curvature is interpreted as uniform in all directions, in every region of space. So, the problem created by an attraction that results from the existence of material sources, causing gross collapse of the universe, vanishes. And simultaneously, the problem created by the introduction of a dubious term into the Einstein field equations of the kind (provided by the cosmological constant) that would cause a general repulsion, vanishes. </div>
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Interestingly enough, the equilibrium is explained naturally, without any <i>ad hoc</i> parameters or <i>new physics</i>. This is a simple example of how globally intrinsic curved spacetime does not cause cosmological expansion or collapse. The beauty of this preliminary model lies in its natural symmetry generated by a globally homogeneous (despite local inhomogeneities) distribution of gravitating bodies (whereas the unrealistic perfect symmetry assumed by our current cosmological models arises as an artifact generated by the big bang/inflation). So, if the universe is observed to be nonexpanding and noncollapsing, in accord with the interpretation of redshift z as a curved spacetime phenomenon, the it is natural to hypothesize that in the past it might have been a cooler place than current theory would have it. That would follow from the idea that all the matter and energy in the universe was not all clumped up together at one time in the past. Since much, if not all, of cosmology depends on the interpretation of redshift z, let's analyze in detail what exactly may be operational.</div>
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<b>Redshift z</b></div>
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Figure G2 below shows how redshift transpires on a Gaussian surface of constant positive curvature (when expanded to four dimensions of course). Here is a schematic 2-dimensional illustration:<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure G2</b></span></div>
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Redshift of Light on a Gaussian Surface of Positive Curvature. This is a reduced dimension static universe with redshift caused by geodesic travel in a curved spacetime manifold. Similar but not identical to the 1917 Einstein universe.</div>
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Figure G2 shows a spherically symmetric surface of constant Gaussian curvature with a positive sign. Redshift of photons increase with distance from the observer by virtue of the fact that photons propagate along geodesic paths (in 'straight' lines), along the line of sight of the observer.</div>
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Globally, geodesic paths upon which all photons would propagate are the shortest distances between two points (on a great circle). These paths are free from the influence of any force. Though these paths appear curved (convex-like) on the surface of a sphere, they are actually straight lines (in the Euclidean sense) when we observe astronomical objects. However, even though these lines are straight the apparent distance from the observer to the object in view is altered (curved, distorted); meaning that proper (Euclidean) distances no longer hold. The change in wavelength (redshift) is directly proportional to the Gaussian curvature, and thus to distance.</div>
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In accord with Figure G2 the wavelength of EMR is progressively redshifted. The spectrum at the source is consistent with the standard spectrum found here on earth for say hydrogen. As EMR propagates away from the source it loses energy as a function of both distance and time; where the redshift reaches its maximum upon arrival. Of course the observer sees only the final result. The diagram shows a snapshot of the entire travel-time.</div>
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In an expanding space of the standard model, the wavelength of EMR is shifted from the start and remains constant throughout the journey, if expansion is a linear function. There is no increasing distortion in expanding space, provided the expansion rate is constant. And of course, too, the observer sees only the final result (which would be the same as at the source). A wavefunction with constant wavelength is indistinguishable form a wavefunction with varying wavelength. (See covariant derivatives in electromagnetism).</div>
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In both cases, an observer located directly at the source (anywhere in the universe) will measure the spectrum of hydrogen to be consistent with the local value measured here on earth. So in both case the redshift is a relative effect.</div>
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One can see in Figure G2 that when the observer is closer to the source, the magnitude of redshift is not as great. Redshift is quasi-nonexistent in close proximity to the source. Likewise, we can see from the spherical representation Figure 1C below, that redshift increases with distance.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure 1C</b></span></div>
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Figure 1C represents a cross-section of a non-expanding globally homogeneous and isotropic four-dimensional spherically symmetric geometrically curved Riemannian (or pseudo-). Riemannian) general relativistic spatiotemporal manifold (i.e., a cross section of the visible universe).</div>
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<b>A few remarks:</b></div>
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<li style="color: #222222; font: 18.0px Times; line-height: 18.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">The observer is located at the origin (O). Every other point of the polar coordinate system is in the look-back time (in 360° on the cross section of the manifold, i.e., in all directions), relative to any observer's rest-frame. An observer located near the horizon would see the universe as if situated at O. So the horizon is an artifact resulting from the loss of energy associated with EMR in the geodesic path and travel time through a non-Euclidean universe, as view from the observers frame of reference.</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 18.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Spatial distances appear, from the observer's rest-frame to become smaller with increasing distance in the look-back time, as seen in the distance between 'spherical shells' (in contrast to Figure 6A, in the hyperbolic case, below).</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 18.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">I have not labeled the distance between 'concentric' circles (spherical shells) centered upon the observer (O), but for the sake of argument, let's assume for now that each spherical shell would be less than 2 billion light years apart (or about 0.5 gigaparsec). So the distance to the horizon would be less than 20 billion light years (Gly) from the observer. This would undoubtedly change depending on model specifications or empirical evidence.</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 18.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Redshift z is plotted more or less in accord with the standard model (out to about z = 1) but diverges exponentially as look-back time tends toward the horizon, where z approaches infinity. This would be analogous to superluminal expansion (where galaxies appear to exceed the speed of light as they disappear beyond the horizon in the standard model). </li>
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From the observer's rest-frame distances appear to become smaller with increasing distance. Time interval appear to slow down with increasing distance from O. This is the relative phenomenon of time dilation. Like the 1917 static de Sitter model, we have a situation where a clock placed at the observer will keep a different time than identical clocks placed elsewhere in the manifold. The timelike intervals depend on distance. The consequence would be that timelike intervals would become smaller for larger distance. In other words, again, clocks would appear to slow down with increasing distance. This is a de Sitter effect in a static universe.</div>
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Interestingly, we have a de Sitter-like effect operational, but with geometrical properties that resemble Einstein's static model. That would be so since in either geometric structure (hyperbolic or spherical) redshift is a curved spacetime phenomenon (regardless of how the manifold is curved, or regardless of the sign, positive of negative).</div>
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Interestingly too, there is a 1926 paper (ApJ 64 321) where E. Hubble derives the radius of curvature of an Einstein static model based on the mass density of nebulae. Hubble's uses the theoretical treatment of Haas (Haas, A. 1924, Introduction to Theoretical Physics, London, Constable & Co.). Even though this displacement toward longer wave-lengths is technically not the same as a de Sitter effect, it is still grounded on a non-expanding world-model. There would be a linear relation with distance over small distances (near the observer) with increasing divergence for larger distances. </div>
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In Einstein’s words; “By reason of the relativistic equations of gravitation…there must be a departure from Euclidean relations, with spaces of cosmic order of magnitude, if there exists a positive mean density, no matter how small, of the matter in the universe” The smallest possible density of matter produces constant positive curvature of space. He continues, “the metric quantities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration.” (1920, see Kerszberg, P. 1989, The Invented Universe, The Einstein-De Sitter Controversy (1916-17) and the Rise of Relativistic Cosmology, p.214). </div>
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If the cosmological redshift is indeed a curved spacetime effect, at least two important conclusions can be drawn: The observed part of our universe is extremely small relative to its actual extent: It follows that the universe is much older than generally believed. There is potentially much more mass-energy (in the form of atoms, stars, galaxies and clusters) beyond our visual horizon than previously suspected; the density of which participates in the overall curvature and distortion of light rays emitted from within our horizon. There would be enough gravitating mass in the cosmos to sustain the redshift-curvature interpretation. The requirement that mass-energy be extremely compact, as for the production of local gravitational redshifts, is not relevant to the global field, since the global curvature is a result of the total mass-energy content of the universe. Photons do not actually climb out of a gravitational well, globally. So there is no missing mass problem on a cosmological scale.</div>
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Indeed, Figure 1C is a general relativistic spacetime manifold not dissimilar to Einstein's 1917 model, i.e. it too has hyperspherical topology and positive spatial curvature, space is neither expanding, contracting, nor flat. The differences are that (1) there is no global instability and (2) no cosmological constant. (3) There is a relationship between redshift z and geometry; a concept which forms the basis for the curved spacetime interpretation for z. </div>
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Like the Chronometric model of Segal, I have not yet excluded a quadratic redshift-distance relation (though I have not embraced it either). Prior to the SNe Ia data a quadratic relation (where redshift increases as the square of the distance) had not been observed (using angular diameters and apparent magnitudes of galaxies), but in light of 'new' evidence, this hypothesis should be tested more rigorously. </div>
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There is no need for a repulsive force to counter the attractive force of gravity <i>locally</i> since there is no absence of motion. There is no need for a repulsive force to counter the attractive force of gravity gl<i>obally</i> since the magnitude of curvature at every point is virtually equal to zero. The universe is viewed as a place where objects move in accord with the mechanics and dynamics of relativity; not an entity itself that possesses a tendency or propensity to expand or contract. Nor would it have an age or a time where it came into being. </div>
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A formal equilibrium is achieved throughout the universe (without balancing gravity and repulsion) in a similar way as equilibrium is achieved locally, despite the propensity to coalesce locally. In other words, locally objects are permitted to form groups due to the 'hills' and 'valleys' inherent in the combined fields, yet there is a limitation (or threshold) beyond which the gravitational fields of objects no longer influence the fields of other objects further removed. This is a physical solution that bypasses the entire debate that has transpired in diverse circles for more than three centuries, culminating with the Einstein-de Sitter controversy (1917) and ending with the onset of expansion (1929). </div>
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The further difference is that the Einstein universe is spatially finite or closed (a three-sphere with a fixed radius r), i.e., with a fixed scale factor (though the model can also be interpreted as infinite spatiotemporally). Figure 1C extends only to the visible horizon, but the universe is considered here to be infinite and without bounds, globally homogeneous and isotropic at any given cosmic time. Like the Einstein model, there is no beginning of time, and there is no big bang in the past (there is no expansion).<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure 1Cb</b></span></div>
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General Relativistic Spacetime Manifold. This is a cross section of the visible universe, with the observer at the center. Oblique angle. This is a stationary, static universe, where redshift z is caused by a globally curved spacetime, with no cosmological constant.</div>
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The observer with knowledge of relativity located at the origin of Figure 1Cb will be able to make two different interpretations based on the observational evidence (redshift that increases with distance and reliable distance measurements). With her knowledge of <i>special relativity</i> she will conclude (1) that objects appear to be moving radially from her rest frame. If she assumes the universe to be homogeneous and isotropic, she will conclude that the universe is expanding, and that it should look the same for all observers. </div>
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With her knowledge of <i>general relativity</i>, our observer will be able to conclude, too, (2) that she lives in a universe where gravity is everywhere present. She will assume the universe is homogeneous and isotropic globally. And she will further conclude (after having obtained reliable distance measurements with techniques such as surface brightness tests, light curve rise times of distant SNe Ia, etc.) that the observed redshift (which she finds increases with distance) occurs as a result of energy loss (an increase in wavelength) associated with the propagation of electromagnetic radiation along geodesics (straight lines in her line of sight). Based on the empirical evidence, she will conclude that she lives in a general relativistic 4-dimensional spacetime continuum that exhibits constant Gaussian curvature with a positive (spherical) geometric signature. Her conclusion is based on the fact that gravity is a curved spacetime phenomenon and that its presence globally rules out a flat geometry. She thus rules out alternative hypotheses based on special relativity (a Minkowski space with time) that treat redshift as a radial motion in a quasi-Euclidean universe, even though she uses SR for local measurements (because SR is the local limit of general relativity).</div>
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From her preliminary investigation she concludes that the universe is spherical (consistent with distances measured in a manifold such as Figure 1Cb). And pending further investigation to confirm distances she left open the idea that curvature could be hyperbolic: All depended on whether objects appeared closer or further from her rest-frame as compared to what she would expect in a flat Euclidean space governed by the laws of special relativity. She had doubts from the start about the sign of curvature, and with the SNe Ia data trickling in, she began to ponder seriously the idea that hyperbolicity (too consistent with GR) might satisfy her ambitions: to learn more about the essence of the physical universe and its evolution in time.</div>
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Initially our observer was tempted by the expansion hypothesis; even to the point of ruling out the hypothesis based purely on GR. After all, her expansion idea (though previously touched upon be the likes of Poe, Kant and Newton himself) seemed new, different from the old loitering idea that had many centuries behind it. This universe was dynamic and evolving. It wasn't static anymore. But then she realized that something was wrong. In order for agreement with all the observations she need to supplement her knowledge of physics with several concepts that hitherto had never been observed: she needed to hypothesize that space itself was expanding. Too, she needed to postulate the existence of a form of matter that was nowhere to be seen in earth-based experiments (not made of ordinary electrons, neutrons or protons, i.e., nonbaryonic and "cold", CDM). Though she would eventually get the government funding she needed to build a large particle accelerator to test her hypothesis it seemed her search would end in vain (she could not produce enough energy; at least not for the time being). But that was the least of her worries, since without a bizarre form of vacuum energy, in addition to the CDM, there was no way to fit the observational data to her hypothesis. Ironically, she would be relieved to find that an <i>ad hoc</i> term originally introduced by Einstein into the field equations to render the universe stable could play the role of dark energy in her dynamic accelerating scenario. </div>
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The rest is history, but there is a point to be made. Upon reflection, when faced with a choice of the kind that presents itself, again, today: whether to choose a model based on Newton, Minkowski space and special relativity, or to choose a model based on a pseudo-Riemannian manifold with constant Gaussian curvature and general relativity, one should not hesitate.</div>
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So what might we conclude? From Figure G2, 1C and 1Cb, in four-dimensions now, we could conclude that light is affected by the global intrinsic curvature of the manifold, whereas material particles, or galaxies, are not. We have a redshift that increase with distance, along with its associated time dilation factor, and does so all the way to the horizon. <span style="font: normal normal normal 18px/normal 'Times New Roman';">Notice that the temporal intervals between two definite events ranges from zero (in the vicinity of the observer) to infinity (at the horizon).</span></div>
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In other words, both global stability and the observed redshift z in the spectra of distant galaxies are a pure products of general relativity, in accord with the basic principles of non-Euclidean geometry. The universe may in fact be free of perturbations that would cause it to collapse or expand. That conclusions can in fact be derived from general relativity and non-Euclidean geometry (without new physics).</div>
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Though most of the discussion in this Redshift z thread at Scienceforums.com has tended to revolve around the notion that spacetime must be hyperbolic (negatively curved) in order to satisfy observations (namely that of redshift z), it is found here that this <i>a priori</i> assumption turns out to be unwarranted, but not untenable (pending further investigation on empirical fronts). It has been found that a spherical geometry is consistent with the notion of cause for redshift z, and that stability can be maintained when the manifold under consideration is geometrically spherical. (Arguments for stability of a hyperbolic universe are explored below).</div>
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The problems of boundary conditions (a mass-horizon), global instability and redshift z are solved rather nicely with a robust analysis of both empirical evidence and the incorporation of such into the framework of general relativity (which itself describes gravity as a geometric phenomenon) leading to a cosmology similar to Einstein's original 1916-17 world-model, yet without the need for lambda. </div>
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Einstein's 1917 spherical model was thought to curve back in on itself, like the surface of a sphere, being spatially finite. Indeed, a 'straight' line on a sphere curves all the way around to form a great circle. But that is a reduced dimension vision of the world. When expanded to four dimensions the problem no longer presents itself. Great circles are nothing more than the analogue of "straight lines" in spherical geometry. In a 4-dimensional universe great circles are straight lines (geodesics). From a physical standpoint all that means is that when an observer looks out into the deep universe objects will appear closer than they actually are. As we saw above, that was because light looses energy as if the photons were 'slowing' down, decelerated. Spherical curvature would not at all change the visual size (or angular diameter) of an object, except in the normal sense that objects further appear smaller and object closer appear larger (than the size an object would be without distortion). And since spherical geometry tends to make objects appear closer to the observer than would be the case in a Euclidean or hyperbolic manifold, objects would appear slightly (imperceptibly) larger with increasing distance. Objects will appear slightly further in a hyperbolic geometry.</div>
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In the following three diagrams, let's look at the location of galaxies (taken from HUDF for convenience) relative to the observer (centrally located). Again, the ten 'concentric' circles represent spherical shells of equal distance centered on the observer.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure ADHU</b></span></div>
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure ADSU</b></span></div>
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure ADEU</b></span></div>
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The point here is that a given galaxy that covers one degree of the sky will cover one degree of sky no matter what the geometry of the universe. That is because the total field of view is always 360° (on any given plane) despite whether the global geometry is hyperbolic, Euclidean, or spherical. Only distances appear to differ, and do so increasingly with distance. Though, angular diameter would seem to change little (and doubtfully observable). Note: proper distances are represented by the Euclidean manifold.</div>
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All curvature means, and by corollary, all redshift means, is that the measure of the relative distance of an object as compared with the objects proper position (where it would be if no distortion of the image in the line of sight were to occur) is inconsistent with flatness.</div>
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All curvature means really, empirically, is that there is a relative change in the distance of an object when measured from an inertial frame (any point at which an observer finds herself, himself or itself, in the universe) as compared to the distance that would be otherwise measured in the absence of curvature. There is no mystery or paradox to be created. </div>
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And all a change in the signature of curvature means is that objects will appear closer to, or further away than, the proper distance of an object, again, if there were no curvature (or distortion). In other words, in a positively (spherically) curved universe object will appear progressively closer than otherwise expected, and in a negatively curved (hyperbolic) spacetime objects will appear progressively further than would be expected in an otherwise flat spacetime. Progressively is a key word here, because it implies a continuous transition, one that increases continually to the horizon (from where light no longer reaches the earth). The only boundary created in both cases is that of the visual horizon: the point beyond which light is distorted enough as to appear either "stretched" to flatness (as figure A above), or "compressed" ('compacted' with distance as figure C above). So, consistent with general relativity, we have in both cases a lengthening of electromagnetic wavelengths (cosmological redshift and time dilation) where spectral lines are shifted towards the red end of the spectrum as photons travel across the universe due to intrinsic global curvature (not cosmological expansion). <br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure SC</b></span></div>
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Figure SC represents a globally homogeneous 2-dimensional static Einsteinian universe characterized by constant positive Gaussian curvature and no cosmological constant. This universe is static and infinite spatiotemporally. The large-scale structures and their gravitational fields are shown as areas of 'concentric' circles (hierarchical structures, inhomogeneities, consisting of solar systems, galaxies, clusters, superclusters, and so on). In this view we are looking at the surface of a sphere. Below we find a similar interpretation.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure SCb</b></span></div>
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Figure SCb, like Figure SC above, represents a globally homogeneous 2-dimensional static Einsteinian universe characterized by constant positive Gaussian curvature and no cosmological constant.</div>
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In order to present a legitimate representation of the observations, the determination of the signature of curvature should be obtained (directly by measurement). This should provided a first order redshift/curvature interpretation for those observations. This cosmology would provide the best possible representation of general relativity. Recall that general relativity is not a cosmology. Being a theory of gravity, however, it should be possible to construct a cosmology. Indeed, many interpretations are possible, but only one interpretation will be correct. In the case above, the model represents a universe that does not expand or collapse, since gravity is not an attractive force, and since the field is globally homogeneous (on average the same everywhere). </div>
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<b>Angular Diameter and Parallel Lines</b></div>
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Note that by virtue of perspective alone, objects in a Euclidean universe will appear to become smaller with increasing distance, just as the ties under the railroad track will appear smaller and closer together further from the observer. And the two parallel tracks will appear to converge toward the horizon; even though we know the tracks are parallel and the space between each tie is the same (they are evenly spaced in the real world). See the figure below which represents train tracks with three different geometries, a, b and c (hyperbolic, Euclidean and spherical, respectively).<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure T</b></span></div>
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Classic one-point perspective of train tracks with three different spatial geometries: (<b>a</b>) negatively curved, (<b>b</b>) flat, and (<b>c</b>) positively curved, extending to infinity.</div>
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First and foremost, in a universe with either of the three geometries, objects of equal size placed along the tracks at equal distances will appear to become smaller, due to perspective alone. The question then becomes how does an observer empirically differentiate between the three geometries. </div>
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Notice that when a photon is emitted from a source located near the horizon of figure <b>b</b> (a flat Euclidean or Minkowski space-time), the intensity of light or other linear waves radiating from a point source (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source; so an object (of the same size) twice as far away, receives only one-quarter the energy (in the same time period); assuming there are no losses caused by absorption or scattering. (<a href="http://en.wikipedia.org/wiki/Inverse_square_law#Light_and_other_electromagnetic_radiation"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>). In the case of diagram <b>a</b> and <b>c</b> the inverse-square law does not appear to hold, since there is in both cases a loss of energy attributed to the geodesic path on which the photons must travel.</div>
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Notice too that the angular diameter of objects (of a fixed physical size) situated anywhere along the track will not differ relative to parallel lines of the tracks (in either <b>a</b>, <b>b</b> or <b>c</b>). The apparent distance and luminosity would differ, but the size difference would be virtually impossible to measure, i.e., it would be imperceptible. So, <span style="text-decoration: underline;">in practice</span> this behavior is <span style="text-decoration: underline;">not observed</span>.</div>
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Intuitively, and in principle, if angular-diameter could be observed, spatial curvature would either increase or decrease the apparent size of an object ever so slightly. With a spherical geometry (<b>c</b>) light rays on a geodesic path would make objects appear closer and therefor slightly larger, while a hyperbolic geometry would have the opposite effect (making objects appear further and slightly smaller). </div>
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But since the difference would be small, and the uncertainty is size would be large, the angular-diameter test would be a little use (if any at all). </div>
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In passing, thought the distances of objects would change slightly depending on curvature, as can be seen by the ties under the tracks (here roughly drawn and exaggerated perhaps), there is an overall difference at the horizon itself. Objects located at the horizon of <b>a</b> and <b>c</b> appear closer and further respectively than objects on the horizon of <b>b</b>. The distance to the horizon appears different in the three manifolds. This difference may not appear to be of much significance. But if each tie in the tracks represents a spherical shell (surrounding an observer) each a distance of 1 billion light years, the deviation in linearity represented by <b>a</b> and <b>c</b> would yield a discrepancy of billion of light years from the observer to the horizon. Simply put, an object at the horizon in a hyperbolic universe (<b>a)</b> would appear billions of light years further than the same objects in a positively curved spacetime (<b>c</b>). </div>
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So the best way to determine the sign of global curvature would to to compare distance measurements of standard candles (inferred from the apparent brightness of high-z SNe Ia) with what would be expected in a flat universe, of the most distant objects in the visible universe. The apparent radius of the visible universe compared to the expected radius of a Euclidean universe (if that could be determined, perhaps via the inverse square law, or Gauss' law) would give us insight as to the nature of the global curvature. The volume of a Euclidean sphere is given by the formula <a href="http://en.wikipedia.org/wiki/Hypersphere"><span style="color: #0536cd; text-decoration: underline;">here, for n-spheres, 3-sphere in 4-dimensional Euclidean space</span></a>.</div>
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<b>Straight lines in a curved spacetime</b></div>
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It may strike the reader as curious that in all three geometries above parallel lines appear to remain parallel (perspective aside), rather than diverging or converging (bent or distorted inwards or outwards, convexly of concavely), as on the surface of a sphere or a saddle shape. The reason is not because the train track is rigid. The tracks could just as well be replace by light beams, and the observer will see the same parallel lines extending to the horizon (or visa versa), as pictured above. The reason is that we are no longer looking at a surface where photons are forced into curved trajectories as they travel a geodesic curved path on a non-Euclidean plane. Nor are rays deflected as they are when they graze the surface of a star, locally (gravitational lensing aside). In four-dimensions, as in two or three dimension, a photon trajectory will be the shortest distance between two points. The difference is that, where in reduced dimensions the trajectory was actually curved (not just geodesically but physically), now, The path of the photons is actually straight from our perspective. We see object along a line of sight. That line of sight is a straight line. Certainly, in a curved 4-dimensional spacetime light travels a geodesic, and certainly distances appear altered, as do the intervals of a clock of distance objects compared to local clocks, but parallel line are the equivalent of two light beams. The line of sight remains intact as straight lines with a difference that the line is either geodesically stretched or geodesically compacted (compared to the Euclidean counterpart). So it would appear that Euclid's parallel postulate holds in four dimension. Parallel transport is path-dependent. Parallel lines do not vary, distances do. </div>
