Saturday, September 5, 2020

Redshift z

Curvature of a globally homogeneous isotropic general relativistic spacetime
and the stability of the cosmos

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.

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 t. (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.

Key words: 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.  


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.

The stability, or rather, the instability 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).

The standard approach to the problem, to some extent, suppresses the distinction between local and global geometry, while the inclusion of an ad hoc 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 t that underlies the definition of the metric tensor). This is an attempt to make the geometric attributes of the spacetime manifold itself (the metric) account for the observed redshift z.

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 stationary, evolving and dynamic universe that is infinite spatiotemporally, in both the past and future directions of time.

The following discussion stems from a thread at entitled Redshift z which began 07/05/2005.

A brief note regarding terminology:

Global curvature 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.

Local curvature 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 geodesic manifold, 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 local refers to the entire visible universe (see here for example). That will not be the case in what follows. 

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?

Figure 1ABC

This diagram illustrates schematically 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 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.

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). 

Which geometric manifold is consistent with observations in the look-back time?


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.)

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.

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' not 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.

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. 

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.

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. 

What shows up most often on search engines are solutions such as static isotropic metric, also called a standard isotropic metric which relates to the Schwarzschild solution (or the Schwarzschild vacuum).

There was an interesting and related work that popped up. It's entitled On the Physical Interpretation of a Solution of a Nonsymmetric Unified Field Theory, 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:

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. [...]

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.

And I'm guessing that the manifold would look something like this:

Figure 2D

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.

Or perhaps this:

Figure 3D

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.

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 Schwarzschild metric; Flamm's paraboloid"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." 

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 global curvature is different from local curvature in that objects are not affected by geodesics, but light will be, 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.

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).

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.

Figure LCDM

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). 

Despite the visual shape of the manifold above, this is actually a 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. We'll come back to the geometric structure observed in this schematic diagram.


Part I

Local versus global curvature in a homogeneous isotropic general relativistic spacetime manifold

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 a priori 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).

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, without the use of an ad hoc cosmological constant-like vacuum energy term to counter the attractive force of gravity.

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.

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 compact space (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). 

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 ad hoc repulsive force is obtained. 

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.

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 general covariance.

It is well known, generally accepted, and confirmed empirically that phenomena such as gravitational redshift and time dilation 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.  

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 relativistic Doppler shift), 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 Lorentz symmetry. 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. 

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. 

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" (source). In other words, one need not change any equations of GR for what is to follow.

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).

Geodesics in a homogeneous spacetime of constant curvature

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.

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.

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.

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.

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.

[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.]

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.

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.

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).

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.

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.

Figure PRMCC

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.

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.

Homogeneous gravitational field

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.

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 Can the notion of a homogeneous gravitational field be transferred from classical mechanics to the Relativistic Theory of Gravity? "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.

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 Crowell ). 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.

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.


Figure ESU

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). 

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.

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.

The Einstein equivalence principle and cosmological considerations

Lets compare two coordinate systems. A coordinate system S1 is accelerating in "empty" space at a rate g in the x direction. The second, S2, is at rest in a homogeneous gravitational field that imparts to every material object in the field an acceleration of –g in the x direction. Einstein observed that: […] as far as we know, the physical laws with respect to the S1 system do not differ from those with respect to the S2 system […] we shall therefore assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.

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. Inertial acceleration  and gravitation appear intrinsically identical.

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 indistinguishable 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.

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.

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.

Locally, observed accelerations of objects in the field, apparent or real, have the characteristic feature that they will all experience the same 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. 

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.

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 the 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. 

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. (Brown, Reflections of General Relativity)

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.

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.

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.

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).   

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).

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.

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). 

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 identifying the purely geometrical effects of non-inertial coordinates with the physical phenomenon of gravitation, but to interpret redshift z and a purely geometrical effect.

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 E isE/c^2 (Crowell, 2010). 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.   