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Note, the above is unrelated to the deflection of light grazing the sun, or gravitational lensing. Clearly in those situations the photon travels a curved path relative to our point of view. Here, we're talking about a situation where the goal is to determine the global curvature of the manifold by means of parallelism or angular diameter. Such could never be accomplished if every (or even one) local deviation from linearity, hump or bump, were taken as a general case. To test global curvature with the parallel postulate (or visa versa) one would have to choose objects not affected by the lensing of foreground objects, which brings us back to Figure T. </div>
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Notice, all parallel lines converge in straight lines toward infinity due to perspective, in the three cases. To assume that we would observe concavity of convexity of the parallel lines (in a positively or negatively curved 4-dimensional spacetime) would be a mistake. This doesn't mean Euclid's fifth postulate is true or false. It means there is no way of testing it by astronomical means. Likewise, there is no way of testing the geometry of the universe by examining the parallelism of light emanating from distant objects. And finally, it means that the test of angular diameter is suppositious at best, and illusory at worst, if the goal is to measure spacetime curvature, i.e., angular diameter is a test without any intrinsic real-world meaning.</div>
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The notion expressed here is consistent with the idea that photons travel globally in straight lines (in the Euclidean sense), but that the actual path itself is spatiotemporally non-Euclidean (a geodesic). Euclidean paths are replaced with non-Euclidean geodesic paths, where photons are<span style="font: normal normal normal 18px/normal 'Times New Roman';"> simply coasting on their geodesic paths</span>. Loss of energy occurs progressively in conformity with the geodesic, which depends on the magnitude of global curvature. But the notion that lines converge or diverge depending on geometry, leading to differences in angular diameter, needs to be abolished.</div>
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Let's visualize the situation from another angle:<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure U</b></span></div>
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Three Geometries from the Observer to the Visible Horizon: <b>A</b> is hyperbolic, <b>B</b> Euclidean and <b>C</b> spherical. This is an oblique angle of half-cross-section of the visible universe viewed from above the line-of-sight of the observer, located at O. This is a schematic representation of three geometries for a static universe, where the circular lines are spherical shells with increments of around 2 Gly each, centered around O. Here we have a 180° view of the sky.</div>
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The core point to make here is that 30° is the same for all observers in all three geometries. The total scope on a plane is 360° in all three geometries, no more and no less. Geometry does not change degrees. The same holds in four dimensions. In another way, an object that occupies 1° of the sky will occupy 1° of the sky regardless of the geometry. The only difference will be the apparent distance of objects in a curved spacetime, relative to the Euclidean distance. However, because a universe with hyperbolic geometry (<b>A)</b> appears larger (and deeper) that the Euclidean and spherical models, 30° appears larger (or wider) in model <b>A</b>. That is an artifact of the illustration, since we are above the plane. To the observer looking through the plane, 30° is 30° in the three cases. Only the distance to the horizon, and objects along the way change with the three geometries.</div>
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Redshift z occurs in illustrations <b>A</b> and <b>C</b> only, increasingly with distance. Objects themselves are unaffected by the distortion of spacetime (globally). </div>
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Note that the three models above are spherically symmetric, homogeneous and isotropic. The Euclidean space <b>B</b> mandates that the electromagnetic radiation passes through the surface in a uniform way, in accord with Gauss' law and the inverse square law. Models <b>A</b> and <b>C</b> do not conform to the inverse square law. If the total flux is known, the field itself can be deduced at every point.</div>
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Notice that the spherical model <b>C</b> does not wrap around on itself (as it does in two dimensions), so it can be considered equivalent to an infinite plane. Clearly there is no reason for concern, if we don't try to define 'size' as the radius <i>R</i> of the sphere. </div>
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If we measure the circumferences of circles (or spherical shells) of progressively larger diameters and divide the former by the latter, all three geometries <b>A</b>, <b>B</b> and <b>C</b> give the value π for small enough diameters (locally) but the ratio departs from π for larger diameters (globally) unless the universe is Euclidean (<b>B</b>). For a hyperbolic spacetime manifold (<b>A</b>) the ratio rises above π. For a spherically curved spacetime the ratio falls below π, since a great circle on a sphere has a circumference equal to twice its diameter.</div>
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The fact would remain though that 30° of the sky is still 30° from the point of view of any observer.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure ADPHU</b></span></div>
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Note that objects in a hyperbolic universe would intuitively appear to become larger with distance, such as in Figure ADPHU. This universe would produce redshift and time dilation. However, this schematic illustration does not take into account perspective: objects appear to become smaller with distance, as in standard one-point or two-point perspective. So the above scenario is not observed.</div>
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__________________</div>
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<b>Cosmology hinges on the interpretation of redshift z</b></div>
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The standard model is a linear or translational kinematic interpretation. It is a description of the apparent motion with space of an object along a line (a trajectory or path) that can be either straight (rectilinear) with a constant or uniform radial acceleration (the inverse of freely-falling) relative to distance from an observer, or curved (curvilinear) with an acceleration that changes in time; leading to a non-Euclidean topology (i.e., the deceleration parameter determines the fate of the universe, whether it's open, closed or flat). </div>
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The interpretation now is a nonlinear or translational static description (at rest with respect to a frame of reference) where an object is observed along a line (a trajectory or path) that cannot be straight (in Euclidean or Minkowski sense) with a constant velocity of light relative to an inertial-frame, but must be geodesically curved ('straight lines') with a distance that remains virtually unchanged in time (intrinsic motion aside); leading to the description of an infinite non-Euclidean spacetime manifold, i.e., where apparent spatial increments and relative time intervals together determine the geometric shape of the visible universe (whether it's hyperbolic or spherical) relative to the observer's reference frame.</div>
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The tricky part is to determine whether redshift is due to a curved spacetime phenomenon, or the expansion of space. If expansion is real then the curvature (to a large extent) is only spurious. If global curvature is real then expansion is entirely spurious. Sometimes (almost always in cosmology) there can be differing interpretations for the same observed phenomena. For example, the apparent superluminal velocities of objects at the horizon of an expanding universe can be interpreted as an effect generated by the curvature of spacetime as the photons propagate towards the observer. The latter does not imply that spacetime at those distance objects in infinitely curved, or that they are traveling faster than light. These affects are only apparent to the observer, relative to her rest-frame.</div>
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The observer located at one of those distant galaxies at the superfluous edge of the universe, typing a few key enthralled words on her computer sitting atop a her Lagrangian-like point peering out into the peaceful heavens on her pixelated screen would see the Milky Way as a tiny spec of real estate breaking the speed of light too. Either that or she could conclude that the Milky Way is immersed in a gravitational potential well so deep that the photons emanating from the luminous objects that make up the Galaxy barely escape in time before the gates of hell close for good.</div>
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Hyperbolic Curvature and the Stability of the Cosmos</div>
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<span style="font: normal normal normal 18px/normal Times;">We've seen above how global <i>stability</i> is maintained in a non-Euclidean Gaussian, or Riemannian manifold that is curved spherically. T</span>he metric components of the globally homogeneous field vanish locally<span style="font: normal normal normal 18px/normal Times;">. B</span>y definition, a Riemannian manifold is flat on a sufficiently small scale; a fact that corresponds to the equivalence principle for the spacetime manifold.<span style="font: normal normal normal 18px/normal Times;"> T</span>here exist necessarily coordinates x,y at any point on the manifold such that the geodesic paths through that point are straight lines. This is why objects do not embark on a collision course towards a <i>Big Crunch</i>.</div>
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The magnitude of Gaussian curvature manifests itself with great distance from any galaxy, or from the origin (just as the curvature of the earth departs from flatness with increasing distance). There is no acceleration due to the global field generated on massive objects. And so the stability of the universe is maintained. Expansion and collapse are avoided rather naturally with geometrical arguments. </div>
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Now let's consider the implications of global stability with respect to 'negative' curvature (hyperbolicity) and simultaneously explore issues regarding the origin of the observed cosmological redshift z in such a manifold. It is shown that the same dynamic mechanism (<span style="font: normal normal normal 18px/normal 'Times New Roman';">the metrical relations of spacetime</span>) involved in the stability and redshift in a spherical manifold is operational in a hyperbolic manifold.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure HUDF</b></span></div>
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Hyperbolic Manifold on HUDF. A hyperbolic slice (or cross-section) of the visible universe with an overlay on Hubble Ultra-Deep Field (for convenience only). This is a static universe representation, where redshift z is caused by the passage of electromagnetic radiation through a curved spacetime<span style="font: normal normal normal 18px/normal Times;">.</span></div>
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A space of constant negative curvature has a geometry called hyperbolic. This geometry is of great importance because it appears to be the geometry that best describes the shape of the universe on a cosmological scale.</div>
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Recall that compared with Friedmann models, the distant SNe Ia are too faint, even for a freely coasting “empty” universe. Light curves appear to be "stretched." Their spectra exhibit slower temporal evolution, by a factor of 1 + z, than nearby SNe Ia. On average, the luminosity distance of high-z SNe Ia are 10-15% farther than expected in a low mass-density universe (without a cosmological constant). (<a href="http://www.blogger.com/%3C/span%3E%3C/span%3E%3Cspan%20class="><span style="color: #5588b1;">http://arxiv.org/pdf/astro-ph/9905049v1">Source</span></a>)</div>
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If the universe was indeed static with negative Gaussian curvature, one could hardly ask for a better empirical demonstration of that fact. A deviation from linearity of the type demonstrated by the SNe Ia data is exactly what would be expected.</div>
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Lobachevsky recognized the universal characteristics of his new geometry, even though the observational evidence would not be available for another 172 years. In his infamous <i>On the Principles of Geometry</i> (1826) Lobachevsky noted that if one were to measure the inner angles of cosmic triangles of great dimension, it would be possible to determine the deviation from the usual 180° experimentally. In his later work <i>New Principles of Geometry With a Complete Theory of Parallels</i>, he put forth the idea that his geometry might apply to the “intimate sphere of molecular attractions.” Clearly, Lobachevskian space (often referred to as hyperbolic space) became the Riemannian space of constant negative curvature that would subsequently find application within the framework of Einstein’s principle of general relativity—more than a half century after its discovery. Astoundingly, today there is still great misunderstanding about the deviation from linearity of spacetime. We shall see how this will play a key role in our comprehension of the evolution of the universe. </div>
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History has it that Gauss too simultaneously, and independently discovered non-Euclidean geometry, but for fear that his reputation would suffer if he were to articulate that non-Euclidean geometry’s were possible withheld his early discoveries from early publication. Bolyai too worked simultaneously along similar lines. Let's examine the situation from the Gaussian point of view.</div>
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The Gaussian curvature of a surface is the product of the principal curvature. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way it is embedded in space. This result is the content of Gauss's Theorema egregium. </div>
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In another way, and in a broader sense, Gaussian curvature of a surface is the product of the principal curvatures of the given manifold. It is an intrinsic measure of curvature because its value depends only on how distances are measured in spacetime, from any point O (by any observer) in the manifold. This result is consistent with Gauss's Theorema egregium as generalized by Riemann.</div>
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A surface of Gaussian curvature is either negative (a hyperboloid), zero (a cylinder), or positive (a sphere). The cylinder is of no particular interest here. The flat surface—a special class of minimal surface on which Gaussian curvature vanishes everywhere—will be of no interest here either, since there would be no redshift or time dilation in a four-dimensional static Minkowskian universe. </div>
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A surface on which the Gaussian curvature <i>K</i> is everywhere positive is called synclastic, while a surface on which Gaussian curvature <i>K</i> is everywhere negative is called <a href="http://mathworld.wolfram.com/Anticlastic.html"><span style="color: #0536cd; text-decoration: underline;">anticlastic</span></a> and is saddle-shaped (in reduced dimensions). </div>
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Again, as described above in the case of positive curvature, this is a generic global field, the most general class of the curvature tensor in Einstein's general theory of relativity. This globally smooth field is an irreducible (not a local product of space) <a href="http://en.wikipedia.org/wiki/Geodesically_complete"><span style="color: #0536cd; text-decoration: underline;">geodesically complete</span></a> n-dimensional homogeneous Riemannian manifold (or pseudo-Riemannian four-dimensional manifold) that contains an inversion symmetry about every point, i.e., a globally Riemannian symmetric space of constant negative sectional curvature where <a href="http://en.wikipedia.org/wiki/Riemannian_symmetric_space"><span style="color: #3b2e7b; text-decoration: underline;">geodesic symmetries</span></a> are defined on the entire manifold.</div>
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This is a class of Riemannian (or pseudo-Riemannian) manifolds with constant negative sectional curvature (-1) over the entire surface, corresponding to hyperbolic space. In four dimensions, an observer measuring distances (relative to her rest-frame) to astronomical objects in such a space of hyperbolic geometry will find that the deviation from linearity increases with distance. In reduced dimension, this is equivalent to deviation in the sum of angels of a triangle from 180°, in accord with Toponogov's theorem which characterizes sectional curvature in terms of how "thin" (in this case) geodesic triangles appear when compared to their Euclidean counterparts. Locally, sufficiently small triangles will appear Euclidean to a close approximation, while with increasing distance non-linearity increases. The larger the triangle under consideration, the "thinner" it becomes. It follows that if an observer measures curvature via triangulation, where one of the points lies near, or at, the visual horizon, it will be found that curvature, or deviation from linearity, attains a maximum value (the magnitude of geodesic distortion is greatest), and redshift z will reach a maxima.</div>
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There are many ways to visualize this scenario for a <a href="http://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane"><span style="color: #0536cd; text-decoration: underline;">hyperbolic plane, or a hyperbolic space</span></a>. Here for practical purposes we choose a saddle-shape. Note though, where in the spherical case above, the observer was situated at any point on the surface, here the observer is situated at the center of the surface. However just as in the spherical case, when we project to four dimensions, all points are equal, so all observers are at the center of the saddle-shape, i.e., as distances are measure radially in all directions. Since all points are equal on the hyperbolic surface the observer (any observer) is entitled (an in fact has no choice but to) consider herself centered on the manifold. This, in four dimensions, is due to the finite velocity of light and the geodesic (the shortest distance between two points on a hyperbolic surface) travel path of the EMR wave-packet emanating from a distant source. On a negatively curved surfaces the inner angle sums of a cosmic triangle are less than their Euclidean counterpart and every point behaves essentially like a saddle point. When the hyperbolic triangle is flattened the surface area is greater than a Euclidean triangle. Thus a hyperbolically curved universe appears larger that a universe with zero curvature or one with spherical geometry. </div>
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There seems to be a common problem associated with extending local non-Euclidean geometry to global topology steming from an inappropriate distinction that separates timelike and spacelike components of a vector, which is not preserved under a Lorentz transformation (the Lorentz boost, given by a symmetric matrix), since the concept singles out a distinction between time and space, separating the two. While this seeming trivial concept may be acceptable under special relativistic considerations, it falls short for the description of the global gravitational field, where general relativity is the rule, rather than the exception.</div>
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The global description of physical spacetime must be 4-dimensional. But as we've noted above, the equivalence principle permits the description of the spacetime manifold as locally flat, just as the description of the local region on the surface of the earth can be considered locally flat. When studying global geometry observers have no other choice but to consider their region locally flat. Each observer, according to general relativity, sees the universe in relation to her own world lines. This is not an artifact of the chosen coordinate system: since any other conclusion would violate observations. And observations will show the trajectory of incoming photons are geodesic straight lines. And since the observer knows, a priori, her coordinate system is only an artifice used to describe nature the way she sees it (i.e., there is nothing special about her rest-frame, in accord with general covariance, also known as diffeomorphism), she realizes her interpretation of observations should apply to all frames. She notices that clocks are nowhere synchronizes with her coordinate <i>t</i>. She knows that there is no universal time (or cosmic time) that can possibly match every clock, regardless of the clock's state of motion. It is not possible to synchronize clocks in a gravitational field. But that's not all, she realizes too that there is no meter-stick anywhere in the universe that matches hers. And when she combines the two in her formulation of spacetime curvature she realizes that there is a direct relation between the spatial distortion and the time variations observed from her frame. Spatial distances are directly proportional to the time dilation factor she observes when she studies distant objects. The further she looks, the more the divergence. The impossibility of synchronization is a physical manifestation proportional to the area (e.g., in spherical shells centered on the observer) of the region synchronization is attempted. And when the area is the entire visible universe (i.e., the area enclosed by light rays) the divergence from linearity is progressive with distance and largest at the horizon (whereas locally divergence approaches zero). That is how she formulates an accurate, general relativistic, coordinate-independent, manifestly intrinsic, geometrical description of the 4-dimensional global topology of spacetime.</div>
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Certainly, hitherto, an accurate description of global topology has resisted precise mathematical formulation. But the fact that we haven't yet placed such a conceptual principle behind a bullet-proof glass in a museum with a mathematical frame around it doesn't make it trivially unimportant.</div>
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The surprise is that such a topology can be described in three equivalent ways: as a gravitational potential, a set of gravitational field lines, and as a gravitational field (a geometric property of spacetime).</div>
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Now lets illustrate a reduced dimension a hyperbolic manifold with negative curvature—with a spatial geometry equivalent to the Figure <b>A</b> diagrams above—then analyze how observations could account for redshift in four dimensional manifold with a negative signature for curvature. And finally, we'll layout the physical mechanism involved in the stability factor.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure H</b></span></div>
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Figure H represents a reduced dimension hyperbolic manifold (here shown as a hyperbolic paraboloid). This is a static, nonexpanding manifold that extends to infinity. The hyperbolic triangle is given by three distinct points joined by geodesics. The sum of the interior angles of a triangle drawn on the surface equals less than 180°. As distances considered become smaller (locally), the hyperbolic plane behaves more and more like Euclidean geometry. Nonlinearity increases with distance, i.e., straight lines become geodesics.</div>
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This is a general relativistic world model in 2-dimensions, not unlike the geometric architecture of the 1917 de Sitter universe. </div>
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An intrinsic property of a Gaussian negatively curved space (also called Gauss-Bolyai-Lobachevsky Space) is independent of the coordinate system used to describe it. (<a href="http://mathworld.wolfram.com/GaussianCurvature.html"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>). In other words, form the point of view of any observer the manifold will appear the same (as if centrally located). This is a space with constant negative Gaussian curvature. Light on such a surface travels straight (geodesic) lines. Unlike Euclidean triangles whose angles add up to 180 degrees or π radians, and unlike spherical triangle whose angles add up to more than 180°, the sum of the angles of a hyperbolic triangle are always less than 180°. The difference, as mentioned above, can be referred to as the defect. The area of a hyperbolic triangle is given by its defect multiplied by R^2. The result is that all hyperbolic triangles have an area less than πR^2. And the circumference of a circle in hyperbolic geometry is greater than π times the diameter. (<a href="http://en.wikipedia.org/wiki/Hyperbolic_geometry"><span style="color: #5588b1;">Source</span></a>)</div>
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The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic triangle equals the deviation of the sum of its angles from π. The sum of the angles of a triangle on a surface of positive curvature will exceed π, while the sum of the angles of a triangle on a surface of negative curvature will be less than π. On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely π. (<a href="http://en.wikipedia.org/wiki/Gaussian_curvature#Total_curvature"><span style="color: #0536cd; text-decoration: underline;">Source</span></a>)<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure 7Ab</b></span></div>
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The cross-section of a homogeneous and isotropic four-dimensional spherically symmetric globally curved geometrically hyperbolic pseudo-Lobachevskian spatiotemporal manifold (a cross section of the visible universe), that mimics observations currently understood as an accelerated expansion in a virtually flat (or quasi-Euclidean) space.</div>
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Events at great distances appear to take longer than in the rest-frame of the observer. Time dilation reaches a maximum at the horizon.</div>
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In looking at the situation from this angle, from the point of origin (the location of any observer), we have a schematic diagram similar to Figure 1C above, albeit, now with negative curvature. </div>
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Distances in a hyperbolic universe can be measured in terms of a unit of length (<a href="http://en.wikipedia.org/wiki/Hyperbolic_geometry"><span style="color: #0536cd; text-decoration: underline;">see here</span></a>). This is analogous to the radius of a sphere in spherical geometry.</div>
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By measuring light curves, the apparent brightness, spectra and peak luminosity of high-z Type Ia supernovae (SNe Ia) it can be inferred that nonlinearity becomes greater with increasing distance, i.e., curvature increases exponentially with distance. The data, which shows broadening of light curves (cosmological time dilation), i.e., a deviation from the expected 1 + z broadening of light curve widths, does not obligatorily imply that the universe is expanding, and doing so at an accelerated rate, in the sense assumed by lambda-CDM. </div>
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General relativity provides a rigorous justification for the interpretation of the globally curved spacetime on all distance scales, and confirms the self-consistency of the conjecture that all electromagnetic radiation propagates a geodesic.</div>
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We found above that there is no special location on the surface of a sphere, and that the same holds when extended to four dimensions. In fact the same conclusion can be drawn for a hyperbolic spacetime. At first sight there appears to be a special location: the rest-frame of an observer positioned at the center of a saddle in reduced dimensions (similar to an inner Lagrange point, L1). </div>
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At second glance, though, every observer would find herself sitting firmly on a saddle (at an L1-like point). Every point in hyperbolic space is a saddle point. All locations are locally Euclidean to a good approximation. This hyperbolic space is the constant negative sectional-curvature (-1) analogue of the Riemann spherical geometry (+1). Globally, this spacetime, too, is maximally symmetric.</div>
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Certainly, if we sprinkle test particles in the vicinity of an L1 point (say, between the earth and sun) they will move in accord with the lines of force, some towards L1 and some away. Sprinkle some beads on a horses saddle, or salt on a Pringles potato chip and the same thing will happen. But the hyperbolic surface we are discussing above is different. First of all, it's not a flattened out surface located on a plane between two massive bodies in rotation, as an inner Lagrangian point. Nor is it a surface where lines of force are pointing towards or away from the origin depending on orientation. This is an intrinsically curved 4-dimensional spacetime within which light travels a geodesic (a "line"), and test particles (galaxy clusters and superclusters) do not partake geodesically (they are not accelerated in any direction), exactly as in the spherical geometric case described extensively above. This is a uniform, homogeneous field where the constant negative Gaussian curvature describes a Riemannian (or pseudo-Riemannian) spacetime manifold. Curvature measurable by all observers. Contrary to popular belief, objects located in such a manifold would not tend to scatter. Only local inhomogeneities (combined gravity fields of massive bodies) induce motion, acceleration. The global field induces no acceleration. This is exactly what we would expect of a cosmology with general relativity at its core. </div>
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However, relativistic partial differential field equations provide only local geometric solutions, they are not sufficient to tell us how to piece these neighbourhoods of spacetime together to form a global topology of the manifold. In other words, there are many possible topologies that can be drawn from general relativity. The hyperbolic model is one possibility that seems to be consistent with astronomical observations.</div>
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It's interesting to note that the Lorentz model (or hyperboloid model) employs a 2-dimensional hyperboloid of revolution (of two sheets, but only using one) embedded in 3-dimensional Minkowski space. This model has direct application to special relativity, since Minkowski 3-space is a model for spacetime, where one spatial dimension is suppressed. The hyperboloid can be interpreted to represent the events that various moving observers radiating outward in a spatial plane (from a single point) will reach in a fixed proper time. The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers. (<a href="http://en.wikipedia.org/wiki/Hyperbolic_geometry"><span style="color: #5588b1;">Source</span></a>).</div>
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If one is to judge by redshift alone, in accord with the equivalence principle, the interpretation of radial motion is a valid one. But one should not be erroneously lead into thinking that radial motion is the only viable solution for redshift z. Nor should one be led to believe that a globally hyperbolic 4-dimensional manifold is intrinsically unstable.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure SA</b></span></div>
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Figure SA represents a globally homogeneous 2-dimensional static universe characterized by constant negative Gaussian curvature, with no cosmological constant. This universe is stationary and infinite spatiotemporally. Redshift z is produced by the propagation of light through a hyperbolic (<span style="font: normal normal normal 15px/normal 'Times New Roman';">or pseudospherical) </span>curved spacetime continuum.</div>
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The large-scale structures and their gravitational fields are shown as areas of 'concentric' circles (hierarchical structures consisting of inhomogeneities such as solar systems, galaxies, clusters, superclusters, and so on). In this view we are looking perpendicularly from a saddle point at the surface of a hyperbolic plane. O<span style="font: normal normal normal 18px/normal 'Times New Roman';">ver sufficiently small spatiotemporal regions, surrounding any given point in spacetime, the coordinate system has a simple Minkowskian form. That is, the metrical relations on the spacetime manifold (over a sufficiently small region) approach arbitrarily close to flatness to the first order in the coordinate differentials.</span></div>
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As is the case for the spherical model above (a "closed" spherical universe with infinite radius R), this universe is spatiotemporally infinite (where K = -1). The Gaussian curvature results from the total mass-energy density of the universe. The sum of the interior angles of a cosmic triangles add up to less than the Euclidean value of 180°. Light propagates along a geodesic paths. Geodesic paths are described by intersections with lines (or planes) through the origin. Though light travels straight lines, the spatial and temporal increments along the line of sight become larger with distance from the observer: redshift z increases with distance as a function of time dilation.</div>
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There is no repulsion provided by the cosmological constant, since there is no cosmological constant. There is no repulsion provided by the slope of gradient of the field, since the slope or gradient of the field has the same value or magnitude everywhere (just as the slope or gradient on the surface of the earth has the same value of curvature everywhere, excluding local humps and bumps) effectively equal to zero locally. There is no "uphill" or "downhill." So a vacuum energy is not required.</div>
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The flow of time throughout the universe is dependent on the observer's frame of reference compared with the influx of photons (and spectral characteristics) emanating from distant sources (local motion of these objects aside), which shows that spatial and temporal dimensions are intertwined. If spatial distortion occurs, temporal distortion occurs. One cannot occur without the other (it would make no sense to distort one quantity without the other, since the invariant velocity c represents a ratio of a distance to a time). So cosmological time dilation and cosmological redshift z are inextricably attached.</div>
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A clock runs faster locally according to any observer's rest-frame relative to any clocks far removed in a globally curved spacetime. Clocks appear to run slower with distance. Light from very remote objects takes longer to reach Earth than would be the case in a flat universe—as if time and space (and the light propagating through it) were continually and increasingly ‘stretched’ with larger distances. Each observer considers herself to be at rest in a quasi-Euclidean spacetime, and (with little knowledge of general relativity) thinks that other observers are embedded in a gravitational field (or radially moving). Each observer says that the other's clock is slower.</div>
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No time or place has special properties that make it distinguishable from other times or places. No direction in space is distinguishable from another. There are no preferred inertial frames. Observations produce relativistic effects of redshift z and time dilation. Distances measurements reveal a non-Euclidean manifold. </div>
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<b>Cosmological Redshift z</b></div>
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In a hyperbolic universe the energy loss associated with a photon as it passes through spacetime is continuous, but there is no reason why it should be linear. The further the photon has to travel, the greater the energy loss, and the higher the redshift. So redshift increases with distance. The redshift increases nonlinearly with an additional (and proportional) time dilation factor of (1 + z). The deviation from linearity will manifest itself hyperbolically (as viewed by an observer) because spatiotemporal increments appear kinematically to increase continually with distance (i.e., distances will appear greater as one ponders objects further removed, and the time increments will be measured to slow down with distance, as compared to local clocks, as compared to what would be expected in a Euclidean manifold). So redshift is a purely relativistic effect directly related to the amount of spacetime curvature contained within the path of a photon. We would expect redshift to be zero (or small) locally and infinite at the horizon.</div>
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According to the field equations of Einstein's general theory of relativity, the structure of spacetime is affected by the presence of both matter and energy. On small scales (say, compatible with that of the Local Group, or within distances of a few Mpc from the Local Group) spacetime appears quasi-Euclidean—as does the surface of the Earth if one looks at a small section. On large scales however, space is 'curved' by the gravitational effect of matter and energy. Because general relativity postulates that matter and energy are equivalent, this apparent curvature effect is also produced by, in addition to matter, the presence of energy (e.g, light and other electromagnetic radiation, and possibly gravity itself). The amount of curvature (distortion or bending) of the manifold depends on the total density of matter/energy present. (Not to mention, the actual curvature of the manifold without mass or energy, an 'empty' universe, which according to de Sitter is curved hyperbolically). That would add to the total effect, if indeed an empty universe is not flat.</div>
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Taken at face value, the presence of the time dilation factor of (1 + z) in the SNe Ia data is simply related to a reduction in the flux density by more than the inverse square law. So the effective distance, from the view-point of any observer (at this time) does not behave like a Euclidean distance with increasing redshift. The SNe Ia results are consistent with hyperbolicity (in a stationary universe).</div>
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In an static regime, the redshift and distance relation of every supernova records <b>not</b> the past change in the scale factor over the inferred time interval, or the expansion rate, but the degree, quantity of value of curvature (the departure from linearity). </div>
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When objects such as galaxies are observed at cosmological distances, and in the look-back time, events and phenomena appear to take longer in our frame of reference than in that of the source (a phenomenon observed for relativistic muons that propagate through our atmosphere). In another way, clocks would appear to slow down with increasing distance (just as the de Sitter effect in a static universe). This is the cosmological manifestation of a phenomenon known as time dilation. See equation 2.31. The result of this equation gives us the expression for redshift z:</div>
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This is one of the most important relations in modern cosmology and displays the real meaning of redshift. Redshift is simply a measure of the scale factor of the Universe when the source emitted its radiation. [...] Note, however, that we obtain no information about when the light was emitted. If we did, we could measure directly from observation the function R(t). [...] Thus, redshift does not really have anything to do with velocities at all in cosmology. The redshift is a beautiful dimensionless number which, as (1 + z)^-1, tells us the relative distance between galaxies when the light was emitted compared with that distance now. <a href="http://books.google.es/books?id=k5lAwbslaG8C&printsec=frontcover&dq=the+deep+universe+saas+fee&source=bl&ots=HOCrD_XUf5&sig=LJ-3ou5cNQE3k3RwAgHHp5-Lyec&hl=en&ei=7m4TTI7dC8iy4QaUvdD1Cw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCEQ6AEwAg#v=onepage&q&f=false"><span style="color: black;">Source: The Deep Universe, M.S. Longair, page 369</span></a>. </div>
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The point is, as you may have guessed, there is a direct empirical correlation between redshift z as interpreted by <b>Longair</b> and the interpretation of redshift z as a curved spacetime phenomenon (with both minor and major differences in physical outcome).</div>
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Both interpretations of redshift z have an associated time dilation factor, and both have nothing to do with velocity at all. All redshift z gives us<span style="font: normal normal normal 18px/normal Symbol;">⎯</span>when interpreted as a general relativistic phenomenon<span style="font: normal normal normal 18px/normal Symbol;">⎯</span>is a clue as to the relative spatial and temporal separation between us (from our rest-frame as we peer into the look-back time) and distant galaxies, when and where the electromagnetic radiation was emitted relative to the observers location and clock, i.e., redshift is interpretated as measure of curvature, since spatial increments and temporal intervals deviate from linearity with distance in the look-back time. So a change in the scale factor is not required.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure PLEM</b></span></div>
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Figure PLEM is a reduced dimension schematic representation of the propagation of light from a distant source to the observer in a non-expanding flat, Euclidean, or Minkowski space-time regime. The wavelength suffers no distortion during travel-time. Redshift does not occur, and time dilation is nonexistent. The speed of light is constant from all inertial reference frames. Yet, intensity is diminished inversely and proportionally to the square of the distance.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure PLSM</b></span></div>
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Figure PLSM is a reduced dimension schematic representation of the propagation of light from a distant source to the observer in an intrinsically spherical spacetime regime. The wavelength tends to lengthen, toward the red end of the spectrum due to the distortion along the entire trajectory. The associated time element is dilated, too, due to the geodesic path. Intensity of light is diminished by a factor greater than the inverse square law. </div>
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This type of geometry could be interpreted observationally as identical to the interpretation that the universe has been accelerating recently and decelerating in the distant past, since high-z objects would appear closer than expected in a flat universe (as if the universe were expanding more slowly near the visible horizon than locally). That is, spherical shells of equal distance centered on an observer appear closer together (more compact) with increasing distance. Redshift z is a function of time dilation. In the spherical case above, as in the hyperbolic case below, source objects located near the <span style="font: normal normal normal 18px/normal 'Times New Roman';">horizon will be extremely redshifted, i.e., will have a highly dilated proper time relative to the observer’s coordinates. At the horizon the amount of time dilation is infinite.</span></div>
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Distance and redshift data obtained by SNe Ia can be interpreted as a physical signature of constant non-linear positive Gaussian curvature (e.g., general relativistic spacetime curvature) and geometrically represents the same manifold Einstein invoked as early as 1916-17.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure PLHM</b></span></div>
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Figure PLHM is a reduced dimension schematic representation of the propagation of light from a distant source to the observer in a hyperbolic spacetime regime. The wavelength tends to lengthen toward the red end of the electromagnetic spectrum due to the distortion along the entire trajectory. There is an associated time dilation factor, too, due to the unique geodesic path traveled between the observer and the source. The intensity of light is diminished by a factor greater than the inverse square law. This 4-dimensional spacetime (as the spherical case above) possesses an intrinsic hyperbolic metric.</div>
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On observational fronts hyperbolicity is revealed, not just by redshift and time dilation together, but by the excessive faintness of distant type Ia supernovae, the brightness of which can be used to calibrate their pseudo-Riemannian distances. The degree of redshift provides a direct measure of the curvature of the universe through which light propagates. The distances of supernovae Ia are measured by comparing their apparent and intrinsic brightness, revealing the time over which those signals have traveled at the speed of light. This <i>deviation from linearity</i> gives the degree of curvature. Therefor, every supernova’s measured redshift and distance records the spatial curvature over the inferred time interval, or, taken together, the global general relativistic spacetime curvature of the universe.</div>
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Just for fun, let's see what happens when a hyperbolic grid is superimposed on the infamous Sloan Digital Sky Survey (SSDS) image.<br />
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure SDSSHT</b></span></div>
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See <a href="http://www.blogger.com/www.sdss.org"><span style="color: #0536cd; text-decoration: underline;">SDSS</span></a> for the original, unadulterated, image.</div>
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________________</div>
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A global Pseudo-Riemannian manifold with local Loentzian submanifolds</div>
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Recall, it is mathematically in the language of differential geometry that general relativity describes the universe of events. </div>
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A Lorentzian manifold is a special case of a pseudo-Riemannian manifold (not of a Riemannian manifold). And a pseudo-Riemannian manifold is a generalization of a Riemannian manifold.</div>
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The distinction is an important one, for several reasons, one of which is that a Lorentzian frame, describes space with or without any gravitational fields (e.g., freely-falling). World-lines of a particles in the absence of gravity are initially parallel and will continue along parallel world-lines. In the presence of gravitational fields, world-lines of free-falling particles are initially parallel and will in general, approach, diverge, or even intersect each other. This lack of parallelism cannot be described as arising from the curvature of the world-lines, since the world-lines of free-falling particles are defined as "straight lines." Instead, the effect can be attributed to the curvature of spacetime itself. </div>
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Furthermore, there is no Lorentz frame for the entire universe. So there is no natural expectation of being able to define a global principle of conservation of momentum. That is why conservation laws are difficult (or impossible) to formulate within the framework of general relativity (i.e, globally). There is no single Lorentz frame that could cover the entire universe. This is one of the reasons why extrapolation of the equivalence principle (EP) to cosmology are so tenuous. In another way, the EP implies that it's always possible to define a local Lorentz frame in a particular neighborhood of spacetime, but it's impossible to do so globally.</div>
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General relativity can indeed be (and is) modeled with a Lorentzian manifold. In many circumstances Lorentzian manifolds are appropriate (perhaps for all submanifolds). Those circumstances revolve around local events. There is little doubt that the universality of local Lorentz covariance, together with the equivalence principle, describes local gravitational phenomena quite well.</div>
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If it can be established that the mass-energy density of the universe affects the topology of a global spacetime manifold, it will follow that the global manifold must have nonzero metrical curvature. Of course, the existence of nonzero metrical curvature at local points of the manifold does not imply nonzero global spacetime curvature, nor does imply a global topology. Lorentzian manifolds, again, do not give you a global topology. </div>
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It's interesting to note, too, that global topology of a homogeneous field of constant curvature does not tell you anything about the local spacetime curvature induced by local inhomogeneities (stars, galaxies, etc.). It can only tell you about its own intrinsic properties locally. For example. Let's say the global topology was spherical (K = 1) like the earth. That topology tells you nothing about how many mountains and valleys there are locally, or anything about the elevations of the mountains or the depths of valleys. All the topology tells you is that spacetime tends toward flatness locally. (I noticed that when I went to the beach a couple days ago and looked at the horizon of the Meditteranean). Likewise, judging from a local region (the observers frame of reference), the mountains and valleys tell you nothing about the global topology.</div>
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So it is very possible, that we have a Lorentzian manifold everywhere-locally (that tells you how objects move in spacetime and how spacetime is curved) and a pseudo-Riemannian topology globally (that describes a Gaussian manifold of constant curvature). This reasoning would apply to either positively or negatively curved topology, but not to a flat pseudo-Riemannian manifold simply because there would be no cosmological redshift is a flat universe, and so would not be consistent with observations.</div>
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And because a Lorentzian manifold tells you nothing about the global topology, it can't determine whether or not spacetime is flat or not on large-scales. That is one reason why a Lorentzian manifold permits a flat Minkowskian universe (amongst other shapes). And indeed, it is customary to treat the general relativistic manifold as an ordinary topological space with the same topology as a 4-dimensional Euclidean spacetime. </div>
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That is one reason why a Lorentzian manifold must be considered a special case of a pseudo-Riemannian manifold. A metric with a Lorentzian signature gives only local attributes of the manifold. It does not tell us the overall global topology, as would a pseudo-Riemannian metric. However, the two together are not incompatible. They can cohabitate.</div>
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A global pseudo-Riemannian manifold of constant Gaussian curvature does not tell objects how to move locally (since it is locally similar to a Euclidean space, no motion is induced locally due the Gaussian curvature). For that, a Lorentzian manifold is required locally. That is one reason why it is so difficult to extend general relativity to cosmology. We should not be misled by believing that a local physical manifold corresponds to the global topology.</div>
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It is entirely possible (or inevitable) that the local manifold has a different topology than the global physical manifold. With this in mind, it is worthwhile to consider very carefully whether a physically meaningful local spacetime topology is necessarily the same as the topology of the global 4-dimensional systems of coordinates. Note, a submanifold of a global pseudo-Riemannian manifold is not obligatorily a pseudo-Riemannian manifold with the same metric (nor does it even need to be a pseudo-Riemannian manifold at all). The submanifold(s) may very well posses a Lorentzian signature yet be 'contained' inside a more general global manifold (just as the mountains and valleys on earth are 'contained' on a spherical globe).</div>
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There certainly are no a priori requirements that a particular global structure can be uniquely determined by a given set of local experiences. If we restrict ourselves to a class of naively realistic local models consistent with the observable predictions of general relativity, there remains an ambiguity in the conceptual framework with regards to the global topology. The situation is complex due to the fact that the field equations of general relativity permit a wide range of global solutions. Some of these solution are unphysical (depending on initial condition, boundary condition, etc.). And so restrictions need to be imposed. The field equations, in this sense, do not represent a complete theory, since these restrictions cannot be inferred from the field equations. Incompleteness is a feature of all physical laws expressed as sets of differential equations, since a wide range of possible formal solutions can generally be extrapolated from such equations. This, by no means is detrimental to relativity. It just requires that at least one external principle (or constraint) be added to yield definitive results. (See for example, this on the topic).</div>
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At present, general relativity does not yield unique predictions about the topological shape of the global manifold. Rather, (once the unphysical solutions are weeded out) it imposes particular conditions on the allowable shapes. The simplest ('well-behaved') global solutions consistent with both general relativity and empirical evidence appear to be that of a pseudo-Riemannian manifold of constant positive or negative Gaussian curvature (yes, with Lorentzian submanifolds). Admittedly, I leave open the possibility that the sign of curvature K can be either or (1 or -1) so as to avoid committing to specific distant correlations, pending a complete model, and empirical verification. But only one of these two possibilities should eventually emerge as a viable topology consistent with physical laws. Obviously, the interpretation of a field theory such as general relativity with a globally flat background spacetime manifold would no longer hold.</div>
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For the above reasons, any thought experiment (in my opinion) that deals only with local phenomenon (with some unphysical extrapolations to the global) is virtually irrelevant for the topic at hand. My contention with this remark is that the global topology, i.e., the global topology alone, can shed light on a possible mechanism for the stability of the cosmos, and for the cause of redshift z. I do not exclude the possibility that the universe is expanding according to Lambda-CDM. I simply point out that there is a viable alternative, totally in line with GR, that needs to be explored further.</div>
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The following illustration is a representation of such a continuum. This manifold has globally negative (hyperbolic) curvature. The same concept could have been presented with positive (spherical) curvature. The local effects are exaggerated for emphasis.<br />
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<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3xOMURel2zMTqEcsVqkxDMMU-XWT8ZAksTNK79N17RlCkfvxv6ESvkN5Eq3KwM6Rsefpy4KG2h98-ZsfZcaGkYEea22PQPCPPV47gEzENhgkpCS7s-4d3dM2BzWU5CtRsonk6lJqh74k/s1600/Pseudo-Riemannian+Manifold+with+Lorentzian+Submanifolds+15cm150dpi.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></div>
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<span class="Apple-style-span" style="color: #222222; font-family: "times"; font-size: 18px;"><b>Figure PRM-LSM </b></span></div>
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Figure PRM-LSM is a schematic diagram representing the global topology, and the local geometric structures, of a general relativistic homogeneous spacetime continuum (a cross-section equatorial slice through the visible universe). This topology is a global four-dimensional maximally symmetric simply connected non-reductive homogeneous and isotropic pseudo-Riemannian manifold of constant negative Gaussian curvature, with everywhere-local Lorentzian submanifolds. (HUDF background)</div>
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The observer is centrally located at the origin of the polar coordinate system of a non-expanding stationary universe. The outer edge of the manifold represents the edge of the visible universe. This is an oblique angle, so the manifold is oval looking as opposed to circular (if it were viewed perpendicular to the polar grid). The circles represent, as usual, spherical shells centered on the observer. Each spherical shell represents an additional 2 Gly from the observer, in the look-back time. So the outer circle is about 20 Gly from the origin. All observer would find themselves located at the origin. This model universe is thus homogeneous and isotropic globally, and inhomogeneous locally, consistent with observations.</div>
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The point of this illustration is to show that the global spacetime continuum is geometrically curved and equipped with a pseudo-Riemannian metric. In this case the global topology is negatively (hyperbolically) curved. The totality of mass and energy (including gravity itself) contained in the universe is the source of the global Gaussian curvature.</div>
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Locally, we have Lorentzian submanifolds that describe the motion of objects and the interactions of such via the curvature of spacetime (gravity) in the vicinity of massive bodies. Local inhomogeneities in the form of mass and energy are the source of spacetime curvature.</div>
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This universe is static, yet the objects in it are dynamically evolving. The cause of cosmological redshift z is attributed to the global curvature. Spectral lines are shifted towards the red end of the spectrum due to the geodesic paths photons must follow as they propagate towards the observer. Global stability is maintained because the pseudo-Riemannian manifold is homogeneous, and thus imparts no net acceleration in any particular direction on objects located on the manifold. In Newtonian term, the gravitational potential is virtually the same at all points on the manifold. In terms of spacetime curvature, all massive objects are at rest relative to the global Gaussian curvature, as all local areas of the global field are, in every practical sense, flat.</div>
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<b>Discussion</b></div>
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I've not yet explored the significance of the CMB observations within the framework presented here. My hunch is that there is an analogue between the interpretation offered by modern cosmology to be draw in the context here, just as there was for redshift z. In other words, where in the former the CMB is interpreted as a remnant of a hot/dense phase, here the CMB would be interpreted as a remnant of stellar activity. This would of course not imply that modern cosmology is dead (shot-down by anti-big bang militant squads ‘Mozambique style’—one bullet to the abstract, one to the main text and one through the conclusion, in a regicidal three-shot power-play), it would just imply that there is a viable alternative that has yet to be fully explored (and understood). As I've written from the outset, these two competing interpretations are both consistent with observations, and both consistent with GR (albeit, one more so that the other). To rule out one hypothesis or the other (static or expanding models) on the basis of observations alone would seem to be futile task. It would seem, then, one way to rule out the solution that is least likely the most accurate representation of what is actually occurring in the cosmos should be based (in addition to empirical verification of distance vs proper distance) on theoretical grounds: based on parsimony, induction, coherentism, and objectivity.</div>
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Recall that parsimony was an important heuristic in the formulation of Einstein's special relativity theory. Though parsimony does not necessarily help toward a rational decision between competing explanations of the same empirical facts, it is rather remarkable how simple the above outline really is. Certainly the simplicity principles were useful heuristics in the formulation of the hypotheses proposed here, but they did not make a contribution to the selection of theories. A theory that is compatible with a person’s subjective world-view is often considered simpler, more logical, and self-evident, than the world-view to which it is compared. It would be a mistake though to quickly reject a world-view simply because it is an overly complex explanation with senseless additional hypotheses (i.e., dark energy, non-baryonic cold dark matter, false vacuums, extra-dimensions and so on). But at the same time, and conversely, it would be a mistake to reject a world-view because it seems to simple to be true.</div>
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The other point to make is that this is not a complete model. It is a qualitative and conceptual way to explain observations consistent with general relativity). Nor is it a complete cosmology: We have not taken into consideration evolution of the observed structures (from the microscopic to galactic superclusters), nor have we considered the origin of the light elements and their isotopes, here. To do so in this context would indeed require a modification of current theory, just as a modification of standard cosmology is required if we are to accept the above relativistic scenario for the cause of redshift z. This is nor the time or place to go into the full world-model. But it would be the logical next step (if it hasn't been elaborated upon already).</div>
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It was written in the first few paragraphs above that this work is a <i>qualitative</i> and <i>conceptual</i> analysis only. I have not yet sat down with, say, the Sloan Digital Sky Survey database and compiled a quantitative match to the predictions of the model. This is actually one of the next steps to take. Of course that doesn't obligatorily preclude investigating other aspects of the theory (CMB, light element production, or the large-scale structure evolution) first. </div>
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I do note, in passing, that the Einstein field equations and the analytical geometrical expressions should not fundamentally change with a switch over. The links that are provided above contain the mathematical proofs upon which these arguments are based (from Gauss to Einstein). I will note too that the claim observations should look the same for these two competing models (i.e., that the curved spacetime interpretation for redshift z should mimic observations currently explained by FLRW models) is based not on a whim or wishful thinking, but upon the contents of several peer reviewed papers, and related books by authors such as:</div>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Ellis, G.F.R. 1977, Is the Universe Expanding?, General Relativity and Gravitation, Vol. 9, No. 2 (1978), pp. 87-94</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Hubble, E. 1929, A Relation Between Distance and Radial Velocity Among Extra-Galactic Nebula, From Field, G.B., Arp, H., Bahcall, J.M. 1973, The Redshift Controversy 173</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Hubble, E. 1936, The Realm of the Nebula 108-201</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Kerszberg, P. 1989, The Invented Universe, The Einstein-De Sitter Controversy (1916-17) and the Rise of Relativistic Cosmology</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Kragh, H. 1996, Cosmology and Controversy, The Historical Development of Two Theories of the Universe</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Daigneault, A. 2003, Standard Cosmology and Other Possible Universes, (SCOPU)</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Segal, I.E. 1976, Mathematical Cosmology and Extragalactic Astronomy</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Segal, I.E., Nicoll, J.F., Wu, P., Zhou, Z. 1993, Statistically Efficient Testing of the Hubble and Lundmark Laws on IRAS Galaxy Samples, Astrophys. J. 465-484</li>
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<li style="color: #222222; font: 18.0px Times; line-height: 22.0px; margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify;">Segal, I.E., 1997, Cosmic time dilation, Ap. J. 482:L115-17</li>
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Indeed, work in the direction of a contemporary relativistic cosmology to replace the old has already begun. In fact, it is a cosmology that had already begun circa 1916, but was cut-short in favor of a model that was deemed dynamically evolving, when it was really just based on a kind of Newtonian instability. The only thing that was dynamic was the change in radius. Both models (expanding and static) are dynamical evolving universes. </div>