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 Einstein equivalence principle, in the absence of a gravitational force (source). It will be shown that the standard concept expressed by Misner et al (1973, p. 5): "Space tells matter how to move. Matter tells space how to curve" 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).

It will be found too that the general theory of relativity does indeed satisfy the conditions of homogeneity, isotropy, and staticity without an additional cosmological term (lambda).

En passant, 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 space itself is expanding 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 Local Supercluster on down. Quite the contrary, all the evidence points to the fact that space does not expand. Things expand into space (e.g., isothermal and adiabatic expansion). General relativity says nothing about expanding space, but everything about curved spacetime!

One more side-note before moving on to geometrical argument:

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 environmental selection 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.

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.

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. 

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. Au contraire. 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: Local-Group tests of dark-matter Concordance Cosmology.

To be continued

Barcelona, June 14-15, 2010


Part II

Disambiguation and the Stability of the Cosmos:
Global curvature of a homogeneous isotropic general relativistic spacetime

Geometrical Arguments

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.

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.

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 Gauss' Theorema Egregium (meaning "Remarkable Theorem" in Latin). Gauss introduced this remarkable theorem in his famous 1827 paper "Disquisitiones generales circa superficies curvas" (General Investigations of Curved Surfaces).

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. 

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 intrinsic curvature 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.

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 intrinsic curvature is essential to the concept presented here.

Intrinsic curvature such as Gaussian curvature is detectable to the "inhabitants" of a surface. Surfaces can have intrinsic curvature, independent of an embedding.

In contrast, extrinsic curvature is defined at each point in a Riemannian manifold. An extrinsic curvature 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. Extrinsic 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 metric tensor relative to any system of coordinates on the manifold. 

Gaussian curvature is, in fact, an intrinsic 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.

Figure xyz

Stereographic projection of a complex number A onto a point αof the Riemann sphere.

Space of three or more dimensions can be intrinsically curved; the full mathematical description is described at Curvature of Riemannian manifolds. The basic idea of intrinsic curvature remains essentially the same in four dimensions.

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.

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 (all points and all directions are indistinguishable). (Same source as above.)

Since mechanics has its foundations in geometry, let's run through some more of the basics.

"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." Source: General Relativity and Gravitation, Canonical quantum gravity, Karel V Kuchai

Gauss discovered many remarkable theorems during his life time, but this one Gauss himself believed to be truly remarkable: he called the result theorema egregium

The reason Gauss was so enthusiastic is that his formula proves the Gaussian curvature of a surface is indeed intrinsic, that is, it is not dependent on the embedding of the surface in higher dimensional space.

Gauss' theorema egregium states that Gaussian curvature 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.

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. "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." (Crowell, 2010)

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 principal curvatures are not.

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 could 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 could tell that the geometry of their universe was different from the that of a flat Minkowski space (one that is expanding or not).

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); source. But not only that, it means that light propagating in such a manifold would be redshifted as it follows the geodesic: Whereas, material objects would not be affected (i.e., massive bodies react as if embedded in a flat spacetime).

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 “spherically symmetric static general relativistic cosmological space-times can reproduce the same cosmological observations as the currently favored Friedmann-Robertson-Walker universes” 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 against the stationary models.

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 anintrinsic 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.

If we want to measure Gaussian curvature 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 finite region of the continuum with reference to which curvature, the Riemannian-Christoffel tensor, essentially vanishes) abides by the laws of special relativity (see The Foundation of the General Theory of Relativity, Einstein, 1914-1917, pp. 176-179, pdf). 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 the gravitational field of an object (which clearly has non-zero intrinsic curvature) cannot simply be transformed away by a change of coordinates.

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-like effect, or as a gravitational redshift-like 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.

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. The 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). 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.

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.

The beauty of applying Gauss' theorema egregium (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. 

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 geodesic symmetries 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).

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. 

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).

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 intrinsic curvature.

Differential geometry, leading up to the mid-nineteenth century, was studied primarily from the extrinsic point of view. Curved surfaces were considered as residing in a Euclidean space of higher dimension (e.g., a the space surrounding a geometric object). 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' theorema egregiumGaussian curvature is an intrinsic invariant.