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I wouldn't expect the presentation here to change anything soon. Modern cosmology is such a ponderous institution that toppling it is not an option. Only it, with its inherent instability, can topple itself. The good news, though, is that an alternative model has burgeoned from the seeds of general relativity, first planted by Einstein and watered by de Sitter (and others), that may one day serve as a viable option against which the standard model can be tested. Now that <a href="http://en.wikipedia.org/wiki/Steady_State_theory"><span style="color: #0536cd; text-decoration: underline;">quasi-steady state cosmology</span></a>(QSSC), <a href="http://en.wikipedia.org/wiki/Halton_Arp"><span style="color: #0536cd; text-decoration: underline;">Arp's hypothesis</span></a>, <a href="http://en.wikipedia.org/wiki/Fritz_Zwicky"><span style="color: #0536cd; text-decoration: underline;">Zwicky's tired light</span></a> scenarios, <a href="http://en.wikipedia.org/wiki/Plasma_cosmology"><span style="color: #0536cd; text-decoration: underline;">plasma cosmology</span></a> and an entire host of other non-big bang or static universe alternatives have been to some extent ruled out by observation, it may be time for the elaboration of a quantitative model that <i>is</i> in line with GR (i.e., in line with non-Euclidean geometry). </div>
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For every new concept that has gained general acceptance, there are many more examples of new concepts that were shown to be invalid. However, most new concepts that <i>have</i> gained consensus were shown to be correct. This is because new concepts are typically presented by an individual. Acceptance involves a large number of individuals verifying and duplicating scientific results. The point is that the entire concept presented here (for redshift z and global stability) is based on concepts that are not new (they are generally accepted). The key had been to interpret GR correctly when switching from the local to the global, and to interpret it literally regarding its geometric considerations of the gravitational interaction. </div>
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<b>Infinity and Stability</b></div>
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Clearly, the notion that astronomical objects are <b>not</b> perturbed gravitationally in a globally homogeneous and isotropic universe is consistent with both Newtonian mechanics and general relativity. In other words, when applied to a uniform, homogeneous universe, both Newtonian theory (in Euclidean space) and Einstein's GR (in pseudo-Riemannian spacetime) yield quasi-identical results. And this would hold in Einstein's original 1917 world-model without lambda (since the latter postulated no boundary condition). The idea that needed to be vacated from Einstein's model is that of a finite spherical distribution of mass, unless, of course, that distribution extends to infinity. For that to be the case one simply needs to consider the geometric structure of spacetime itself as spherical and infinite and discard the notion of a finite spherical distribution of matter. The former does not imply the latter (and visa versa). The outcome of Einstein's quest to find static solutions for the field equations for a finite universe, with a spherical distribution of matter (with or without a boundary; with or without lambda) and constant geometrically positive curvature could have terminated no other way than with failure (for the same reason's Newtonian theory would fail under similar conditions in a Euclidean space).</div>
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The problem arose as a result of confusion and difficulties when applying or extrapolating GR to the whole universe. An erroneous assumption stemmed from considering a mass-distribution with a fixed radius. This could have been avoided from the start had the notion of Riemann spherical geometry (of constant positive curvature) been correctly extrapolated (or projected) onto a four-dimensional continuum with an infinite radius. Indeed there is no boundary in a spherical manifold, and there is no spherical distribution of mass required in a homogeneous universe (i.e., a finite distribution of mass, by definition, implies the universe is not homogeneous), whereby an escape velocity is required of a test-particle as in the local case of a gravitational field surrounding massive bodies. The erroneous assumption was to limit GR to a small finite region of space where Newton's equations applied, and where the metric coefficients that determine the curvature at each point in space are made equivalent to a single Newtonian gravity potential that varies in space and time for an isolated cosmic sphere with a homogeneous distribution of matter. (<a href="http://books.google.es/books?id=-8PJbcA2lLoC&pg=PA323&lpg=PA323&dq=infinite+newtonian+universe+gravity+is+the+same+in+all+directions&source=bl&ots=T3cSL4yTuY&sig=fSQz6Gb0JFNAb91TA2CwNUfR5aA&hl=en&ei=9TEjTP6JJtS7jAee96hB&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCEQ6AEwAw#v=onepage&q=infinite%20newtonian%20universe%20gravity%20is%20the%20same%20in%20all%20directions&f=false"><span style="color: #0536cd; text-decoration: underline;">E. R. Harrison, 2000, Cosmology: the science of the universe,</span></a> p. 334).</div>
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All theories or hypotheses that would subsequently emanate from such an erroneous assumption or bold extrapolation (e.g., the FLRW models with a flat expanding space, scale-factor that changes in time and begins with a bang) would be doomed to failure; if the intent was to describe accurately the essence of the physical universe and its evolution in time. Only a cosmological constant-like term together with a strange form of matter could salvage the model, but there was nothing natural about these types of additives.</div>
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The subsequent 'natural' generalization of Einstein's field equations—that allow the radius of curvature <i>R(t)</i> to be a function of time, and with its notion of cosmic temporal structure analogous to the conception of pre-relativistic physics—may not be natural after all. The three solutions to the field equations (FLRW) are in fact be exact solutions to the field equations, but they may have no (or little) corollary in the natural world. In other words the solutions found may not be the only solutions consistent with general relativity, when GR is interpreted in the context of cosmology (in a different light). The solutions found were indeed unstable, there is no denying that fact, and there is some agreement inherent within the FLRW models today with observations, thanks in part to the equivalence principle and in part to the flexible parameters involved, e.g., DE and CDM. (For a discussion about the pre-relativistic temporal structure inherent within FLRW models consult <a href="http://philsci-archive.pitt.edu/archive/00000800/00/The_Arrow_of_Time_in_Cosmology.pdf"><span style="color: #0536cd;">The Arrow of Time in Cosmology, M. Castagnino, O. Lombardi and L. Lara</span></a>).</div>
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What I hope to have shown is that the universe can (and does) remain stable. Global instability is not something that should be regarded as acceptable in light of the ambiguous nature of observations concerning redshift z, the physical interpretations of which may very well be consistent with general relativity. Surely, our conclusion leaves open the question of global stability and its relation to local instability. Though even local instability is questionable to some extent, since gravitating systems often seem to find natural ways of maintaining quasi-equilibrium configurations. The basic strategy for approaching the problem of global stability should be base on physical and geometric arguments in the absence of preconceived boundary conditions or finite spherical mass distributions that inevitably lead to catastrophic collapse. The formulation of a complete cosmology against which observations can be tested unfortunately largely exceeds the capabilities on a single individual. However, the general elements of this proposal should remain valid. The problem will consist of finding, empirically, the difference between two interpretations for cosmological redshift z, both of which grounded on physical arguments. The formulation of an accurate representation of how massive objects respond, or not, to the presence of a general relativistic globally curved homogeneous and isotropic four-dimensional spacetime manifold seems less problematic. </div>
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<b>On equilibrium and the cosmological constant: Λ</b></div>
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Up until now, little has been mentioned about the involvement of pressure (the force per unit area applied in a direction perpendicular to the surface of an object, or even pressure as a scalar which has no direction on any object) throughout the cosmos, and its relation to global stability. Several simplifying assumptions were made, chief among which are that both linear momentum and kinetic energy are globally conserved (in accord wit the <a href="http://en.wikipedia.org/wiki/Ideal_gas_law"><span style="color: #0536cd; text-decoration: underline;">ideal gas law</span></a>). In cosmology it is common to consider galaxies, stars, planets, rocks and dust (amongst other things) as a gas. In a static gas, the gas as a whole does not appear to expand or contract. The individual molecules of the gas are, however, in constant motion. Because we are dealing with an extremely large number of molecules (or galaxies) and because the motion of the individual molecules (or stars) is more or less random in all directions, there is no overall motion (globally). When a gas is enclosed within a container, pressure in the gas is detected from the molecules colliding with the container walls. We can put the walls of a container anywhere inside the gas, and the force per unit area (the pressure) is the same. The size of the container can be very small or very large and the pressure has a single value. Therefore, pressure is a scalar quantity, not a vector quantity. It has magnitude but no sense of direction associated with it. Pressure acts in all directions inside a gas. The extrapolation to a stationary universe is straight forward.</div>
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While pressure is usually considered positive, there are situations where negative pressure arises. When attractive forces (such as van der Waals forces) between the particles of a fluid exceed repulsive forces, results are usually unstable and the particles will move closer together until repulsive forces balance attractive forces. This has very little (if any) net effect on the dynamics of gravitating systems, and so will be considered irrelevant with respect to global stability.</div>
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Basically, what we have globally is situation where the net forces acting on galactic clusters is virtually zero: the competing influences are balanced. This is a necessary and sufficient condition for galaxies to remain is <a href="http://en.wikipedia.org/wiki/Mechanical_equilibrium"><span style="color: #0536cd; text-decoration: underline;">mechanical equilibrium</span></a>. The vector sum of all external forces is zero. In Newtonian terms the net torque (moment of force) and net force on every galaxy supercluster is zero (or practically zero).</div>
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When evolution is considered in a forthcoming post, we will discuss the series of quasi-static equilibrium processes that follow a succession of equilibrium states, where both objects and the surroundings are irreversibly altered and the system traverses successive states which differ from its initial state. Though this has more to do with evolution of galaxies, it could become part of a world-model the idea that an <a href="http://en.wikipedia.org/wiki/Equations_of_state"><span style="color: #0536cd; text-decoration: underline;">equations of state</span></a> could describe the universe itself as a systems characterized by very slow change (if measured on the incremental time scale). Simply put (for now), the quasistatic equilibrium world-model approximates change as a series of equilibrium processes.</div>
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In a quasistatic or equilibrium processes, an adequately slow transition of a thermodynamic systems from one equilibrium state to another transpires in a way that the state of the system is close to equilibrium at every moment in time. During the process a system can attain equilibrium much faster than its physical parameters vary. It will be shown that, in both the case of the universe itself and the large-scale structures, the quasistatic process is necessarily an irreversible one. The only requirement is that the properties of the system under consideration be homogeneous and isotropic at any instant during a process. One of the questions is; if entropy was lower in the past, why hasn't it attained a maximum value today (after all, if the universe had no beginning there would have been an infinite amount of time to attain equilibrium)? Alas, there is no rigorous definition of entropy for systems coupled to the gravitational interaction, but simple estimates can be made, and will be made in a forthcoming post.</div>
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I'd like to finish (what has almost become a manifesto) with a few words about one of Einstein's fabulous inventions. Something that has alternated between playing a leading role in modern cosmology, and one more subdued. I had always believed (up until recently) that Einstein's introduction of <b>lambda</b> into the field equations was justified, and would actually have contributed physically to the maintenance of global stability, just as he had originally hoped. I believed there was a natural form of pressure inherent in the vacuum of space that countered gravity, even to the point of thinking it might actually be the opposite of gravity. But not gravity with a different sign. The opposite of gravity would be NO gravity. This was a concept not dissimilar to one Eddington (a fervent believer in the cosmological constant) proposed in 1939. The problem Eddington set out to solve was to find out exactly how the equilibrium between gravity and Λ is brought about. Eddington writes: <i>“If we are contemplating a limited region of space, it is natural to take emptiness as the standard zero condition—the energy in a region is that which we should have to take out in order to leave it a complete vacuum.”</i> </div>
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So the stress of the gravitational field was exactly countered by the stress of the vacuum (Λ), consistent with energy conservation laws along with symmetry principles. And the assumption (mine) was that lambda was operational locally as well as globally, that equilibrium was generated or attained by the interaction between gravitating bodies, the field and empty space. Here, of course, there was a reasonable assumption that physics within our solar system is the same as physics in any other region of the cosmos, but too, that lambda must somehow be attached to a physical law (perhaps as a fundamental constant of nature).</div>
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“Either the universe has a center, has a vanishing density everywhere, empty at infinity where all thermal energy is gradually lost as radiation; or, all the points are equivalent on average, and the mean density is everywhere the same. In either case, one needs a hypothetical constant Λ, which specifies the particular mean density of matter consistent with equilibrium." [Preferring the second possibility, Einstein continues] "Since the universe is unique, there is no essential difference between considering Λ as a constant which is peculiar to a law of nature or as a constant of integration.” (Einstein, 1918, from Kragh, 1996, p. 10) </div>
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Words are carefully measured and devoid of fiery rhetoric—nothing more. At that time, 1918, Einstein wrote of the cosmological constant as “peculiar to a law of nature.” Peculiar in this phrase probably meant ‘belonging exclusively to’ or ‘identified with,’ but it’s other meaning, unusual, strange or unconventional is revelatory that Einstein himself didn’t know exactly what he had touched upon or how his constant fit into the fundamental laws. The key point is that he suspected then that something in nature was responsible for stabilizing systems against gravitational collapse or disintegration.</div>
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In another way the Λ-term could have been interpreted as a unique equilibrium state with a nonnegative, non-positive absolute value of curvature (i.e., equal to zero for all time). So we would have a stable Minkowski-like background as a kind of stage or substratum upon which all events would transpire.</div>
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For this reason I believed that Einstein's greatest blunder had not been the introduction of lambda, but the discarding of lambda (when expansion was 'discovered'). Obviously it could now be argued that its introduction was indeed Einstein's greatest blunder, but not for reasons generally believed.</div>
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The cosmological constant is traditionally thought to have the same effect as an intrinsic energy density of the vacuum, with an associated pressure. A negative vacuum energy density resulting from lambda implies a positive pressure, and vice versa. If the energy density is positive, the associated negative pressure drives an accelerated expansion of empty space. There are other potential causes for an accelerating expansion (e.g., quintessence) but the cosmological constant is thought to be the most parsimonious solution. Of course, it follows from the present text that the most parsimonious solution is to do away with lambda all together. </div>
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The discrepancy between cosmological measurements and quantum field theories predictions for the value of lambda have been referred to as "the worst theoretical prediction in the history of physics!" This is called the cosmological constant problem. It is currently the worst problem of fine-tuning in contemporary physics, since there is no known natural way to derive the small cosmological constant employed in lambda-CDM from particle physics. That too now can be put to rest.</div>
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Einstein wrote in 1945, "the introduction of [the cosmological constant] constitutes a complication of the theory, which seriously reduces its logical simplicity". Indeed, recent astronomical data (gained by studying distant SNe Ia that seem to indicate an accelerating expansion) have caused most scientists to abandon closed models, but there seems to be some lack of appreciation for the damage an open universe [with a nonzero cosmological constant] does to the epistemological strength of general relativity (see Brown, Reflections on Relativity).</div>
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<b>Concluding remarks</b></div>
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In order to explain current observations, non-baryonic cold-dark matter, dark energy and inflation (not to mention the big bang, with its primordial creation, or inaccessible domains within an ultra-large-scale multiverse) are no longer the sole protagonists. The model laid out here, contrary to the concordance model (lambda-CDM), has a firm basis in well established physics, notably that of general relativity (without the cosmological constant), thereby providing a more natural solution for the observed phenomena. It is a cosmology (not based on ad hoc assumptions) that resides in the realm of known physics. There simply is no dark energy or nonbaryonic cold dark matter.</div>
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Will it be difficult to distinguish between models based on redshift and distances alone? Yes it will. But fortunately there are other ways to rule out the model. Predictions for the theory would include observational data. For example, the rate of evolution in the look-back time would appear far slower than currently hypothesized. There should be observed stars and galaxies as old as those found in the Local Group all the way to the visual horizon (no "Dark Age"). Indeed, there should be observed near the visible horizon (in addition to ongoing star-formation, protogalaxies and so on) metal-rich stellar populations consistent with intermediate-age populations and older populations with high element rations. Galaxy morphologies at these early times should also be consistent with mature galaxies found locally. Observations of this kind are crucial since it is here that the major differences stand out between competing models. In the coming years, stellar spectroscopic measurements will become increasingly available for distant galaxies for which we only have photometric estimates at present.</div>
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This preliminary model is by no means complete. For now it is simply the continuation of a preliminary model that had slowly evolved since the time of Newton and dramatically furthered by Einstein, with his general theory of relativity as a guideline. Indeed, it could be argued that this preliminary cosmological model is entirely based upon GR. It is not an extension of it, but an interpretation of it, adapted to empirical evidence. The remainder of the model, touched on only briefly, if at all, here, should comprise all observations. In addition to redshift z and global stability are the issues of the cosmic background radiation (its origin and thermal evolution), the creation of the light elements and their isotopes (and in particular the creation of hydrogen), galaxy formation, dynamics and evolution (including rotational curve observations), formation and evolution of the large-scale structures, <i>entre autre</i>. </div>
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The implications of a full blown model are evident: nothing is free from the natural laws. Yet, with all of the answers that it could offer, the preliminary model outlined above, and its offspring, should set up a central question from the outset: Yes or no, is the rediscovery of universal stability, of quasi-equilibrium, of symmetry (broken it is true with the temporal increase of entropy) enough to bring together the contradictory inherent within the unbounded imagination? By this is meant, of course, the unification of general relativity and quantum mechanics, and too, understanding the differing values of the fundamental constants of nature, or even something as apparently simple as a vacuum state. I tend to answer in the affirmative.</div>
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The scenarios discussed in this text involve the invocation of geometry, not just within a visibly accessible domain, but within an infinite spatiotemporal universe: one with no beginning and no end. This is a universe that has been evolving forever, and will continue to do so for an infinite amount of time. For justifiable reasons, the reliance on the properties of regions outside the observable domain and the difficulty in falsifying such extrapolations may make physicists reticent. But untestable predictions plague all theories. For this reason theories should be judged on the grounds of testable predictions.</div>
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Falsifiability or refutability: there exists the logical possibility that the assertions, regarding cosmological redshift z and global stability presented here, could be shown false. <i>If t</i>he contentions discussed in this work are false, then the falsehood can be demonstrated, simply by showing (1) that electromagnetic radiation is <span style="font: normal normal normal 19px/normal Times;"><i>not</i></span> redshifted as it propagates along geodesics in a manifold of constant Gaussian curvature (whether hyperbolic or spherical). Or (2) that material objects would all coalesce towards one another in such a Gaussian manifold. </div>
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In the final analysis, the hope is that the preliminary model has definitive consequences for the observed structures, in such a way as to provide an explanation for how the observable section of the universe has arisen via natural means. The goal is that through further laboratory testing and astrophysical observations, additional predictions can be made that will help rule out inaccurate representations of the physical universe and its evolution in time. Further investigation, only, will demonstrate whether such a program represents a productive aspiration or a unreasonable presumption. </div>
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The total preservation of the physical laws (at all times, in all places and at all scales) is the point of Coldcreation. This central theme is not a dream, or an illusion of the imaginary as real, so that everything becomes fabulous, manufactured and therefore mythical. Coldcreation simply places the laws in the foreground, and at the core of its treatment of spacetime, of cosmology, of physics, of life. </div>
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A continued discussion on this topic may be found here: <a href="http://scienceforums.com/topic/3031-redshift-z/page__view__findpost__p__300079"><span style="color: #5588b1;">Redshift z </span></a>at scienceforums.com</div>
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Alex Mittelmann<br />
Coldcreation</div>
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June-August 2010, Barcelona, Spain<br />
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email:<br />
alexmittelmann@yahoo.com<br />
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Alex Mittelmann, alexmittelmann@yahoo.comhttp://www.blogger.com/profile/03457606761033752726noreply@blogger.com0tag:blogger.com,1999:blog-7565568908021398484.post-11309251762881882142011-01-24T01:54:00.000-08:002011-01-25T11:43:46.348-08:00<div style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; font: normal normal normal 18px/normal Times; line-height: 18px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-height: 23px; text-align: justify;"><div style="text-align: center;"><b><span class="Apple-style-span" style="color: #666666;"><span class="Apple-style-span" style="font-size: x-large;">Large-Scale Structures - Superclusters and Supervoids</span></span></b></div><br />
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This section, in fact this entire text, is nothing more than a documentary covering the evolution of the universe. It is designed to uncover the truth about what is transpiring around us. In presenting empirical evidence objectively and in an informative manner, without inserting highly speculative material (or dark energy) to make observations fit the theory, the goal is that a better understanding of nature (in the widest possible sense of the term) can be obtained. The ultimate goal is the understanding of <i>the essence of the physical universe and its evolution in time</i>.<br />
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It may turn out the truth is embodied in big bang cosmology. Or it may not. If the evidence were indisputable I would certainly not, as a pragmatic empiricist (and Bright), be wasting time writing about an alternative model, let along one that is static. The fact is, there are no indisputable facts inherent in big bang cosmology. Redshift z is not necessarily a relativistic Doppler effect, nor is the CMB necessarily a relic of a hot dense phase of the early universe. There may not have been an epoch of primordial creation at the outset, or even an outset at all.<br />
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The fact is too, there exists at least one viable alternative consistent with general relativity (where gravitation is treated as a curved spacetime phenomenon) that is dynamic and evolving, and is consistent with observations (to the extent that those observations have been interpreted within the outline of the model).<br />
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Let me sum up what has been discussed so far in the context of a general relativistic static universe model that possess constant Gaussian curvature of the global spacetime continuum. A few concepts not extensively covered so far will be explored below regarding the problems and solutions of diverging gravitational potential in a homogeneous stationary universe inherent in gravitational theory since the time of Newton.<br />
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<div style="text-align: center;"><b>Spacetime Curvature</b></div><br />
Curvature is the central theme of theoretical and observational cosmology. Spacetime curvature in the presence of matter, introduced by Einstein in the general theory of relativity, is the physical concept that replaced the profound mystery of Newtonian action at a distance. To a large extent today the concept of spacetime curvature is a profound mystery.<br />
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The concept of curvature found its way into general relativity following the lead of Saccheri, Lobachewsky, Gauss, Bolyai, Clifford. Subsequently, Riemann, Christoffel, Ricci and Levi-Citiva developed non-Euclidean geometry into a robust analytical discipline.<br />
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The premise upon which all observational cosmology rests was built by Gauss in his 1827 "Disquisitiones Generales Circa Superficies Curvas." Curvature (a number) can be calculated for any arbitrary manifold from data obtained by measurement of the metric properties internal to that manifold. The departure from flatness is called curvature. See Gauss' Theorema Egregium.<br />
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The surface of a sphere is a 2-space example with constant positive curvature. Curvature remains the same along the entire surface in all directions (every point is the same). It is often the difficulty of visualizing this intrinsic 2-space curvature in 3-space that causes intuition to fail. I've added a schematic diagram below to help in this visualization (a cross section of a static four-dimensional universe: Figure ESU).<br />
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Ultimately, the curvature of 3-space must be expressed by equations that contain quantities observed telescopically: Fluxes (apparent magnitudes), redshift z (the spectral shift relative to laboratory wavelengths of a particular spectral line), angular diameter and surface brightness. All of these observables are affected by Gaussian curvature of 3-space. Note that in a spacetime manifold of the type under study here—a static universe of positive Gaussian curvature k = 1—the volume at a given distance interval appears smaller than that of a Euclidean or hyperbolic 3-space (k = 0, k = -1). In addition, time intervals appear to run slower with increasing distance from the observer (from any arbitrary origin of the manifold centered on an observer). Again, spacetime curvature is determined by the mass-energy density of the manifold, in accord with general relativity.<br />
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The point of the current discussion (see above), is in part to show that photons propagate along 'true' geodesic paths in a four-dimensional spacetime continuum (three space and one time), consistent with great circle arcs (in reduced dimension, i.e., on a spherical surface of constant positive Gaussian curvature). Electromagnetic radiation is redshifted due to the geodesic path. Thus, in a static universe of constant positive Gaussian curvature cosmological redshift z is observed to increase with distance from the observer in the look-back time. Redshift is a measure of geodesic distance between the emitting source and the observer in a homogeneous and isotropic curved spacetime. Redshift is due to a geodesic distortion along the path of the photon wave between two epochs. Cosmological redshift along with the associate time dilation factor is a direct measurement of constant intrinsic Gaussian curvature of the spacetime continuum.<br />
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The universe remains non-expanding and non-collapsing (i.e., static) due to the physical properties of a spacetime continuum of constant positive curvature. The fact that massive objects do not follow the same geodesic trajectory as photons—they are indeed restricted to local gravitational field geodesic motion—implies that all objects in the universe are not impelled to accelerate, gravitate or free-fall towards one another.<br />
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Massive objects are bound to follow local geodesics determined by the gravitational fields of neighboring objects (such as the orbit of the earth around the sun, the sun around the Galaxy, the Galaxy relative to the Local Group, the Local Group relative to the Local Supercluster). The question is whether galaxy superclusters are stable gravitationally bounded systems. Unfortunately, the dynamics of galaxy clusters cannot be directly observed since the rotational times are typically hundreds of millions (or billions) of years. Galaxy clusters and superclusters appear as motionless systems, but their morphology, orientation relative to neighbors, and physical connections in a web-like or sponge-like network imply otherwise.<br />
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<div style="text-align: center;"><b>Equilibrium Solutions</b></div><br />
Fortunately, methods for computing the N-body problem have grown exponentially in computational efficiency. These methods have been used to simulate the dynamics of systems with as many as 10 billion particles. (<a href="http://www.galaxydynamics.org/linernotes.html" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a>). Not only do clusters persist as clusters when they fall into a larger host, but groups of clusters retain their identity for long periods within larger host superclusters. Line-of-sight velocity dispersions for individual clusters show large variance which depends on viewing angle. Kinematics is strongly viewing angle dependent. The point is that clusters of galaxies, even after merging with a larger host cluster (or supercluster), can retain their identity for several Gyr. Equilibrium solutions are possible for scales compatible with superclusters. Exceedingly large tangent velocities are not required in order that superclusters remain in stable equilibrium relative to one another. The larger the scale of the clusters, the greater the distance that separates clusters, and as the mass density profile becomes less concentrated, the slower the clusters need to move relative to one another in order for the maintenance of stability. Whatever the initial orbital shapes of superclusters relative to one another, isotropization of their relative orbits can (and here argued, <i>does</i>) occur. Observed should be a lack of significant evolution in the orbital anisotropy. It is essential to tighten the current constraints on the orbital evolution of superclusters (both nearby and high-z) in the future, and to re-assess such dynamics as a function of both time and supercluster mass. Only from this type of analysis will it be possible to obtain a thorough understanding of the dynamics and hierarchical assembly history (evolutionary trends) of galaxy superclusters as bounded gravitating systems. Within a supercluster, the individual galaxy clusters orbit the supercluster's center of mass, roughly as the sun orbits the Galaxy. Of course, such an orbital velocity would be difficult to observe, since both the intrinsic redshift and blueshift due to orbital motion would be dwarfed by the cosmological redshift (i.e., a blushift induced by rotation towards the observer would simply be seen as a redshift, and the intrinsic redshift would make objects looks further away when coupled with cosmological redshift z).<br />
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Locally, where cosmological redshift is negligible, investigations into cluster dynamics is more robust. Observations inferring total mass in galaxies and clusters, based on the distribution of orbital velocities of components within these systems, suggest that much of the mass is non-luminous. Dynamical mass estimates are usually 5-10 times larger than mass estimates derived by summing the luminous components (and hot gas, in the case of clusters). The existence of nonbaryonic cold dark matter is hypothesized to explain such observations (Postman 2006, Distribution of Galaxies, Clusters, and Superclusters). Of course the possibility that much of this dark matter is solely baryonic exists (e.g., in the form of neutron stars, white dwarfs, brown dwarfs, massive planets, dilute plasma, neutral gas). But something still appears to be missing. Certainly, in a static universe, times scales are sufficiently large to explain the presence of vast quantities of non-luminous objects. An additional possibility (which should be compounded with the greater baryonic density than current estimates) is that the dynamics of these systems is determined to a large extent by the larger systems still, within which are embedded the objects in question. The larger the scale, the greater the effect. For example, gravitational effect on the solar system due to the Galaxy are small. Effects on the dynamics of the Galaxy due to the Local group are small but relatively larger. The dynamical gravitational effects on the Local Group due to the Local Supercluster would be large in comparison. This would mean that the density perturbations associated with superclusters affects the dynamics of its components. Very large and extremely massive superclusters would have a greater affect on the dynamics of their subsystems than would smaller less massive superclusters. Orbital velocities or rotational curves should appear faster than gravitational effects of luminous matter would permit for components of very large massive systems. Whether this mechanism is operational remains to be tested. But is seems that the CDM contribution may not be necessary if this hypothesis is valid. The question of how best to use observations regarding excessively large orbital velocities and rotational curves to obtain optimal mass estimates from the relevant equations is one I hope to address in future work.<br />
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On large scales, beyond which the lumpiness seen in the large-scale structure of the universe appears isotropized and homogenized as per the <a href="http://en.wikipedia.org/wiki/Cosmological_Principle" style="color: #2288bb; text-decoration: none;" target="_blank">cosmological principle</a>, the distances that separates systems (great walls, filaments, supervoids and associated superclusters) are so large and time-scales so vast that orbital velocity is insignificant (or irrelevant). Orbital velocity tends to zero as distance and time-scale increases, up to the scale where the universe appears homogeneous and isotropic. That would be so due to the finite speed of the gravitational interaction (something not present in Newtonian gravitation). And on scales compatible with the observable section of the universe, no orbital velocity is required for the maintenance of stability.<br />
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The terms void and supervoid here corresponds to the lowdensity regions on the physical scales of clusters and supercusters respectively. Supervoid structures are roughly 100-150 <i>h</i>^1 Mpc (<i>h</i> is the Hubble constant in units of 100 km s^−1 Mpc^−1); a scale below which the universe is inhomogeneous, and above which the matter distribution tends toward homogeneity.<br />
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<div style="text-align: center;"><b>The Local Group</b></div><br />
Only with respect to the <a href="http://en.wikipedia.org/wiki/Local_group" style="color: #2288bb; text-decoration: none;" target="_blank">Local Group</a> and the <a href="http://en.wikipedia.org/wiki/Local_Void" style="color: #2288bb; text-decoration: none;" target="_blank">Local Void</a> is it possible to distinguish between the redshift of galaxies due to the Gaussian curvature of the universe and the local relative motions caused by the way matter is clustered together, with its consequential gravitational effects. It has been found that galaxies flow in streams, with coherent flows caused by large distant attractors and eddies caused by modest nearby attractors. The currently favored standard model of the universe (Lambda-CDM) with dark matter and dark energy does not allow for voids that are as large as inferred for the Local Void. (<a href="http://www.astronomy.com/en/sitecore/content/Home/News-Observing/News/2007/06/Milky%20Way%20moving%20away%20from%20void.aspx" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a>).<br />
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<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhB93lmoGgP3FcRvyJqEPkSDpDE01yj_EeaaZjI-91d1JZgz0an6Xg9d1o8whGR1FzPVkKq8EolAzZuR13UN4cc_BOjN8Af2b7l0RZHRi_63pM2SsFktwIFlgiM-_7nVm6O2hPsExVuQaU/s1600/Local+Group+motion+to+Local+Void+and+Virgo+14cm150dpi+ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; -webkit-user-select: none; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /><br />
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The distribution, in supergalactic coordinates, of galaxies in the region of the Milky Way.<br />
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(Source: <a href="http://www.ifa.hawaii.edu/cosmicflows/" style="color: #2288bb; text-decoration: none;" target="_blank">Institute for Astronomy University of Hawaii</a> and <a href="http://adsabs.harvard.edu/abs/2008ApJ...676..184T" style="color: #2288bb; text-decoration: none;" target="_blank">Our Peculiar Motion Away from the Local Void, ApJ 676, 184</a>)<br />
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Each dot represents a galaxy (on average containing 100 billion stars). The colors indicate the relative motions: green and blue indicate motions toward the Milky Way. Shades of yellow, orange and red indicate motions away from us. The closest galaxies have small relative motions (see right panel). The Galaxy, along with all the Local Group, are moving together toward the lower right corner of each diagram. As a result, all galaxies in the lower right appear to be moving toward us, while those in the upper left appear to be moving away. The motion of the Galaxy is represented by the orange arrow. There are two suspected causes for this motion. The Virgo Cluster on the right of the figures causes an attraction indicated by the blue vector (in the exploded panel on the right). The red vector is the remainder which represents a motion of ~260 km s−1 (600,000 miles-per-hour) away from the Local Void. "given the velocities expected from gravitational instability theory in the standard cosmological paradigm, the distance to the center of the Local Void must be at least 23 Mpc from our position. The Local Void is extremely large."<br />
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Such a problem does not arise in this general relativistic static model, since cosmological redshift is not interpreted as resulting from expansion, i.e., a large component of the observed redshift is not due to radial motion away from the local supervoid.<br />
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The <a href="http://en.wikipedia.org/wiki/WMAP_cold_spot" style="color: #2288bb; text-decoration: none;" target="_blank">WMAP Cold Spot</a>, a cold region in the microwave sky, is highly improbable under the Lambda-cold dark matter model (ΛCDM). This supervoid could be the cause the cold spot, but it would have to be improbably large: a billion light-years across, according to some estimations.<br />
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<div style="text-align: center;"><b>The Large-scale Structures</b></div><br />
The differentiation between models must be made by studying the characteristics that would arise in the case where superclusters are gravitationally bound, in comparison with properties in the case where they are not gravitationally bound. Recall, it is currently believed, by virtue of expansion, that superclusters are not gravitationally bound systems (so there should be little observed interaction, if any at all, between these groups). The observed nonlinearity in the redshift and rise times of distant Type Ia supernovae interpreted as an acceleration of expansion is usually attributed to a mysterious negative pressure (dark energy) that causes the gravitational potential between large-scale structures to decay, lowering the amount of interaction between superclusters. The supervoids that separate these structures grow with time. Regularity in the structure of large-scale density perturbations would be disrupted with time (with expansion), resulting in irregularity in the distribution of superclusters.<br />
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Conversely, in a static universe where these large-scale structures are gravitationally bound there should be observed a great deal of interaction and regularity in the distribution of superclusters. After all, these objects would be connected gravitationally, just as clusters themselves (in either model). There should be a straightforward relation between clusters and the supercluster substructure, and there should be a relation between superclusters themselves, e.g., there should be observed a tendency for superclusters to be alined with their closest neighbor as well as with other superclusters that reside in the local vicinity (the friends of friends approach), as well as large-scale filamentary structures connecting superclusters. There should be a relation between the dynamic internal state of superclusters and the large-scale environment. The morphology of superclusters must be consistent with systems that exhibit rotation on smaller scales (e.g., flattening). In sum, there should be observed a coherent orientation effect and a strong link between the dynamical state of superclusters and their large-scale environment. The size of the supervoids that separate superclusters remains relatively constant over time. In other words, there should be observed a regularity on scales of superclusters (which are considered gravitating systems in a static universe), i.e., the supercluster-supervoid network should be highly structured, just as there exists non-random highly structured regularity in the distribution of gravitating systems on smaller scales, where systems are known to be gravitationally bound. In fact, in a static universe all systems are bound gravitationally. Albeit, gravitational influences are strongest for those objects that neighbor one another, but non-negligible for friends of friends. Structural coherence should be observed across a vast region of the visible universe.<br />
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Einasto <i>et al</i> (1997, The supercluster-void network II. An oscillating cluster correlation function) compared correlation functions derived for popular CDM-models of structure formation (as well as simple geometrical models with randomly and regularly located superclusters) with the empirically <i>observed</i> cluster correlation function. The quantitative tests showed that overdense regions in the universe punctuated by rich galaxy clusters are distributed more regularly than expected, and that, consequently, "our present understanding of structure formation needs revision." Note: the discrepancy between galaxy formation models and observation would be even greater in a universe dominated by dark energy. "The fact that the amplitude of oscillations near the last maximum is still rather large suggests that the coherence of positions of high-density regions extends over very large separations (at least 10% of the diameter of the observable Universe)." Einasto <i>et al </i>pose a good question: "Can the observed correlation function of clusters of galaxies be reproduced by conventional models of structure evolution? If not, what changes in models are needed to reproduce the observed function?"<br />
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Early models of structure formation based on a universe dominated by CDM had predicted a trend, according to Marc Postman, towards higher correlation lengths as the mass of galaxies and clusters increased. But "these same models did not come close to predicting the actual observations in which the richest clusters are 10-20 times more strongly clustered than are galaxies." (Postman 2006)<br />
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In contrast to the standard expanding picture, the dynamical times of superclusters in a static universe are very large. Some, if not all, superclusters should be relaxed, thereby bearing the imprints of systems in dynamic equilibrium. Too, since expansion is not ripping these systems apart, the large-scale structures should bear the signature of the physical density fluctuations that were dominant during the formation process.<br />
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Certainly, the merging or colliding of components of superclusters (even of superclusters themselves) as well as the scattering or dispersion of superclusters are ongoing processes (this can occur in superclusters just as it does in smaller clusters). According to the standard model galaxy clusters (not superclusters) are the largest systems known to have reached dynamic equilibrium. In a non-expanding universe gravitationally bound structures exist on larger scales. Indeed, in a static universe superclusters, supervoids, great walls, sheets and filaments (some of which, in excess of 10^16 solar masses, are known to span 200 million light-years, or several hundred Mpc: <a href="http://www.astro.caltech.edu/~george/ay21/eaa/eaa-lss.pdf" style="color: #2288bb; text-decoration: none;" target="_blank">see here for example</a>) have had amply sufficient time to reach dynamic equilibrium. In addition, these structures need not be governed by the gravitational potential of ubiquitous nonbaryonic cold dark matter.<br />
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"Although early versions of structure formation theory based upon a universe dominated by cold dark matter (CDM) did predict a trend towards higher correlation lengths as the galaxy and cluster mass increased, these same models did not come close to predicting the actual observations in which the richest clusters are 10–20 times more strongly clustered than are galaxies." <a href="http://www.astro.caltech.edu/~george/ay21/eaa/eaa-lss.pdf" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a><br />
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All the material objects in the universe are subject to their own mutual gravitational field interactions. The fields of massive objects affect the local motion of other neighboring objects, proportionally to the gradients of the local gravitational fields. Inhomogeneities in the mass distribution induce motions (peculiar velocities) of galaxies, clusters and superclusters that are unrelated to the expansion of space, and unrelated to the globally non-Euclidean geometry in the case of a static universe where the cause of redshift z is attributed to the geodesic distortion in the photon path in a manifold of constant Gaussian curvature.<br />
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Preliminary evidence suggests that there do exist bulk flows of 600-800 (±150-300) km s−1 that extend to scales exceeding 100 Mpc. Though these ongoing surveys have not yielded conclusive directions for these flows. To date, there appears no inconsistency with the conclusion that superclusters are gravitationally bound systems in relative motion with respect to one another.<br />
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"If very largescale bulk flows turn out to be a reality, then present models for structure formation in the universe will require substantial revision (if not outright rejection)." <a href="http://www.astro.caltech.edu/~george/ay21/eaa/eaa-lss.pdf" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a><br />
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The impetus that all neighboring galaxy superclusters should merge gravitationally over time is vacated. Global collapse to a Big Crunch-like event is not an option, even in the absence of an expanding regime.<br />
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Let's take a break and look at this breathtaking 4-D simulation of the large-scale structure evolution. (Note, this model universe is not rotating. The geometrical setup of the camera path, the observers viewpoint, is in motion relative to the structures.)<br />
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<div style="text-align: center;"><a href="http://www.youtube.com/watch?v=yIvmiucMOvU" style="color: #2288bb; text-decoration: none;" target="_blank"><b>Simulation</b></a></div><br />
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This simulation uses a uniform time-dependent background radiation field, revealing the superclusters, voids, sheets, walls and filamentary structures formed so far. This video is shown in the Virtual Reality facility of the new Turin Planetarium. (See K Dolag, M Reinecke, C Gheller and S Imboden, Max-Planck-Institute for Astrophysics, <a href="http://iopscience.iop.org/1367-2630/10/12/125006/fulltext#nj280000s4" style="color: #2288bb; text-decoration: none;" target="_blank">Splotch: visualizing cosmological simulations</a>). The compressed demo version of this simulation can in principle be <a href="http://www.mpa-garching.mpg.de/galform/data_vis/index.shtml#movie11" style="color: #2288bb; text-decoration: none;" target="_blank">download here</a>, along with a host of other simulations.<br />
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The importance of all this is fundamental, as it shows that in a static four-dimensional universe, where positive curvature is due to the mass-energy density of the universe, objects do <b>not</b> all coalesce geodesically towards one massive big crunch. Global curvature does not cause gravitational instability, since objects are not bound by the same geodesic path as the photon. In another way, the global topology of the manifold does not affect the motion of gravitating systems within it. Yet the global topology is determined by the mass-energy content.<br />
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The example was given that the earth has settled into a stable orbit. So too the other planets of the solar system. Galaxies, though at time exhibit chaotic behavior, are just about everywhere present in the universe, i.e., they are relatively stable gravitating systems. Galaxy custers are relatively stable bound self-gravitating systems. These objects do not all gravitate towards one another, since their motion (angular momentum) is in general sufficient for the maintenance of dynamic equilibrium (see references below). So far so good.<br />
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<div style="text-align: center;"><b>Superclusters</b></div><br />
Superclusters, on the other hand, are thought to be dynamically unstable systems, not bounded gravitationally. The reason given is usually because the universe is thought to be expanding, i.e., superclusters are moving away from one another, radially, in all directions. It is thought too that the velocity at which they are moving away from one another is sufficient to prevent them form collapsing back onto each other gravitationally. I will argue that superclusters are most definitely gravitationally bound systems.<br />
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In a static universe it has been suggested that the orbital velocity of superclusters is insufficient to prevent wholesale collapse of such structures. But this conjecture is not justified by empirical evidence. In addition, the larger the scale considered, the greater is the distance that separates objects, and the smaller is the force of gravity between them. This is like the classic analogy of an figure skater spinning on her axis. While her arms extend outwards the velocity of rotation decreases. In the case of superclusters, distances that separate galactic components (and often superclusters themselves) are so vast that orbital velocities sufficient for dynamic equilibrium are practically negligible. Large orbital velocities would not be required for stability to be maintained, even for the most massive superclustes (though again, interactions or merging will occur when in close proximity and when there is insufficient relative velocity). Indeed, the distances that separate galaxies in a cluster are much larger (in general) than distance that separate stars in a galaxy. In other words, distances that separate superclusters (on average) are relatively much larger than the distances that separated clusters. The larger the system under consideration the slower the orbital velocity required for equilibrium to be maintained (or the longer the collapse time). This is why we would not see (even if we could) peculiar velocities of superclusters in excess of peculiar velocities observed for clusters. See below, this hierarchal structuring of the cosmos has been used to undermine the diverging Newtonian gravitational potential in a homogeneous universe. But it is not the only method for dealing with the problem. Other methods have been shown more conclusive.<br />
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See below too for a further explanation of stability on scales compatible with the observable universe (or the universe in its entirety).<br />
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So the conclusion that <i>over large distances there are no stable orbits because there are no tangent velocities sufficient to attain orbit</i> is not justified by empirical evidence.<br />
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Empirical evidence confirms that the orientation of galaxies inside superclusters are generally alined in the directions of the major axes of the galaxies. Galaxies are significantly correlated. It has been known for many decades (Brown, 1964, 1968) that the minor axis of superclusters are oriented along the direction of the flattening of the supercluster as a whole. The same phenomenon is observed in the Local Supercluster, associated with the rotation established by de Vaucouleurs (1959). The broad conclusion that superclusters rotate is permitted. The rotation on these large scales cannot be explained by local effects, or tidal effects. Rotation on such scales is exceedingly difficult to incorporate into the standard model of galaxy formation, which suggests clusters are formed relatively rapidly via density fluctuations as the universe expands (<a href="http://adsabs.harvard.edu/abs/1974IAUS...58...85O" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a>). In a static universe this problem does not arise since density fluctuations have no time limit, i.e., pregalactic, precluster and protosupercluster density inhomogeneities can grow freely and interact in a variety of ways over extended (virtually limitless) time periods.<br />
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Due to motion, the distances involved, and due to non-instantaneous interaction (or finite velocity) of gravity, superclusters during the formation process and therafter remain gravitationally bound systems free of the need of to attain large peculiar velocities (their angular momentum is sufficiently large and operational), and free of the propensity to grossly collapse. On scales compatible with the observed superclusters, supervoids, sheets, filaments and walls, the universe is in dynamic equilibrium. At least, that possibility remains a viable option, in principle. Though again, to establish the rotational velocity of distance superclusters is quite a difficult task, since we would have to disentangle the intrinsic redshift (due to the rotation towards us) and the blueshift (rotation away from our viewpoint) from cosmological redshift z (which at great distances dwarfs the effect of peculiar motion). The observed redshift contains contributions from both gravitationally induced velocity and a cosmological component (whatever its cause). In another way, orbital velocities distort the spatiotemporal positions determined from cosmological redshift measurements, resulting in a highly elongated appearance of clusters and superclusters oriented toward the center of 2-D and 3-D maps of the sky. This is an artifact, dubbed "the fingers of God," of large orbital velocities (close to 1000 km s^-1) around the center of clusters. If distances could be plotted independent of redshift the fingers of god would vanish. (Source: <a href="http://www.astro.caltech.edu/~george/ay21/eaa/eaa-lss.pdf" style="color: #2288bb; text-decoration: none;" target="_blank">Distribution of Galaxies, Clusters, and Superclusters, IoP</a>).<br />
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Implications from the relevant data are that the larger the correlation length (i.e., the greater the mean separation between objects) and the larger the mass of the system, the stronger the clustering. (See source directly above). The question is whether this trend continues to scales of superclusters. My hunch is that it does. The standard model for structure formation dominated by CDM had predicted a this correlation, but the predictions didn't come close to actual observations, which show that the richest clusters are between 10 and 20 times more strongly clustered than galaxies.<br />
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On scales exceeding 150-200 Mpc the universe tends toward homogeneity, so the idea that velocities are required to maintain equilibrium configurations is no longer relevant. There is no need to stipulate that larger systems still (ad infinitum) and greater velocities are required for stability to be attained. Or simply, there appear to be no structures larger than 150 Mpc that would need to rotate around something. In principle, this fact (if indeed it is a fact) would set an upper limit on the peculiar velocity of objects in the field.<br />
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There is a recent tendency for larger structures to be recognized. So it's quite possible that the top of clustering hierarchy has not yet been established, i.e., the largest inhomogeneities in the universe may be, to date, unidentified as individual structures. The Great Wall, for example, and its full extent is still being assessed. Whenever a volume of space is sampled, "there always seems to be structure with a dimension comparable to that of the volume surveyed. This has led to considerations of fractal structures (identical forms repeated on ever-increasing scales) occurring in the universe. If this is correct, although one would gain a geometrical interpretation to the nature of the structures, it would make a physical explanation extremely difficult." (<a href="http://nedwww.ipac.caltech.edu/level5/ESSAYS/Fairall/fairall.html" style="color: #2288bb; text-decoration: none;" target="_blank">Source: Fairall</a>). The discovery of larger structures still would bolster the prediction that smaller structures (superclusters in this case) are gravitationally bound systems.<br />
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On scales where inhomogeneities in the mass distribution are observed, motions of clusters and superclusters are induced by mutual gravitational forces (or the local spacetime curvature) in proportion to the local gradients in the composite fields. If matter were distributed homogeneously locally no motion would be induced, since the gravitational forces would cancel on average. However, matter is not quite homogeneously distributed on scales the size of a supercluster.<br />
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Observations of the CMB and the large-scale structures indicate the universe is relatively homogeneous on scales in excess of 150 Mpc and up. The fluctuations become weaker the larger the scale. Weak fluctuations imply that the cluster distribution is very close to homogeneous. The larger the scale, the weaker the fluctuation, as homogeneity is approached.<br />
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By symmetry considerations (discussed below) we will se that the universe in its entirety is a stable system in dynamic equilibrium.<br />
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<div style="text-align: center;"><b>Mounting Evidence for Gravitationally Bound Superclusters</b> </div><br />
Direct evidence that superclusters rotate was obtained by Vauclouleurs (1958), when it was observed a high degree of flattening of the Local Supercluster. The link between flattening and orbital motion had been found by Oort with respect to the Milky Way many years prior to the suggestion by Vaucouleurs that the same principle holds for clusters and superclusters.<br />
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Indirect evidence was found by Reinhard and Roberts (1972) when it was observed that the orientation of the minor axis of spiral galaxies correlates with the direction of the poles corresponding to the flattening of the Local Supercluster. Said differently, the angular momentum vectors tend to coincide with the rotational axis of the Supercluster itself (again; <a href="http://adsabs.harvard.edu/abs/1974IAUS...58...85O" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a>).<br />
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A significant flattening of a large number of other superclusters supports the hypothesis of supercluster rotation, in addition to observed correlated orientations of galaxies along the direction of extension of superclusters, the nonrandom correlation between the main axis of radiosourses with the central supergiant cD-galaxies, and the rotation (and flattening) of clusters within superclusters: for example, the direction of rotation (indicated by the orbital planes of spiral galaxies within clusters) coincides with the apparent rotation of superclusters.<br />
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The shape of the Draco supercluster, a very isolated, extremely rich supercluster, resembles a pancake with axis ratios 1:4:5 (Einasto <i>et al</i> 1997 and references therein); precisely what would be expected of a rotating system.<br />
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Note, when using the word 'rotation' relative to superclusters (or even clusters), one should not imagine orbits like that of 3-body system (say, the earth-moon orbiting the sun), to be smooth, quasi-circular and of relatively constant velocity. These rotations or motions would be not be isolated.<br />
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After all, some are conglomerates roughly 10^15-10^17 solar masses (representing thousands of galaxies). Superclusters are typically separated by 100 Mpc, which means there are about 10 million superclusters in the visible section of the universe. From their size and mass inferences it has been calculated an average free-fall time of 40 billion years. According to the standard model the universe is only 13.7 Gyr, which implies that superclusters are dynamically young. A typical galaxy cluster will not have passed through the system for the first time (A. Jerkstrand, Superclusters).<br />