Recall that Einstein's general theory of relativity is expressed in the language of differential geometry. Accordingly, the universe is considered a smooth manifold endowed with pseudo-Riemannian metric. This metric describes the curvature of spacetime. The spacetime metric is the field. 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.

The intrinsic point of view is useful in general relativity when spacetime manifold cannot be taken as extrinsic (what is there beyond?). The global concept of curvature from the intrinsic 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 extrinsic difference between the two, i.e., intrinsic measurements available in general relativity are not capable of detecting an arbitrary change of coordinates (Crowell, 2010).

In order to reconcile the two different views, extrinsic curvature can be considered as a structural addition to intrinsiccurvature. (Source). 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.

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.

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.

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 apparent curvature, 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 relative, 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.

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. 

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. 

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 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 crests) must increase. (Source)

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. (Source)

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.

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.

Geodesics and redshift

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). 

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). 

Let's exemplify the situation with a schematic diagram:

Figure G

Three spherical triangles 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.

Remark: Figure G 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 Figure 1C 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 here, for example).

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 great circle (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 does not affect objects (i.e., objects are not displaced by the global curvature). 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. (Source)

"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 Geometrical theory of Spacetime, section 1.5.2).

The angles of all planar triangles (or spherical triangles) 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 this or this). 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 cosmic distance ladder).

Notice what happens in Figure G: when an observer studies regions locally (or nearby) she measures distances and angles that lead to a quasi-Euclidean geometry (see triangle a). The further she looks, the greater the distance, the greater the departure from linearity, the greater the curvature (see triangle b in Figure G). Notice the curvature with respect to the triangle formed by points AB and C. 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). 

The amount by which curvature exceeds 180° is called the spherical excess. 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).

For hyperbolic triangles (discussed below) excess is replaced by defect. 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 Gauss-Bonnet theorem and Euler characteristics for a further discussion and proofs.  

Here is one more example of spherical triangles, only this time closer to home (mine that is):

Figure G3

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°2820N 3°3339W. The map is of Cannes, France at 43°3305N 7°0046E. GoogleEarth images with high-resolution aerial photography data. (The 60° figure is only a rough estimate).


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 gravitational time dilation and frequency shiftthat 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.

So, the metrical distance 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.

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.

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.

This concept finds application in Lagrangian mechanics, where it plays a conjugate role to generalized coordinates, by deriving equations of motion 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 Lagrange–d'Alembert principle in accelerating systems and the principle of virtual work for applied forces 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.  

A Natural Fine-Tuning

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 conflated. 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.

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 visa versa) 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.

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.

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.

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 Principia Newton wrote, "The fixed stars, being equally spread out in all points of the heavens, cancel out their mutual pulls by opposite attractions." 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. (E. R. Harrison, 2000, Cosmology: the science of the universe). 

It is though nevertheless often considered that the equilibrium described by a Newtonian universe is unstable. For example Cromwell writes, "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 r, then its gravitational is proportional to r^3 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." 

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.

Figure SB

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.

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.

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).

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.

Such as collapsein a homogenous and isotropic universe without a center towards which objects would acceleratedis 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 gravitational collapse (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 universeincluding space and timewould 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.

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.

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).


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). 

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. 

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 Gaussian curvature 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 embedded 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 without a boundary. (Source). Note: this would be in contrast to a compact Euclidean n-sphere, 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.

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. 

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 t (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).

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. 

Interestingly enough, the equilibrium is explained naturally, without any ad hoc parameters or new physics. 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.

Redshift z

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:

Figure G2

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.

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.

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.

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.

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).

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.

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.

Figure 1C

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).

A few remarks:

  • 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.

  • 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).

  • 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.

  • 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). 

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.

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).

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. 

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). 

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.

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. 

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. 