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Superclusters form part of a web-like network of extensive walls, supervoids, clouds and filaments, all interacting gravitationally with their neighbors in a vast and extensive series of local minima and maxima in the combined gravitational fields of an exceedingly complex N-body system (or systems) extending perhaps 100 Mpc or more. Even the regions of galaxy underdensity (supervoids) are nonspherical in nature. To give an idea of the scales we're talking about, consider that the Local Group, a loose aggregate of several dozen galaxies (including the Milky Way), resides within a volume of radius about 1 Mpc. The Local Group itself lies asymmetrically on the outskirts of a flattened structure with a radius of about 15 Mpc, known as the Local Supercluster, centered on a rich cluster of galaxies; the Virgo cluster.<br />
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To complicate the matter significantly, all of these systems are embedded in a manifold that is either expanding nonlinearly or of constant positive Gaussian curvature globally. Note the topologically complex structure pictured in the diagram below. No simple characterization describes the distribution of galaxies or superclusters:<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1RH-1CXYwFl3ZGZy782IiNzKQDzhXkDSAFOvfdoosPgJGqGwayYBhHJ0exRGNdWO0drt4_E-KH78gX95SILLhG98oThi9cDlNvnYoAfrefAN4LUOD9k56i1aAeQCeqA0-kIDSvTqDi7U/s1600/I03-02-attractor2+8cm150dpi+ok.jpg" imageanchor="1" style="color: #2288bb; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1RH-1CXYwFl3ZGZy782IiNzKQDzhXkDSAFOvfdoosPgJGqGwayYBhHJ0exRGNdWO0drt4_E-KH78gX95SILLhG98oThi9cDlNvnYoAfrefAN4LUOD9k56i1aAeQCeqA0-kIDSvTqDi7U/s320/I03-02-attractor2+8cm150dpi+ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; border-width: initial; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px; position: relative;" width="318" /></a></div><br />
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<div style="text-align: center;">(<a href="http://universe-review.ca/F03-supercluster.htm" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a>)</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFg8Pd2zq-PgLS3iTjDPXki7y4fXxjBP1Ct832VxGNbED56YjpMqHoteFJ2u2ca43-92Zk3Oh3kDIEa1T-CcWiyaaGpUY0oGGYhgyEpdipFI8lr-XYbUovriu3rxT6yWgnpdnQMpsseSY/s1600/Fig3b+10cm+150dpi+okk.jpg" imageanchor="1" style="color: #2288bb; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" height="301" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFg8Pd2zq-PgLS3iTjDPXki7y4fXxjBP1Ct832VxGNbED56YjpMqHoteFJ2u2ca43-92Zk3Oh3kDIEa1T-CcWiyaaGpUY0oGGYhgyEpdipFI8lr-XYbUovriu3rxT6yWgnpdnQMpsseSY/s320/Fig3b+10cm+150dpi+okk.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; border-width: initial; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px; position: relative;" width="320" /></a></div><br />
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Space density and velocity fields derived from the PSCz survey. The amplitude of the velocity vectors is on an arbitrary scale.<a href="http://icc.dur.ac.uk/index.php?content=Links/PSCz/PSCz" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a><br />
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It can be seen from these simulations that the density and velocity fields of N-body systems are very complex. There are no clear-cut orbits (as that of the earth around the sun). The point is that superclusters can be in motion relative to one another, close to dynamic equilibrium, without possessing a well defined orbital path or velocity relative to neighboring superclusters. Yet these large-scale structures may still be gravitationally bound systems.<br />
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The distribution of superclusters, according to Einasto <i>et al</i> (1997) is far from homogeneous. "Most of very rich superclusters are located along rods of a quasi-rectangular cubic lattice with almost constant step, and form elongated structures - chains. These chains are almost parallel to axes of the supergalactic coordinate. [...] Several data sets suggest that giant structures seen in the Southern and Northern sky may be connected, and superclusters form sheets or planes in supergalactic coordinates. One example of such connection is the Supergalactic Plane, which contains the Local Supercluster, the Coma Supercluster, the Pisces-Cetus and the Shapley superclusters [...] This aggregate separates two giant voids - the Northern and Southern Local supervoids." (Einasto et al 1997, The Supercluster-Void Network I, A&A Suppl. Ser. 123, 119-133).<br />
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Evidence that superclusters are connected clearly does rule out the possibility that these systems are gravitationally bound. Quite the contrary. These findings are consistent with the large-scale dynamics of a non-expanding universe, whereby all neighboring objects are gravitationally bound (and so too friends of friends). Tully <i>et al </i>(1992) had already described the the supercluster-void network as a 3-dimensional chessboard, due to the presence of superclusters nearly orthogonal to the Supergalactic plane. Einasto <i>et al </i>(1997) find that these structures (delineated by rich superclusters) are not only orthogonal but distributed quite regularly. Too, isolated clusters and poor superclusters are distributed very closely to rich superclusters, belonging to outlying portions of superclusters (i.e., they do not form a random population in voids). Superclusters form intertwined systems that appear regularly spaced. "Thus the mean separation of high-density regions across voids is almost identical for all observed samples." (Einasto et al 1997) The characteristic separation scale of rich superclusters resides around 120 h^-1 Mpc, corresponding to the distance between superclusters across the supervoids. Evidence that the supercluster-void network is regularly structured is indicated by the small scatter of this characteristic distance. Einasto <i>et al</i> also conclude that there exists no larger preffered scale in the universe, i.e., superclusters and supervoids should be the upper end of the hierarchy of galaxy distribution. Rich superclusters are often found in pairs (e.g., Fornax-Eridanus and the Caelum superclusters, the supercluster in the Aquarius complex and others), a surprising feature, for any expanding universe model. The pairing of gravitationally bound objects is observed across a very wide range of scales, from planets and satellites to binary stars on up to superclusters.<br />
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This manifestation of hierarchical (ordered) structure from galactic scales up to the scale of superclusters would indeed be a very surprising feature in a universe blowing apart at the seams, where one would expect a discontinuity for systems not gravitationally bound.<br />
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In fact, the ordered structure is observed in simple two- and three-body systems. Arguably, the classic geometric configurations discovered by Lagrange operational on scales compatible with planets and satellites can be observed on scales of the largest known structures. See for example Coldcreation 2008 <a href="http://scienceforums.com/topic/14217-physical-mechanism-of-gravity-the-spatiotemporal-ground-state/" style="color: #2288bb; text-decoration: none;" target="_blank">The Physical Mechanism of Gravity - The Spatiotemporal Ground-State</a>.<br />
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<a href="http://scienceforums.com/topic/13839-dynamic-equilibrium-of-the-universe-and-subsystems/" style="color: #2288bb; text-decoration: none;" target="_blank">Dynamic Equilibrium of the universe and Subsystems, 2008</a>: In Théorie des fonctions analytiques (1796), more than one hundred years before Albert Einstein and Hermann Minkowski, Joseph-Louis Lagrange referred to dynamics as a “four-dimensional geometry.” Without doubt, such illustrations aimed at their audiences, with physical intent, were certainly not geometric in any ordinary sense; but unquestionably they were geometric in their concern with planimetric space of gravitating systems, and in the fundamentals of their relationship with matter. He believed now another basic factor was to be understood and exploited: space. He saw the understanding of space as a legacy left by the invention of Newton. In Lagrange’s system imbued with the notion of dynamic continuity, is stressed the importance of unbroken rhythms and completed movements by circular field lines; now with the spatial factor introduced.<br />
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It is difficult to overestimate the importance of the Lagrange discovery. Not only does it confine the number of possible structures that can exist, it demonstrates a regularity and pattern among those systems that do exist. The Lagrange system reveals how interacting gravitational fields of massive bodies generate periodicity, just as the Pauli principle generates the periodicity among atomic elements in the case of electrons orbiting atomic nuclei—as in the Mendeleev periodic table of elements—and the systematic pattern among quark clusters that represent the subsistence of a profound layer of reality on the smallest scales. (The forces that cluster quarks together are not yet fully understood but some of the patterns and features have already been identified). Our ability to recognize that Nature forms regular patterns at all scales (from quarks to superclusters) and limits the number of available structures, rather than giving way to disorganized chaos, is essential if we are to make any progress at all in cosmology.<br />
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<div style="text-align: center;"><b>Limits of the Cosmological Principle</b></div><br />
On the scales of superclusters the cosmological principle does not hold. Deviations from pure Hubble expansion are expected, just as deviations from constant Gaussian curvature would be expected in the static model. Nevertheless, for convenience, it can be argued that any observer can consider the universe as homogeneous and isotropic on sufficiently large scales. There is no preferred spatiotemporal location or epoch.<br />
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With increasing scale galaxy-clustering and cluster-clustering become weaker. At some scale the correlation function (characterizing the distribution of galaxies) tends to zero. On scales of superclusters, it has been found a series of almost regularly spaced maxima and minima, corresponding to superclusters and supervoids (around 115 h^-1 Mpc). The correlation function of galaxy clusters, on large scales, depends on the geometry of the distribution of superclusters. The correlation function has an oscillatory behavior when superclusters are distributed quasi-regularly. (Einasto et al, 1997, The supercluster-void network III. The correlation function as a geometrical statistic).<br />
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<div style="text-align: center;"><b>Redshift Surveys and the Large-scale Structure</b></div><br />
The web-like (or sponge-like) structure of the cluster-void network was discovered when the first 3D maps of the universe were made in the late 1970's based on galaxy redshift surveys. In 1976 Stephen A. Gregory and Laird A. Thompson decided to create a 3D map of a large volume of space in a slice across the sky that stretched 21 degrees from the Coma cluster to the cluster Abell 1367. They hypothesized that if the two galaxy clusters (Coma and Abell 1367) are members of the same supercluster, the 21 degree span between them should be filled with galaxies that bridge the gap between the two clusters. Gregory and Thompson measured the redshifts at Kitt Peak National Observatory, made a 3D map, and were astonished to find that not only was their hypothesis confirmed regarding the "bridge of galaxies connecting Coma and Abell 1367, but more importantly, the galaxy distribution within the entire 21 degree slice of the sky was very filamentary with large empty regions of space throughout the survey volume." (<a href="https://netfiles.uiuc.edu/lthomps/www/webpage-2010/void.html" style="color: #2288bb; text-decoration: none;" target="_blank">see Gregory S. and Thompson L.</a>). Astrophysical Journal published their work in June 1978. In this manuscript, Gregory and Thompson introduced the word "voids" to describe (for the first time) the large empty regions seen in the 3D redshift map. Based on previous studies, Gerard de Vaucouleurs properly suggested that the spatial distribution of the local galaxy distribution (out to a redshift of about 2000 km/s) was quite irregular. But the true nature of the supercluster-void network wouldn't become clear until large-scale surveys were projected onto a 3D map.<br />
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Gregory and Thompson wrote in their 1978 manuscript: "It is an important challenge for any cosmological model to explain the origin of these vast, apparently empty regions of space. There are two possibilities: (1) the regions are truly empty, or (2) the mass in these regions is in some form other than bright galaxies. In the first case, severe constraints will be placed on theories of galaxy formation because it requires a careful (and perhaps impossible) choice of both omega (present mass density/closure density) and the spectrum of initial irregularities in order to grow such large density irregularities. If the second case is correct, then matter might be present in the form of faint galaxies, and an explanation would have to be sought for the peculiar nature of the luminosity function."<br />
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Redshift surveys are important in that the measurement of peculiar velocities (and bulk motion) allows the reconstruction of the underlying density field, which can be compared with that derived from the distribution of the visible galaxies. Measurements to date suggest that either (a) there exists a large component of invisible cold dark matter and that superclusters are not gravitationally bound systems (in the expanding case), or that (b) clusters and superclusters are gravitationally bound and that these large-scale structures are moving relative to one another (in the static case).<br />
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In the latter scenario, where superclusters are gravitationally bound systems, there should be observed interrelated connections between superclusters (e.g., in the form of luminous bridges of interconnected clusters).<br />
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In the past few decades large galaxy redshift surveys have revealed that there are connections between superclusters. In 2005, Proust <i>et al </i>(Structure and dynamics of the Shapley Supercluster, A&A 447, 133–144 (2006), DOI: 10.1051/0004-6361:20052838, ESO 2006) present results of our wide-field redshift survey of galaxies in a 285 square degree region of the Shapley Supercluster (SSC), based on a set of 10,529 velocity measurements (including 1,201 new ones) on 8,632 galaxies obtained from various telescopes and from the literature. The data reveals that the main plane of the SSC extends further than previous estimates, filling the whole extent of our survey region of 12 degrees by 30 degrees on the sky. There is also a connecting structure associated with the slightly nearer Abell 3571 cluster complex. These galaxies seem to link two previously identified sheets of galaxies and establish a connection with a third one. The remarkably rich Shapley Supercluster is one of the most massive agglomerations of galaxies in the local universe. Background structure appear to be linked to the SSC by filaments (see Figure 4 of Proust et al 2006). Too, superclusters SCL146 and SCL266 seem to be associated with radial extensions of the SSC. The Shapley Supercluster is clearly linked to other huge superstructures, as shown in the Figs. 9 and 10 of Jones et al. (2004, MNRAS, 355, 747). Apart from the radial connection to the Hydra-Centaurus complex, a tangent bridge of galaxies extends in the direction of the Sextans supercluster. The inferred mass of the Shapley supercluster is large enough to effect gravitationally the observed motion of the Local Group. Its morphology is generally flat, and extends further than previously estimated, linking the Hydra-Centaurus foreground region (D. Proust, et al, 2006).<br />
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The fact that superclusters tend to be grouped together, linked by filaments and walls forming a weaved fabric throughout space, suggests that superclusters are indeed gravitationally bound systems moving relative to one another in the static universe.<br />
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Bacall (1990, ASPC, 21, 281B) finds a weak correlation for r=100-150 Mpc, suggesting superclusters may be themselves grouped together with a characteristic separation of this scale.<br />
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Superclusters are not completely isolated in vacuo (Einasto et al 1997, The Supercluster-Void Network I, A&A Suppl. Ser. 123, 119-133), separated by voids or supervoids, as could be expected in an expanding universe where superclusters are not gravitationally bound systems. Neighboring superclusters are connected by galaxy and cluster filaments forming a single network (Einasto et al 1997), just as would be expected in a static universe where superclusters are bound gravitationally.<br />
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Recently, it was suggested that the relative orientation of neighboring clusters within superclusters reflects an underlying formation mechanism. Numerical simulations have demonstrated that clusters formation is intrinsically connected with supercluster systems that characterize the large-scale structure of the universe. There has been found a correlation between the relative orientation (parallel or filamentary alignment along the major elongation axes) of neighboring clusters and the angular momenta of clusters. Major elongation axes and the lines pointing towards neighboring clusters shows a strong deviation from random orientation (Faltenbacher et al, 2002, Correlations in the orientations of galaxy clusters). Angular momentum of ellipsoidal systems tend to align with minor the axis. Closely neighboring cluster pairs tend to have similar or higher angular momenta compared to clusters further removed (or to the global average).<br />
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The observed distribution and dynamics of matter in space may hold the key to understanding the essence of the physical universe and its evolution in time. The large-scale structures and spatial distribution can place important constraints on formation and evolution as function of time. That is, the spatiotemporal distribution of galaxies, groups of galaxies, clusters and superclusters within the bubble-like networks of sheets, filaments, walls, voids and supervoids (along with spectrographic and morphological properties) place important constraints on how these systems formed, and can be used to test the viability of cosmological models, whether static or unstable. Much progress has been made but much has yet to be explored.<br />
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The evidence to date relative to the kinematics of the large-scale structures seems to suggest that at least a few (if not all) superclusters are gravitationally bound systems. Observations both direct and indirect do not contradict this possibility. Again, in the static case these large-scale structures (some in excess of 100 Mpc) have had ample time to reach dynamic equilibrium (without the need of cold dark matter).<br />
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Interestingly, the CMB measurements imply density fluctuations of 0.001%. According to theory, a characteristic time for the gravitational settling at the center of a clump where the density fluctuation is 1.7% would be 1 billion years. A fluctuation of 0.3% would require 13 Gyr. And for fluctuations consistent with the CMB (0.001%) the settling time would be 1000 times greater than the supposed age of the universe. Only by adding cold dark nonbaryonic matter to enhance the fluctuation can the gross inconsistency be appeased. In 2004 (<a href="http://astro.berkeley.edu/~mwhite/modelcmp.html" style="color: #2288bb; text-decoration: none;" target="_blank">Large Structure Formation, Comparison of Models</a>) several measurements of galaxies and clusters deep in the look-back time (in the "early" universe) indicate that the structures are larger than predicted by the standard dark-energy cosmology (lambda-CDM). The problem revolves around the inability of a dark-energy dominated universe to create such large structures within such a short time; 1/5 of the present age. (<a href="http://universe-review.ca/F03-supercluster.htm" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a>).<br />
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Further observations (with the next generation of telescopes) may refute the dark-energy model if large-scale structures are observed as far back in time as our telescopes can see (i.e., more research will likely refute the dark energy model. Obviously the prediction here is that superclusters are present at all redshift distances; all the way to the horizon. :)<br />
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<div style="text-align: center;"><b>Time-scales and the large structures</b></div><br />
So far, observations appear to show large-scale structuring at High-redshift. For example, high-z quasars and radiosources show clustering above correlation levels of galaxies, which means large-scale structures exist at early times, in sharp contrast with standard model predictions. Strong indications for the existence of both clusters and superclusters at high-z have been known for several decades. Quasar clustering had already been quantified in 1988 (Iovino and Shaver). This was shown to be associated with superclustering by Bacall and Choksi (ASPC, 21, 281B). It was found (West, 1991, ApJ, 379, 19W) that radio galaxies, in addition to clustering, show spin axis alignment pointing in the direction of nearby radio galaxies and quasars. Pencil-beam surveys at intermediate redshift (z < 2) indicate that superclustering is not a recent development. At redshift z = 2.38 a 100 Mpc long string of galaxies was observed, in sharp contrast, too, with time-scales considered in hierarchical computer simulations. Observations of large-scale structure at very high-z is not compatible with the time-scales typically required by ΛCDM model. The universe at early times is simply not old enough to account for observations (A. Jerkstrand). This insufficient time-scale would be a problem for both the hierarchical and monolithic models of formation.<br />
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Large-scale structures are potentially very old. Tully (1886) suggests that the large dimensions of these objects (or groups of objects), some possibly in excess of 300 Mpc, must have taken a hundred billion years to form. Tully finds that the entire Local Supercluster conglomeration has an elongated dimension of at least 386 Mpc (for H0 = 70 km s−1 Mpc−1. A similarly large dimension is also found for the sheet-like configuration attached to an agglomeration of superclusters in the region of Aquarius (Tully, 1986 Astrophys. J. 303, 25).<br />
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These elongated cellular walls formed by supercluster complexes are perhaps indeed ancient, as suggested too by Lerner (1991). His conclusion is that these cellular walls are at least one hundred billion years old. (Lerner, E.J., The Big Bang Never Happened). (See too Hoyle, F., Burbidge, G., Narlikar, J.V., 1997, Mon. Not. R. Astron. Soc. 286, 173).<br />
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All high redshift surveys indicate that large-scale structures are present at great distances in the look-back time. These structures either formed exceedingly rapidly after the big bang (even though expansion dilutes density fluctuations, thereby the ability to cluster), or they've been around for much longer than currently suspected. No doubt, I'm banking on the latter. Forthcoming high-z data in the next decade should rule out one model or the other.<br />
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The existence of rotation (or bulk motion) on scales compatible with superclusters cannot be ignored by any model of structure formation. Indeed, data regarding the dynamical age of the Local Supercluster (and others) present serious difficulties for the standard model, with its hierarchical model of galaxy formation from primeval density fluctuations (a 'bottom-up' scheme). In fact, the existence of the observed structures, from their size alone, seems to rule out the standard model, i.e., their hasn't been enough time since "creation" for such large-scale structures to have formed (gravitationally bound systems or not) from minute density clumps in the Radiation Era which would have acted as seeds for the growth of galaxies early in the Matter Era.<br />
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Clearly, the large-scale structures including the supervoids (not just locally but at great distances in the look-back time) are far more advanced than predicted by the concordance model (ΛCDM).<br />
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Cosmic Structure Formation: Hierarchical and monolithic collapse vs the top-down model in a static universe<br />
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One of the beauties of the static model of constant Gaussian curvature, contrarily to the standard hierarchical model or the picture of monolithic collapse, is that the formation of the large-scale structures and galaxies themselves occurs over very long time periods and can do from the top-down. In other words, the density fluctuations from which individual galaxies and clusters are formed cover vast regions of space (perhaps 100 Mpc or more). These density fluctuations are on scales of superclusters. The over-dense regions would splinter into smaller substructures consistent with cluster scales, and subsequently individual galaxies would form with well defined spatial orientations relative to neighboring galaxies, the clusters themselves, and the parent superclusters. The location, orientation, morphology, and dynamics of galaxies (from the early formation process onwards) are all related to the topological shape and size of the supercluster denstity fluctuation, as well as the relative proximity of neighboring superclusters, which can affect all of the above properties. The fact that (i) planes of galaxies tend to be lined up perpendicularly with the primary plane of the parent supercluster, and that (ii) projections of rotational axes on the primary plane of superclusters are oriented toward the primary structure, and that both the above can be interpreted as perpendicularity of the galaxies plane to the radius vector, lend support to the top-down scenario; where supercluster density fluctuations are formed prior to those of galaxies (P. Flin, 1996, <a href="http://www.informaworld.com/smpp/content~db=all?content=10.1080/10556799608203023" style="color: #2288bb; text-decoration: none;" target="_blank">The Alignment of Galaxies in Superclusters</a>, Astronomical & Astrophysical Transactions, 10:2, 153 - 159). Certainly there is little evidence, if any, that would suggest otherwise.<br />
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The point is that density fluctuations occur on a variety of scales. The smaller fluctuations that lead to the formation or stars and planets and satellites, star clusters, galaxies and galaxy clusters, are superimposed on the largest fluctuations, i.e., these smaller fluctuations form part of the larger ones from the outset. Superclusters are relatively old objects from which galaxies and clusters are formed.<br />
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Resulting from shocks generated by supersonic motion and adiabatic compression during virialization and shell collapse a gas (in the form predominantly of hydrogen) permeating the superclusters gravitational potential well is heated and ionized and emits through bremsstrahlung in the X-ray band, consistent with observations. Baryonic and leptonic matter represents 100% of the mass content of the universe; much more in the form of baryonic dark matter than currently suggested.<br />
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[Note: It's worth pointing out that it is precisely this combined total mass-energy density of the universe to which is attributed the globally homogeneous gravitational field of constant positive Gaussian curvature].<br />
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Coherent orientation effects from the scale of superclusters to that of galaxies support the contention that superclusters are gravitationally bound systems within which clusters, smaller groups (such as the Local Group) and individual galaxies form via the top-down scenario.<br />
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Certainly there are characteristic features that will overlap between the two scenarios (the top down and the bottom up). That is because primordial density fluctuations come is all sizes up to a scale beyond which the universe tends toward homogeneity and isotropy. Within large overdense regions (compatible with the scales of 150-200 Mpc or more) of the underlying matter field at very early times there are, too, overdense regions on scales all the way down to individual atoms. Gravitational influences due to the large-scale fluctuations of the matter field impel individual atom to move closer together, thus overdensity increases across a wide range of scales as a function of time. This accretion process responsible for the formation material structures is simultaneously responsible for the formation of voids. As some areas of the primordial substratum condense under gravitational influences, other areas become underdense. The less dense these regions become, the less accretion will occur. Eventually, most of the material in these regions will have migrated gravitationally towards denser areas. According to this scenario of gravitational instability for the evolution of the observed cosmic structures the universe (perhaps hundreds or thousands of billion years ago) was practically smooth, except for spatially extensive density variations (which were small in amplitude) with respect to the overall background. The current size and mass of superclusters observed today is directly related to the scale of the fluctuations. The scales and variations in amplitude of the observed thermal CMB blackbody spectrum would mimic the large-scale distribution of these density fluctuations. The CMB would be (not a relic of a hot dense phase) a product of baryonic matter and its spatial distribution in time. The thermalization of the blackbody spectrum in a static universe will be discussed in a separate dedicated post, as well as the formation of hydrogen and the other elements.<br />
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It is also predicted by some hierarchical models (Plionis, 1994) the alignment of galaxy clusters with their closest companions (since these are gravitationally bound) as well as with other clusters that reside within the same supersluster. Alignment is suspected to occur as a result of a property of Gaussian random fields which interact between density fluctuations of various scales (West et al 1991, Faltenbacher et al 2002, and others). Flatter systems tend to show lower velocity dispersions (the range of velocities about the mean velocity). However the dynamical evolution of clusters and the intergalactic medium, along with post-merging relaxation time corresponding to clusters remain open issues, i.e., the existence of substructure does not imply that clusters are dynamically young (Plionis 2001, and references given therein).<br />
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There are primary difference between the two views on the origin of cosmic structures: The Coldcreation top-down view is that superclusters are not young cosmic objects that formed after the merging of smaller systems (cannibalism), whereby larger systems form from smaller ones, and the baryonic matter does not follow some other dominant specie of matter (CDM) during the collapse phase. In other words, density perturbations on stellar and galactic scale do not collapse to form stars and galaxies first, then subsequently merge to form galaxy systems followed by the large-scale filamentary superstructures. Nor do galaxies form by monolithic collapse in a relatively recent given epoch in cosmic history (the "epoch of galaxy formation" just after the "dark age"). The process is long, drawn-out and ongoing.<br />
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Star formation is thought to be triggered primarily by violent merging events, where by the gravitational fields of dense cluster regions vary rapidly. Active galaxies usually reside in high-density environments, so merging may not be the sole contribution to star formation activity. An interesting feature is that increased cluster sphericity appears to be associated with rich massive systems where the velocity dispersion is large. The increase in cluster velocity dispersion is larger for systems that are further removed from one another (Ragone et al 2004). In analogy to galaxy formation from larger density fluctuations associated with protoclusters, individual stars are formed in regions of larger density fluctuations associated with protogalaxies. So too are objects such as planets formed from larger fluctuations consistent, in this case, with the scale of the protoplanetary disc itself, and fragmentations thereof.<br />
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It should be noted that a perfectly smooth homogeneous background was never an option. The very early universe, whether expanding or not, whether infinite or not, whether formation of the structures occurred from the top-down or the bottom-up, could not have been perfectly homogeneous and isotropic. This would have been so from the large-scale all the way up to the smallest. It could not be said that that the universe was inhomogeneous on one scale and not the other. So what we really have is both a top-down and a bottom-up formation process which occurs relatively simultaneously. The important thing to recall is that gravitational interactions are not instantaneous. It takes a relatively long time for a large-scale fluctuations to produce significant contraction on scales compatible with components that would form stars and galaxies in a static universe. But it would take a lot longer to produce the observed large-scale structures (from the bottom-up or top-down) in an expanding universe.<br />
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Notice the evolution pictured in the simulation below:<br />
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</div><div style="text-align: center;"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIWbMZPwoXazt5rKtOSMECnlAeDIyQN2bsQNWylXpvTnNhGjMXBFaOsDInbrQGAg952xkKHLVRqieMCfaaRTEI8zDCPRiQekZnCW7jU-YXU_W6HT7-quwo_rW1i4H6iRHNhFvuBA45X0Q/s1600/Formation+of+the+large-scale+structures+13cm150dpi+ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; -webkit-user-select: none; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></div><div style="text-align: center;"><br />
</div><div style="text-align: center;"><b>Figure FCLSF</b></div><br />
The formation of clusters and large-scale filaments in the Cold Dark Matter model with dark energy. The frames show the evolution of structures in a 43 million parsecs (or 140 million light years) box from redshift of 30 to the present epoch, corresponding to redshift in accord with the Hubble law (upper left z=30 to lower right z=0). <a href="http://cosmicweb.uchicago.edu/filaments.html" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a>.<br />
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And here is the first frame modified by Coldcreation:<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgnCdr3ai8S5Nn019iHxMUNMTzhShdbKaMKbX2LuLdxl4QAjR9aHtLe0on7JJGBfRkRNkmctETX2S3vMLg6_xSSYLCBBDT7rPoR93OSncJKIg23hrgfBKXLAJkvqE5rgQL00iEE_v_k5w/s1600/Formation+of+the+large-scale+structures+1+Detail+4cm150dpi+ok.jpg" imageanchor="1" style="color: #2288bb; margin-left: 1em; margin-right: 1em; text-decoration: none;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgnCdr3ai8S5Nn019iHxMUNMTzhShdbKaMKbX2LuLdxl4QAjR9aHtLe0on7JJGBfRkRNkmctETX2S3vMLg6_xSSYLCBBDT7rPoR93OSncJKIg23hrgfBKXLAJkvqE5rgQL00iEE_v_k5w/s1600/Formation+of+the+large-scale+structures+1+Detail+4cm150dpi+ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; border-width: initial; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px; position: relative;" /></a></div><br />
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I modified the first frame because it showed a homogeneous distribution of hot-spots. Notice in the original frame (top left) the color of the background is identical to the color of condensed areas of the other frames. In a static universe these hot-spots, or condensed areas are not cold dark matter, but galaxies, clusters and superclusters. What we see now in the first frame, with its modification, is the absence of hot-spots; there are no stars or galaxies yet. But what we see too is that the background is not perfectly smooth. There are inhomogeneities (over- and underdense regions) on all scales. The large-scale inhomogeneities are already present. The same density fluctuations present in the first modified frame are also present in the last (something that would not be observed in an expanding hierarchical model where galaxies merge to form clusters and subsequently superclusters). These large-scale fluctuations will determine where and how galaxies are located and oriented relative to others. The background in the modified frame consists of a very cool hydrogen gas (the origin of which will be discussed in a subsequent post). With time, the overdense regions become denser, and the underdense regions less dense. As the process of accretion continues with time the structures move closer toward dynamic equilibrium. Though just as a perfectly smooth homogeneous background is unattainable in the past, so too is a perfect dynamic equilibrium unattainable in the future. [Note, this simulation was performed with dark matter in mind, but one can just as well picture these structures as being composed of ordinary baryonic matter.]<br />
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<div style="text-align: center;"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgl4zjUV2A62x0zgpCXt_iOghlJuF0UWcIVCuqV40xV3ooG_c2Nxx6VDkDnO641YkRUbra6_KqQF4pA5O9DF6VXp9ngJArKMVJPHFXfZTpA_VYzZzKVvKXXplmnfpzY60x-NoojdQLizj0/s1600/evolution+13cm150dpi+ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; -webkit-user-select: none; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /></div><br />
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</div>The above simulation (<span class="Apple-style-span" style="line-height: normal;"><a href="http://icc.dur.ac.uk/index.php?content=Research/Topics/O6" style="color: #2288bb; text-decoration: none;" target="_blank">Source</a></span>) confirms the scenario presented here. The original large-scale density fluctuations within which galaxies are embedded at later times are present from the outset. The change in size of these fluctuations over time is negligible (there is no expansion in this case), but the density within both the voids and condensing regions does change slowly with time, relative to the mean density. Galaxies are formed within the density fluctuations present at very large physical scales, in contrast to the hierarchical model.<br />
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<div style="text-align: center;"><b>Top down model of galaxy formation in a static universe</b></div><br />
The top-down scenario proposed here is also in contrast to the monolithic collapse saga (a kind of <i>wham-bam thank you ma'am</i> approach) since the latter postulates that all galaxies were formed by gravitational collapse of a primordial gas cloud in a single event shortly (in relative terms) after the big bang. This is far more inappropriate than casual serendipitous <i>tidal encounters</i>, <i>ram pressure stripping</i> or <i>galaxy harassment</i>.<br />
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Large galaxies located at the cluster core need not form by merging or cannibalistic processes (pathological colliding galaxies). The location, size and densities of large active galaxies (along with the rate at which they will evolve) depends upon favorable conditions present within the large-scale Gaussian density fluctuations. These are not linearly uniform spherically symmetric fluctuations that cause the free-fall of galaxies towards the central massive dominant galaxy of a given cluster. Nor do clusters ultimately collapse for similar reasons. Just as the rotational characteristics of spiral and elliptical galaxies differ, so too the rotational characteristics of clusters and superclusters differ. While some clusters and supercluster exhibit organized rotational structure, others will be dominated by random motion. The stability of superclusters relative to themselves and adjacent structures is not put into question, since their motions (orbital or random), implied by morphologically flattened structure, would be sufficient for the maintenance of stability. The typical dispersion velocity is consistently fitted by a Gaussian distribution for systems in dynamic equilibrium. For poor clusters (20-30 members) the velocity dispersion is on the order of 500 kms^-1, while for rich systems, such as the Virgo cluster (about 2500 galaxies), the typical velocity distribution is about 1000 kms^-1 (Popesso, 2006, <a href="http://www.imprs-astro.mpg.de/Alumni/2005_Popesso_Paola.pdf" style="color: #2288bb; text-decoration: none;" target="_blank">The RASS-SDSS Galaxy Cluster Survey</a>. Correlating X-ray and optical properties of Galaxy Clusters).<br />
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So far, all observational attempts to distinguish between the two competing big bang models for the formation of galaxies and the large-scale structures have failed. The question of how galaxies form and evolve with time thus remains one of the most important unanswered questions of contemporary astrophysics (Popesso 2006).<br />
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Because superclusters are often large (up to 10% of the horizon scale) they are extremely difficult to study statistically. The answers to the questions of how superclusters form, how their morphology evolves, multiplicity (richness), sizes and structures evolve as a function of time, are not yet known (<a href="http://iopscience.iop.org/0004-637X/652/2/907/64788.text.html" style="color: #2288bb; text-decoration: none;" target="_blank">Wray and Bahcall</a>, et al, 2006). The lack of statistical data regarding these large-scale objects makes it difficult to determine whether they are gravitationally bound systems or not. Increasing the linking length in numerical simulations often causes neighboring superclusters to be connected (joined into one), resulting in many large, complex structures with central cores joining multiple lower density filaments. Long linking length find higher percentages of of large rich superclusters. Conversely, shorter linking lengths show small superclusters with low multiplicity. At early times, in accord with the hierarchical model (wherein clusters migrate toward one another gravitationally over time) there is a reduction in supercluster density. With increasing redshift there is a significant drop in the simulated number density of superclusters. Abundance of rich superclusters drops off very quickly. (Wray and Bahcall 2006). The problem, at the telescope, is that the resolution of high-z clusters drops off with distance, and the identification of these faint distant clusters as members of superclusters becomes more and more difficult with increasing z, until they become unidentifiable (very faint galaxies cannot be observed). Massive clusters are more easily identified in high-z surveys, yet they will appear less rich with increasing z. Generally thin filamentary superstructures will tend to be undetectable too with higher z. This drop-off with distance can easily mimic significant evolution in the look-back time. One should not be misled in this respect when statistical uncertainties are so large.<br />
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In short, it seems rather natural, given sufficient time scales, that large-scale fluctuations lead to the formation of small-scale structures such as galaxies. Density fluctuations of characteristic scale between 100 - 200 h^-1 Mpc are related to superclusters and supervoids. Short density fluctuations of wavelength consistent with several Mpc give rise to the formation of small galaxy systems and individual galaxies, interspersed by small voids. Intermediate scale density fluctuations give rise to galaxy clusters and voids. Perturbations larger than superclcusters and supervoids (if indeed they exist, and they likely do) are of much lower amplitude, and therefor only modulate densities and masses of smaller gravitating systems (Frisch et al 1995, A&A 296, 611). Thus, as scale increases, the amplitude of the perturbation decreases. Taking this to the extreme, we could conjecture: <i>as scale tends to infinity, the fluctuation amplitude(s) tends to zero</i>.<br />
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Properties derived from large-scale high resolution cosmological simulations (alternatively with static coordinates in place of comoving coordinates) and more sophisticated high resolution observations to be carried out in the coming years should provide direct information on the physical size, richness, morphology and evolution of superclusters, along with testable predictions that can distinguish between static and expanding models; between models that describe these systems as dynamically and gravitationally bound conglomerations and models which do not.<br />
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Just as one would expect observational evidence to support the conclusion that superclusters are gravitationally bound systems in a static universe, one would expect supervoids and voids to form hierarchical systems resembling the hierarchy formed by superclsuters, galaxy clusters and galaxies. We would also expect, in a static universe, an upper limit of this void hierarchy to be compatible with scales of the largest material structures (high-density regions, or supervoids).<br />
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<div style="text-align: center;"><b>Voids and Supervoids</b></div><br />
What is the role played by voids and supervoids in the cosmic dance? How can the structure and dynamics of supervoids be used to distinguish between competing cosmologies?<br />
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Supervoids are large underdense voids (with diameters of about 100 h^-1 Mpc) interspersed with superclusters, defined as low-density regions of the universe that do not contain rich galaxy clusters, though they can contain poor clusters and small galaxy systems. They are completely devoid of certain types of objects, e.g., galaxies of a specified morphological type or luminosity limit, or rich or poor galaxy clusters. These are the largest known voids in the universe, and they are connected in a regular chess board-like structural network (Linder et al 1997). In other words, supervoids are not necessarily completely empty, nor are they isolated. The size and properties of supervoids, along with smaller lowdensity regions (voids) and void-walls are related, i.e., voids form hierarchical systems that depend on the large-scale environment, void diameter and the luminosity of galaxies contained within. The presence of Multi-branching systems in the galaxy distribution appear to be the principle mechanism that creates the hierarchy in the void and supervoid distribution (see Linder et al 1995, The Structure of Supervoids, Astron. Astrophys. 301, 329-347).<br />
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Recall that de Vaucouleurs (1970) had advocated the presence of a hierarchy of galactic systems. That voids, too, form hierarchical structures has been confirmed since. Supervoids may be divided into a network of faint galaxy systems and smaller voids (3 - 10 time smaller than supervoides, which may exceed 100 Mpc). Comparing structural differences (if any) of galaxies found in high- and lowdensity regions may provide insight on galaxy formation and the dependence on environment. The walls interspersed between superclusters consist of numerous smaller poorer galaxy systems that appear structured, forming a thin web of filaments. These are not large structureless clouds of galaxies, or a smooth distribution of isolated galaxies. The size, morphology and luminosity of galaxies (from poor systems to rich superclusters) are related to the environmental properties of the system within which they reside (Linder et al 1995).<br />
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Note that <i>hierarchical structuring</i> is not confirmation of the <i>hierarchical model</i> (the bottom-up scenario) of galaxy formation. It simply means that populations of clusters, voids, superclusters and supervoids are not randomly distributed. For both high- and lowdensity regions there is observed the presence of a fine structured network of system consisting alternatively of both poor and rich clusters. The void-filament structure is observed on a wide variety of physical scales, up to at least 130 Mpc. Voids located in lowdensity environments of superclusters are larger than voids located in highdensity environments. Any realistic galaxy formation scenario must explain these observed properties (Linder et al 1995).<br />
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One of the current views is that while overdense regions of superclusters continue to collapse, the underdense regions interspersed amongst them, supervoids, continue to grow, becoming increasingly underdense and larger with time. These superclusters and supervoids are, respectively, undergoing collapse and expansion. One consequence of this interpretation is that superclusters cannot be gravitationally bound systems; a property that distinguishes them for their constituent cluster counterparts. The physical scale of voids that separate these constituent clusters remains constant in time (since cluster are bound gravitationally), or changes insignificantly.<br />
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The empirical evidence is consistent with the scenario that large-scale (primordial) density fluctuations (maxima and minima on scales of superclusters and supervoids) splinter in course of time to form clusters and subvoids at later epochs, i.e., there was less substructure at early epochs than observed tonight. The existence of (i) hierarchical structure on scales compatible with superclusters and supervoids, (ii) the morphology and physical distribution of objects within the supercluster-supervoid network, and (iii) the existence of faint structures inside supervoids, supports the conclusion that superclusters are gravitationally bound systems in a non-expanding universe, i.e., observations are consistent with the dynamics and evolution of structures expected in a static universe.<br />
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The average physical scale of large underdense supervoid regions is about 100 <i>h</i>^1 Mpc; a scale above which the universe approaches homogeneity (Granett et al 2008).<br />
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The distribution of these large-scale structures affects the cosmic microwave background (CMB) in both expanding models and static models, by heating or cooling photons as they travel through over-and underdense regions; a phenomenon described as a late-time integrated Sachs-Wolfe effect (ISW). The effect on the CMB is small variations (anisotropies) in temperature between regions of corresponding to 100 Mpc. The consensus in the relevant literature is that the presence of dark energy (DE) in a geometrically flat ΛCDM universe induces the ISW effect on the CMB via the decay of gravitational potentials as the DE dominated universe expands and accelerates.<br />
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It's of particular interest to note that the presence of hot spots and cold spots in the CMB correspond to regions of over- and underdensity (superclusters and supervoids). This is exactly what would be predicted in a static universe where the CMB is produced by hydrogen burning stars over time-scales that dwarf the suspected age of the universe. (The physical mechanism operating that thermalizes the radiation released through hydrogen burning will be discussed in a forthcoming post). The reason is straightforward: in vast underdense regions there are fewer stars emitting radiation, thus the thermal spectrum of these areas appears cooler, and visa versa for the dense regions.<br />
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This is exactly what would be predicted, too, in a static universe where the top-down scenario for galaxy formation is operational. The large-scale structures form from large-scale density fluctuations which determine the morphology, orientation and location of galaxies and clusters.<br />
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Detecting the ISW effect in the framework of a static universe of constant positive Gaussian curvature has nothing to do with dark energy. By definition, hot and cold spots are present throughout the cosmos in overdense and underdense regions respectively, though observing such anisotropies at high-redshift would remain a daunting task. Low- to moderate-redshift superclusters and supervoids would dominate the spectrum, explaining most of the thermal fluctuations observed on the CMB. Foreground objects skew background fluctuations.<br />
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The larger the light-crossing time the greater the deviation from smoothness. On scales of clusters and voids such an effect would be present but smaller. Photons passing though regions of high and low gravitational potentials (through deep gravitational wells and areas of negligible local curvature) become slightly hotter or colder. The larger the system and the deeper the well, the greater the observed correlation between galaxy catalogs and the CMB anisotropies. Local effects should be extracted in order to reveal the imprints of individual superclusters and supervoids at significant cosmological distances. Without local signal extraction (or averaging) the CMB fluctuations will appear larger than superclusters and supervoids.<br />
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<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjF2_ohBB8dCQG4hFyRMJx1-W6Ut1cJbpOFzd2Yit04ch7xgZWJmWpjUICw-QWw1ZKD1Pl3ug8NXG3iCCBZcL3CoLYuux_ZCB_2BpTz4MtN0pbdRiq9VqYSeeaHuBPOCaBpo7ODKmU5dvo/s1600/A+map+of+the+microwave+sky+over+the+SDSS+area+15cm+150dpi+ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; -webkit-user-select: none; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /><br />
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Figure 1. A map of the microwave sky over the SDSS area (diagram from <a href="http://arxiv.org/pdf/0805.2974v2" style="color: #2288bb; text-decoration: none;" target="_blank">Dark Energy with Supervoids and Superclusters</a> Granett et al 2008).<br />
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Einasto <i>et al</i> 1994 suggested supercluster and voids [supervoids] form a rather regular network characterized by peak distance separations 110 - 140 h^-1 Mpc. (See too Einasto et al 1997, The Supercluster-Void Network I, A&A Suppl. Ser. 123, 119-133).<br />
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For distances that are small in comparison to the characteristic scale of a given density fluctuation, with respect to a globally homogeneous manifold of constant Gaussian curvature, gravitational phenomenon are virtually indistinguishable from those in Minkowski spacetime. The properties of the spacetime manifold, globally, are homogeneous; meaning that (i) there are no privileged points in space or time, (ii) there are no privileged directions, and (iii) there are no privileged inertial frames. There is a generalized equivalence of all arbitrary frames of reference relative to the background spacetime. The background spacetime is a Riemannian (or semi-Riemannian) manifold of quadratic differential form whose symmetry group is the group of arbitrary transformations (it is thus not the absolute space of Newton), it is an Einstein manifold, where gravity is represented by its curvature and its metric tensor is governed by the Einstein field equations.<br />
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Material structures present in the manifold are thus not dynamically affected by the geometric properties intrinsic to the manifold of constant Gaussian curvature. I.e., the dynamical character of the metric are irrelevant. All objects, and groups of objects (galaxies, clusters, superclusters), since they are all considered at the origin of the manifold where the magnitude of curvature asymptotically approaches zero (and attain it), are freely-floating relative to the homogeneous background curvature. The globally homogeneous gravity field is everywhere locally Minkowskian. All objects can be, and must be, considered at the origin of a polar coordinate system. In passing, all clocks tick at a slower rate, relative to clocks at the origin, wherever the observer is located, and do so at an increasingly slower rate with increasing distance. (Cosmological redshift z, an affect on clocks and light signals, is a visual embodiment of this global spacetime curvature; which results from the total mass-energy density of the universe).<br />
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However, locally, spacetime has a nontrivial topology, i.e., it is not asymptotically flat. The metric is governed by the locally anisotropic inhomogeneous matter distribution through the gravitational field equations. Therefor the curvature of the metric is altered in the vicinity of massive objects. The metrical spacetime structure acts on the matter distribution, and the matter distribution acts back on the metric structure. Though gravity technically has an infinite range, the gravitational field of, say, the Sun, extends slightly beyond the solar system (around one parsec). For the Galaxy this 'limit' is relatively closer, extending to the outer reaches of the Local Supercluster (approximately 100 Mpc). Beyond this range the gravitational field is dominated by other superclusters (see Sokolowski 2008, Stability of a Metric f(R) Gravity Theory Implies the Newtonian Limit).<br />
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<div style="text-align: center;"><b>Re-Redshift z:</b></div><br />
The origin of the observed redshift z in the spectra of distant objects is attributed to the fact that photons propagate along geodesic paths (great circle arc <i>equivalents</i> in reduced dimension) in a non-expanding, non-contracting, four-dimensional spacetime continuum. These paths are the shortest distance between two points (from any source, to any observer). These paths appear straight along the line of sight, but there is a constant distortion that increases with distance and look-back time, just as the curvature of the earth increases with distance (e.g., the distortion along a great arc from LA to Paris is larger than the distortion between LA and NY, along a great circle arc). The distortion, or curvature we are talking about begins to manifest itself as redshift z at the outer limits of the traditional boundary of the Local Group, beyond which redshift increases with distance proportionally with the constant positive curvature of the manifold.<br />
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<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4tMQpiNilJUspmM4BBFHgnJV-C1aVpPc_1K6Jf0WBh1Sj1BhtxoxCtW2rLn37smdWboUJRbYbrU9up2HFqi1D6JYNKMub0Rp0C_yC7UwcoUC5hrbiyOyliYQbzXitwImVkFq7MsasIco/s1600/Einstein+Static+Universe+of+Constant+Positive+Gaussian+Curvature+15cm150dpi+3ok.jpg" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0976562) 1px 1px 5px; -webkit-user-select: none; background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(238, 238, 238); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(238, 238, 238); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(238, 238, 238); border-top-style: solid; border-top-width: 1px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;" /><br />
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<div class="separator" style="clear: both; text-align: center;"><b>Figure ESU</b></div><br />
Figure ESU represents an equatorial slice of the visible universe (on an oblique angle) with the observer centered at the origin of the past light cone. A slice through any point of origin, and at any angle will be identical. All distances are in look-back time from the observers view-point. Each concentric circle around O represents about 2 Gly. This universe has a spherical topology, where light is redshifted due to the propagation of photons along (great circle arcs in reduced dimension) geodesic paths (shown here schematically as straight lines converging towards the origin). Stability is maintained locally due to motion. Massive objects move along local geodesic paths determined by spacetime curvature in the local vicinity of the objects under consideration, just as the earth remains stable against gravitational collapse into the sun. (Local curvature is not shown in this diagram: see Figure PRM-LSM for an example of local curvature, inhomogeneities or deviations in a negatively curved spacetime). Stability is maintained globally because all points in the Riemannian manifold of constant positive curvature are equivalent.<br />
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Massive objects remain unaffected by the non-Euclidean geometry of the background field since the origin (anywhere in the universe) is effectually flat. There is no motion or acceleration imparted on massive objects resulting for the topology of the spacetime continuum, again, since all points and directions are the same in the globally homogeneous field of constant positive Gaussian curvature. The radius of curvature, the scale factor or the size of the universe does not change with time (i.e., there is no expansion or collapse) since the mass-energy density remains constant, i.e., the conserved quantity ensures a constant radius of curvature over time (excluding for now evolution in the look-back time). This equilibrium is not of the unstable kind, e.g., a pencil balancing on it's point, or a roller coaster at the summit of the track poised to steal the world speed record. To elaborate on the latter analogy, this equilibrium is stable since the roller coaster track is as if curved around the surface of the world (on what amounts to a great circle arc in reduced dimensions): the coaster is essentially on a flat surface relative to the geodesic arc. Hills and valleys that induce acceleration are only present locally (inhomogeneities in the local spacetime surrounding massive objects). Note, we still need to discuss and solve the problem of diverging integrals of the Newtonian gravitational potential.<br />
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Notice too in Figure ESU light propagates in what appears to be straight lines in the Euclidean sense. Curvature no longer appears as it does on the surface of a sphere, where light follows the curve of the surface along great circle arcs. Now, in a four-dimensional relativistic spacetime, the photon paths are essentially straight geodesic lines (excluding local gravitational effects, such as lensing and deflection in the vicinity of massive bodies). The curvature, or distortion, occurs along the path itself. Though the path is straight from the viewpoint of the observer, there is a distortion, plainly visible in the schematic diagram above. And the spatiotemporal distortion becomes increasingly apparent the further the distance considered (increasingly with look-back time). This is exemplified by the cross-sections of the spherical shells centered on the observer, which appear to become closer together with distance. The volume of this positively curved universe appears smaller than those of its Euclidean or hyperbolic counterparts.<br />
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Recall that in reduced dimension, i.e., on the surface of a sphere, light propagates along the very same geodesic lines towards the origin (centered on the any observer, always located at what would look like a North or South pole: compare with Figure PRMCC). The distortion in the path is the cause of cosmological redshift z in a static Einstein universe. There is a loss of energy associated with increasing distance of propagation from the observer in the non-Euclidean manifold: the result is redshift z and time dilation. This is exactly what would be observed from the rest-frame of any observer located anywhere in the Einstein static four-dimensional manifold of constant positive intrinsic Gaussian curvature.<br />
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The fact that redshift occurs in such a universe appears intuitively far less problematic than the maintenance of equilibrium globally in such a universe. Locally there seems to be no problem, since objects such as the solar system, the Galaxy, the Local Group and so on, are observed to be relatively stable systems due to a process of 'environmental selection' (objects of orbital velocity superior to, or less than required in order for stability to be maintained either disperse of gravitationally collapse). Fortunately for us many gravitationally bound systems manage to survive.<br />
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So the problem seems to be with regard to global stability (intuitively): How can the universe itself, if curved positively, negatively, or curved not at all, remain stable in the face of gravity, an attractive force <i>a la</i> Newton, or a curved spacetime phenomenon <i>a la</i> Einstein that causes objects to merge along geodesic paths? Well, we've already seen how gravitationally bound systems remain locally stable. Notice, here we assume galaxy superclusters are gravitationally bound systems as well. So what about the universe itself? Isn't a non-expanding universe a gravitationally bound system subject to the same laws and principles as local systems, meaning that the universe too should eventually gravitationally collapse into one great massive fireball?<br />
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Yes to the former, and no to the latter. The universe is an agglomeration of bounded gravitating systems and as such can be treated as a gravitationally bound system unto itself. Yes too, the universe itself is subject to the same laws and principles as local systems (local physics is global physics) and it is precisely for this reason that the universe itself remains free of global instability. And no, the universe doesn't collapses due to the globally curved spacetime.<br />
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Globally, equilibrium is maintained as mentioned previously not just because its parts are in motion, thus avoiding gravitational collapse up to scales consistent with groups of superclusters, great walls, and supervoids, but because on larger scales still, compatible with the visible universe (and beyond) the curvature of the universe itself plays a fundamental role in the maintenance of equilibrium. Locally, the universe is inhomogeneous. Objects move geodesically relative to gravitational fields of neighboring massive objects. Globally, the universe is homogeneous, and the larger the scale considered, the more homogeneous it becomes. The gravitational field that permeates the entire spacetime continuum is homogeneous and isotropic, while locally the fields of individual objects and clusters of objects remains inhomogeneous. This difference is key to understanding the essence of the physical universe and its evolution in time.<br />
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A homogeneous field of this type is the same at all points. All points are equivalent in a gravitational field of constant curvature. There is no "inward" direction towards which all material objects will collapse. There is no global geodesic path upon which all objects will gravitate, as for photons. Curvature of the manifold vanishes locally (i.e., on scale compatible with galaxy clusters the topology is virtually flat) just as curvature vanishes locally on the surface of the earth.<br />
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<b>Figure SUCGC</b></div><br />
Static general relativistic Einstein universe of constant positive Gaussian curvature. This universe is non-expanding. Redshift z is a curved spacetime effect. This is a reduced dimension schematic diagram. Galaxies, clusters and superclusters reside on the surface of the reduced dimension sphere. The actual structure of the spacetime continuum is positively curved in four dimensions: 3 spatial and 1 temporal.<br />
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While locally, a test particle relative to the inhomogeneous gravitational field of a nearby object will freely-fall, or accelerate towards the gravitating object, globally this does not happen. There is no free-fall or acceleration is any particular direction relative to a homogeneous gravitational field of constant curvature. And since massive objects do not follow the same geodesic path as photons (the equivalent of great circle arcs on a reduced dimension spherical manifold of constant positive Gaussian curvature) all objects do not impart on a journey towards one another leading to a big crunch.<br />
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In another way, each supercluster is embedded inside what amounts to a four-dimensional Minkowski spacetime relative to the globally homogeneous field of constant positive curvature. Essentially, curvature vanishes locally and does so everywhere (again, just as curvature vanishes on the surface of the earth locally, at all locations). Interestingly, this would mean that the geometric structure of universe (e.g., a homogeneous gravitational field of constant positive Gaussian curvature) imparts no acceleration in any direction whatsoever on any object in the manifold. Massive objects are <i>freely-floating</i> relative to the background topology. Notice here that Ernst Fischer's "tension" (the physical embodiment of lambda) is not present in the discourse. We'll come back to this shortly.<br />
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It is of interest to note too that <i>space tells matter how to move, and matter tells space how to curve</i> (Misner <i>et al</i>, 1973, p. 5) in this model. Or, more precisely, <i>locally inhomogeneous gravitational fields impel matter to move, while the globally homogeneous field of constant curvature does not. And matter tells space how to curve both locally and globally</i>.<br />
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The universe remains stable against gross expansion or wholesale deflation.<br />
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In sum, cosmological redshift z in a static universe is caused by the geodesic trajectory of photons in a curved spacetime. Generally speaking, the large-scale geometric structure of the spacetime continuum distorts the wavelength of every photon in direct proportion to the curvature of space in the elapsed time interval. Redshift is interpreted as a function of both distance and time. According to any observer located at the origin O (the rest-frame of an observer) in a static curved spacetime continuum, the distance that separates the source and O will not equal the proper distance (or actual distance) between the emitting source and O. The wavelength of each photon is lengthened (as if 'stretched') between emission and reception. In another way, the energy of every photon is reduced, degraded, in direct proportion with the spatial distance traveled and the amount of time it takes for the photon to arrive at the observer.<br />
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In this static model, redshift occurs not because galaxies a moving away from the observer, nor because of the expansion of space itself, but because of the curvature of spacetime in accord with Einstein's geometric interpretation of gravitation. The curvature of this Einstein manifold is directly related to the gravitating mass-energy density contained in the universe.<br />
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Olbers' paradox is solved in such a non-Euclidean universe because of the relation between energy-loss and distance. The greater the distance, the greater the energy-loss associated with the lengthening of the photon wave packet. The energy carried per photon upon arrival is always less than the energy upon emission. What seemed before a profound cosmological statement, and method of determining whether the universe was static or expanding, is actually a trivial observation. The fact that the night-sky is dark was used as a viable argument (by Hawking and others) for ruling out all static universe models. But in reality, the only static model that failed to resolve the paradox would be geometrically flat (k = 0).<br />
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<div style="text-align: center;"><b>Converging Integrals in a Homogeneous Universe</b></div><br />
Structurally similar to Olber's paradox (the darkness of the night sky) is a problem that had affected the cosmology of both Newton and Einstein.<br />
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Recall briefly the problem: The gravitational force exerted on a test body in a Newtonian universe (a linear theory of gravitation) is the resultant of the forces exerted by the totality of the masses present in the universe, which are assumed to be homogeneously distributed. The force is computed by an integration over all masses. The integration does not converge. Any value may be obtained depending on how the limit of integration over all of space is approached. (Norton 1999, The Cosmological Woes of Newtonian Gravitation Theory).<br />
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In other words, if an infinite 3-dimensional Euclidean universe is filled with a uniform and isotropic distribution of mass the net gravitational force on a test mass located at any point can take on any nominated magnitude and direction. (See also Norton 2002, A Paradox in Newtonian Gravitation Theory II). In laymen's term a homogeneous Newtonian universe would be gravitationally unstable. I.e., Based on diverging potential and the Poisson equation, no Newtonian cosmology with a homogeneous distribution of matter is possible without some non-local extension of Newtonian theory.<br />
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Although this problem was known by Newton (as shown in the exchange of letters with Bentley) it was not clearly stated until the late 19th Century by Seeliger. Einstein tackled the problem within the context of his own theory of general relativity but the inconsistency had not yet disappeared. Milne and McCrea (1934) discovered that relativistic cosmologies gave results similar to that of Newton.<br />
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One way of dealing with the problem, historically, had been to simply ignore it. This had arguably been Newton's tactic. It had also been this author's tactic, to some extent, just as Olbers' paradox could be ignored in the context of a static non-Euclidean universe. But the problem continued to raise it's ugly head and so had to be addressed a new.<br />
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According to Norton (2002) there are three types of responses by theorists to the inconsistency of Newtonian cosmology: (1) They are unaware of the inconsistency and derive their results without impediment. (2) They are aware of the problem but ignore the possibility of deriving results that contradict those that seem appropriate. (3) They find the inconsistency unacceptable and attempt to modify assumptions of the theory in order to restore consistency.<br />
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The latter had been the most common approach. "At one time or another, virtually every supposition of Newtonian cosmology has been a candidate for modification in the efforts to eliminate the inconsistency. These candidates include Newton's law of gravitation, the uniformity of the matter distribution, the geometry of space and the kinematics of Newton's space and time itself." (Norton 2002)<br />
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An expanding universe solves the problem (albeit not of instability), just as it solves Olbers' paradox, but it is not the only solution.<br />
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A spherical mass distribution of arbitrarily large yet finite size (in an otherwise empty infinite universe) solves the problem by avoiding the complications of the infinite mass case.<br />
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As part of an ambitious analysis of Newtonian gravitation in non-Euclidean spaces, Josef Lense (1917) postulated a closed spherical-eliptical spatial geometry with a uniform (homogeneous) mass distribution, corresponding to a finite total mass. Convergent integrals resulted from Lense's work, which gives constant values for the potential and tidal forces, and a vanishing gravitational force for the first dependence. The second law gives a constant value for the potential and vanishing gravitational and tidal forces. (see Norton 1999, section 10, pp. 314-315)<br />
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Similarly, Ernst Fischer solves the problem by postulating a finite homogeneous mass distribution in a universe of constant positive curvature.<br />
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In the case of homogeneous, isotropic cosmologies where the mass distribution is non-expanding, symmetry considerations would require the vanishing of the net gravitational force on a test particle, regardless of its location. Seeliger (1895, 1896) noted that adding an attenuation factor to the inverse-square law of gravitation would be sufficient to solve the problem. The larger the distances, the faster the force of gravity would fall-off relative to the standard inverse-square law. Such an attenuation factor would be negligible effects on scales compatible with the solar system, making empirical verification very difficult (Norton 2002), but not impossible.<br />
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The removal of flat (absolute) space (Newtonian or otherwise) along with the instantaneous propagation of gravity (absolute time) and its replacement with a geometrically curved structural spacetime manifold and the influence of gravitation proceeding at finite speed consistent with general relativity provides a natural and compelling path to the removal of the problem of diverging integrals inherent in Newtonian cosmology. The assumptions of homogeneity, isotropy, and static mass distribution in an infinite (or finite) universe can be retained, and in fact allowable, provided the manifold is of constant Gaussian curvature. This is how stability is maintained on scale above and beyond superclusters. Or rather, this is why the instability associated with a static Newtonian universe does not carry over to a homogeneous and isotropic general relativistic cosmology.<br />
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There is another (related) transient point to make, to be discussed in a detailed subsequent post, regarding evolution, homogeneity, and infinity. Philosophically, the universe may be considered homogeneous at any given cosmic time. But cosmic time does not exist in the physical universe. A universe that evolves from initial density fluctuations to the observed structures (in either expanding or static models) is necessarily inhomogeneous over time. In the static case, such would be equivalent to an "island universe" where the matter density (in the form of stars, galaxies, clusters and so on) diminishes in the look-back time, leading to an epoch dubbed the "dark age." In an expanding model this epoch occurs relatively shortly after the big bang. In a static (Coldcreation) model, this epoch transpires hundreds of billions of years ago. One of the consequences of such evolution in the past is that the gravitational potential diminishes with distance (in the past), since the mass-density diminishes in the past. So too does the magnitude (or radius) of Gaussian curvature. As time tends to minus infinity, the gravitating mass-energy density, along with the magnitude of curvature, tends toward zero (without ever attaining such a value). The Newtonian problem diverging potentials is resolved in this model by virtue of cosmic evolution. Here we tread treacherously beyond the domain of empirical verifiability, since the conjecture resides well beyond the limits of the observable universe. However, there are means by which such a comprehensive model can be tested; thus the precipitous boundary (superfluous as it often seems) that separates science and metaphysics can be avoided.<br />
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Indeed, this is a kind of "finite infinity' proposal for the general relativistic case of a Newtonian universe. GFR Ellis <i>et al </i>write in a similar context: "that is, to isolate the considered local system by a sphere that is far enough away to be regarded as infinity for all practical purposes but, because it is at a finite distance, can be investigated easily and uses as a surface where boundary condition can be imposed (and the residual influence of the outer regions on the effectively 'isolated' interior can thus be determined)" The gravitational potential fades away outside some bounded region, at some epoch, allowing a boundary condition at the expense of denying the philosophical assumption of spatial homogeneity (Ellis et al 2008, Newtonian Evolution of the Weyl Tensor).<br />
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(How this deviation from the inverse-square law affects the problem of rotational curves and excessive orbital velocities of galaxies within clusters remains to be explored, as a possible resolution of the missing-mass problem).<br />
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An interesting feature to emerge from this scenario is that the need to introduce the cosmological constant into the Einstein field equations (as either a repulsive force of the vacuum a la de Sitter, or tension associated with the mass-energy density a la Fischer) is explicitly excluded.<br />
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Symmetry considerations are still valid (per the cosmological principle), yet due to evolution as a function of time there exists an asymmetry between past and future (i.e., we have an 'arrow of time').<br />
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<div style="text-align: center;"><b>Homogeneous (background) gravitational field in general relativity, the problem of diverging integrals, and the inverse-square law of gravity</b></div><br />
Consider the situation of test particles placed in a four-dimensional manifold of constant positive Gaussian curvature in reduced dimension. The surface of a sphere represents our manifold. We begin by placing one test particle, A, at rest on the surface of the sphere. Since all points are the same on our manifold particle A is not impelled to move in any direction on the background field of constant curvature. It remains at rest. We then introduce another particle, B, in motion relative to test particle A, at some other point on the manifold. Depending on various factors, particle B will either collide with A, disperse away form A (pending a potential close encounter at a later date), or find a configuration whereby the two particles can remain bound in a quasi-equilibrium configuration. In the latter case we'll say these particle are gravitationally bound. Now we place another test particle, C, arbitrarily in motion into the manifold. This particle too will be confronted with the three possibilities: collision, dispersion, or gravitational binding. We could continue this process until billions, (or even trillions) of test particles are present on the manifold. Some particles will form groups, or clusters and superclusters, others will not (pending potential close encounters at later dates).<br />
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Notice, during this entire process, the background field of constant positive Gaussian curvature had no affect on the outcome of the experiment. The dynamics of all particles is determined by local gravitational effects, the relative directions and relative velocities of particles placed in the field, in relation to others. At no time does the spherical surface influence (gravitationally) the entire population of particles to coalesce at one point.<br />
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The other important point to make is that the gravitational force between each particle on our spherical surface is not the same as the gravitational force between particles located on a flat, Euclidean, Minkowskian, or Newtonian manifold. Just as the propagation of light is affected by the Gaussian curvature, the gravitational force between particles diminishes with distance at a greater rate than predicted by the inverse-square law (ISL). In another way, the gravitational potential exerted on a test particle located in a manifold of constant positive Gaussian curvature with a homogeneous distribution of material particles converges with increasing volume. Integrals, rather than diverging to some arbitrarily large value (or even to infinity) as would possibly occur on a flat manifold, converge due to the geodetic propagation of force on the background manifold of constant positive intrinsic Gaussian curvature. There is a diminution of gravitational force with increasing distance, greater than inversely proportional to the square of the distance.<br />
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Just as a curved spacetime manifold solves Olbers' paradox by modifying the inverse-square law for the propagation of light, so too does a curved spacetime manifold solve the problem of diverging integrals by modifying the inverse-square law of gravitation. It is no miracle that the quantity by which the inverse-square law for the propagation of light is affected is exactly identical the quantity by which the inverse-square law of gravitation is affected. The deviation from the inverse-square law (for light and gravity) is directly proportional to the Gaussian curvature.<br />
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In another way, there is a divergence from the inverse-square law of gravitation (1/r^2) directly proportional to the magnitude of curvature of the manifold, and thus with increasing distance from the origin. The problem of diverging integrals inherent in a geometrically flat Newtonian manifold differs thus from the situation in an Einsteinian manifold of constant positive Gaussian curvature in such a way that the entire problem vanishes naturally.<br />
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At sufficiently small distances, where spacetime curvature of the manifold is negligible, the ISL holds, to a good approximation (i.e., there will be found no deviation from Newtonian physics locally, e.g., within the range of the Local Group, beyond which cosmological redshift manifests itself). The deviation from 1/r^2 manifests itself increasingly with distance. The greater the distance from the origin, the greater the deviation from 1/r^2.<br />
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Einstein's views on Newtonian cosmology was expressed in a 1917 paper (Cosmological Considerations on the General Theory of Relativity). "...the density of matter becomes zero at infinity...the mean density p must decrease towards zero more rapidly than 1/r^2 as the distance r from the center increases. In this sense, therefor, the universe according to Newton if finite, although it may possess an infinitely great total mass."<br />
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Of course, viz the cosmological principle, such a universe described by Einstein is homogeneous and isotropic at any given cosmic time. The 'island universe' surrounded by empty space is not at all incompatible, as it would first appear, with homogeneity and isotropy. This is simply a (static) universe that evolves in time from a relatively smooth background ('empty space') to the observed structures. The universe described by Einstein in 1917 could be observed, in principle, only in the look-back time and only if one could 'see' that far.<br />
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Of course too, if it were possible to see the empty space at infinity now (in cosmic time), it would appear as does the local universe, with a similar galaxy distribution. Such a universe has no center. The density of stars, galaxies, clusters and superclusters always appears to be at its maximum from the point of view of an observer. All observers appear as if located at the center of an island universe. And as we proceed outward from the origin (the observer) the density of galaxies and clusters diminishes in the look-back time, and continues to diminish, until eventually, at distances far beyond the range of telescopes (and at times far beyond the suspected age of the universe), the mass-density galaxies and clusters (or rather proto-galaxies, proto-clusters and proto-superclusters) tends towards zero. The stellar universe is thus a finite island in an "infinite ocean of space," static, homogeneous, isotropic and evolving with time. I might add that such a static universe possess a curved spatiotemporal manifold, i.e., this is a 4-dimensional static universe with constant positive Gaussian curvature (by virtue of the nonzero mass-energy density) and of infinite spatiotemporal extent. It is consistent with both Newtonian theory and Einsteinian theory in that there exists a globally homogeneous gravitational field.<br />
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So far, Einstein's analysis of Newtonian cosmology is impeccable. What we have here is a solution to the problem of diverging integrals in a static Einsteinian universe compatible with homogeneity and isotropy considerations. The gravitational force converges. Gravitational force diminishes with increasing distance from the origin (and asymptotically vanishes as time tends to minus infinity). Einstein then fell into error. Ironically, he criticized such a universe as unsatisfactory, speculating that (i) an island universe would lose radiation to infinite space and that (ii) the energy of motion distributed statistically among stars of the 'island' would impel stars (once they acquire enough velocity) to escape the island's gravitational pull. Such a universe, in contradiction with the presumed static nature of matter on large-scales, would 'evaporate' Einstein suggested, using Boltzmann's analysis of the statistical physics of gas molecules in a gravitational field. (See John D. Norton, 1999, The Cosmological Woes of Newtonian Gravitation Theory, Goenner et al. (Eds.), The Expanding Worlds of General Relativity, Einstein Studies, volume 7, pp. 271-323, The Center for Einstein Studies).<br />
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The problem Einstein laid out (either the mass distribution was concentrated in an island or it was homogeneously distributed) was a false dilemma. The "island" would only be apparent (in principle) due to the limited speed of light in vacuo. An observer situated at the "edge" of the island, relative to an observer situated at the origin, would see the universe as if, herself, centered on the island. In other words there is no center. All points are equivalent. The "island" exist only as a function of time, and thus of distance. With cosmic time there is no island at all (i.e., when all clocks in the universe are synchronized the universe is homogeneous and isotropic).<br />
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There would be no loss of radiation to infinite space. By symmetry considerations there would be no stars escaping the island's gravitational pull, no evaporation. Such a universe was indeed consistent with the presumed static nature of matter on large-scales. It's difficult to comprehend how the inventor of general relativity (amongst numerous other contributions to science) could have made such a mistake. Perhaps Einstein leveled this criticism against Newtonian cosmology in the context of a geometrically flat Euclidean manifold with absolute space and absolute time, in which case he would have been correct after all: the island universe (i.e., Newtonian cosmology) is untenable in the absence of general relativistic principles.<br />
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Newtonian theory (along with special relativity, Minkowski spacetime) would be a good approximation only in the local neighborhood of an observer, i.e., when distances are small and time scales short.<br />
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Einstein would attempt to resolve the problem of Newtonian gravitation by modifying Poisson's equation, thereby admitting an acceptable cosmological solution with a homogeneous distribution of matter. The result was a spatially closed relativistic model of spacetime with a static mass distribution (known as the Einstein universe). In order for such a model to satisfy the field equations of general relativity Einstein felt it necessary to introduce a supplementary term, "perfectly analogous to the extension of Poisson's equation". Consistent with symmetry considerations the average gravitational force (the force per unit volume at a point) was zero everywhere. Though perfect the cosmological constant was not. Einstein's initial satisfaction with these modifications were twofold: the integral would converge rather than diverge, and Einstein's dream of a Machian universe would be realized. What Einstein had (apparently) not seriously contemplated was how such a static universe with a uniform mass distribution would evolve as a function of time. Einstein had shown that a slight modification of the inverse-square law of gravitation permitted a cosmology consistent with staticity and homogeneity (dissimilar in form but with the same result as Seeliger 1895, 1896, 1909, and Neumann 1896). The fact too that Newtonian gravitation could be adjusted to readmit cosmology by other means was immediately apparent (Norton 1999, pp. 297-303).<br />
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The most obvious solution was to modify cosmological assumptions and leave Newtonian gravitation unchanged, rather than visa versa. Such was the solution of Wilsing (1895). Charlier (1908) developed a scheme that led to a vanishing mass density when averaged over space that required no preferred center, free of gravitational divergence. The idea was to consider the hierarchical grouping of matter into clusters on different scales. The spacing between each scale of clusters would increase as the scale of the cluster considered increased, indefinitely. The escape from the gravitational problem, and Olbers' paradox in passing, occurred as the mean mass density would vanish over infinite space, i.e., there is a dilution of mass density with increasing volume (Norton 1999, pp. 304-309).<br />
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Certainly, the big bang model thwarts the problem of instability associated with a homogeneous distribution of matter. But at what price? The universe is simply unstable; expanding, and loaded profusely with an unidentified form of dark matter along with an unphysical form of energy or pressure.<br />
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Fortunately, the big bang model is not the only solution; there have been many viable solutions over the relevant years. One of these solutions is consistent with a wide variety of observations, ranging from phenomena such as the formation of large-scale structures, cosmological redshift z, the CMBR, and the abundance of light elements and their isotopes.<br />
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<b>In sum</b>, a variety of solutions had emerged for both the homogeneity of Newtonian cosmology and the gravitational potential problem, including the kinematic solution (the expanding model of Milne and McCrea, 1934, that mimics the dynamics of a relativistic Friedmann universe).<br />
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"Perhaps the most astonishing part of our story" according to Norton, "is that both of the greatest figures of cosmology and gravitation, Newton and Einstein, stumbled on the same problem. When presented with the problem, Newton seemed so sure that his cosmology would be well behaved that he saw no need to think the problem through. Einstein also was overly hasty, seeing in the problem a dilemma for Newtonian cosmology that others showed to be a false dilemma." (Norton 1999, pp. 315, 316)<br />
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In any case, what we are left with is a cosmology (or several cosmologies) the consequences of which are testable in principle. The one that stands out here is a static general relativistic universe of constant positive Gaussian curvature. This universe is homogeneous and isotropic at any given time on very large scales and possibly infinite in spatiotemporal extent (in both past and future directions of time), i.e., there is no beginning or end. The universe does not collapse or expand as a function of time. Stability is maintained on the largest scales since all points are the same; there is no preferred direction in the 4-dimensional spacetime manifold of constant curvature. Material objects are not impelled to move due to a gradient of the global manifold, since the gradient is the same everywhere locally (equal to zero at every point), just as the curvature of the surface of the earth (in reduced dimension) is zero at all points. Curvature manifests itself with increasing distance and time as measured from the rest frame of any observer (the origin). Cosmological redshift z is the result of this curvature, since photons are obliged to travel a geodesic path in a curved spacetime manifold.<br />
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On scales relatively smaller than the observable universe (somewhere around or below 300-200 Mpc), homogeneity and isotropy break-down to form a hierarchical network of superclusters-supervoids, cluster-voids, galaxy groups and smaller voids, galaxies, star clusters, stars, planets and so on down to the realm of quantum mechanics, where clumps of matter and voids are present still. It would have been nice to delve a little deeper into the quantum world here, for there is much to be said and written in relation to all that has been discussed so far, but that will be for a subsequent post. Perhaps just as a foretaste for what will come let it just be said that phenomena such as Bose-Einstein condensation (BEC), Brownian motion, superfluidity, supercondictivity, zero-point energy ZPE, Casimir force, are all of interest and to some extent on-topic here. Though there still exists a rather large discrepancy between GR and QM, there are laws of nature common to both the macro- and micro-scopic domain that permit both reversible and irreversible phenomena, but irreversibility is the rule rather than the exception. Symmetry-breaking brings the universe from a static geometrical configuration to one whereby space and time are shaped by the events and objects in the evolving system. Order and coherence of the system are the extraordinary features to emerge from the spontaneous transition from the simple vacuum to the complex structures observed today (a process that has transpired over many hundreds of billions of years). Despite the random fluctuations of thermal motion performed by a large number of particles, the coherent self-organization leads to the emergence of complex behavior and far-from-equilibrium states. It is this self-accelerating process that has opened the gateway to systems capable of self-reproduction: the most fundamental property of life. Yes, life, in animal and plant form, humans, are all part a cosmology. The history of the universe is the history of life, its origins, its evolution.<br />
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At the heart of nature lies the deep concepts of symmetry and geometry, the competition between order and disorder, of energy and entropy, self-organization, erratic and coherent behavior at large length scales, frictionless, resistanceless flow of ground-energy, phase transitions, density fluctuations at many scales, and the critical points at which new phenomena occur.<br />
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</div>Alex Mittelmann, alexmittelmann@yahoo.comhttp://www.blogger.com/profile/03457606761033752726noreply@blogger.com0