There is no need for a repulsive force to counter the attractive force of gravity locally since there is no absence of motion. There is no need for a repulsive force to counter the attractive force of gravity globally 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. 

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). 

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).

Figure 1Cb

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.

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 special relativity 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. 

With her knowledge of general relativity, 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).

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.

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 ad hoc 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. 

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.

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. Notice that the temporal intervals between two definite events ranges from zero (in the vicinity of the observer) to infinity (at the horizon).

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).

Though most of the discussion in this Redshift z thread at 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 a priori 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).

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. 

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.

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.

Figure ADHU

Figure ADSU

Figure ADEU

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.
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.

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. 

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).

Figure SC

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.

Figure SCb

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.

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).     

Angular Diameter and Parallel Lines

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).

Figure T

Classic one-point perspective of train tracks with three different spatial geometries: (a) negatively curved, (b) flat, and (c) positively curved, extending to infinity.

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. 

Notice that when a photon is emitted from a source located near the horizon of figure 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. (Source). In the case of diagram a and c 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.

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 ab or c). The apparent distance and luminosity would differ, but the size difference would be virtually impossible to measure, i.e., it would be imperceptible. So, in practice this behavior is not observed.

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 (c) 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). 

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). 

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 a and c appear closer and further respectively than objects on the horizon of 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 a and c 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 (a) would appear billions of light years further than the same objects in a positively curved spacetime (c). 

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 here, for n-spheres, 3-sphere in 4-dimensional Euclidean space.

Straight lines in a curved spacetime

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. 

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. 

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.

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 simply coasting on their geodesic paths. 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.

Let's visualize the situation from another angle:

Figure U

Three Geometries from the Observer to the Visible Horizon: A is hyperbolic, B Euclidean and C 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.

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 (A) appears larger (and deeper) that the Euclidean and spherical models, 30° appears larger (or wider) in model A. 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.

Redshift z occurs in illustrations A and C only, increasingly with distance. Objects themselves are unaffected by the distortion of spacetime (globally). 

Note that the three models above are spherically symmetric, homogeneous and isotropic. The Euclidean space 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 A and C do not conform to the inverse square law. If the total flux is known, the field itself can be deduced at every point.

Notice that the spherical model C 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 R of the sphere. 

If we measure the circumferences of circles (or spherical shells) of progressively larger diameters and divide the former by the latter, all three geometries AB and C 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). For a hyperbolic spacetime manifold (A) 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.

The fact would remain though that 30° of the sky is still 30° from the point of view of any observer.

Figure ADPHU

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.


Cosmology hinges on the interpretation of redshift z

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). 

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.

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.

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.

Hyperbolic Curvature and the Stability of the Cosmos

We've seen above how global stability is maintained in a non-Euclidean Gaussian, or Riemannian manifold that is curved spherically. The metric components of the globally homogeneous field vanish locally. By definition, a Riemannian manifold is flat on a sufficiently small scale; a fact that corresponds to the equivalence principle for the spacetime manifold. There 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 Big Crunch.

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. 

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 (the metrical relations of spacetime) involved in the stability and redshift in a spherical manifold is operational in a hyperbolic manifold.

Figure HUDF

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.

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.

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). (">Source)

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.

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 On the Principles of Geometry (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 New Principles of Geometry With a Complete Theory of Parallels, 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. 

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.

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. 

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.

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. 

A surface on which the Gaussian curvature K is everywhere positive is called synclastic, while a surface on which Gaussian curvature K is everywhere negative is called anticlastic and is saddle-shaped (in reduced dimensions). 

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) geodesically complete 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 geodesic symmetries are defined on the entire manifold.

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.

There are many ways to visualize this scenario for a hyperbolic plane, or a hyperbolic space. 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. 

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.

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 t. 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.

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.

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).

Now lets illustrate a reduced dimension a hyperbolic manifold with negative curvature—with a spatial geometry equivalent to the Figure A 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.

Figure H

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.

This is a general relativistic world model in 2-dimensions, not unlike the geometric architecture of the 1917 de Sitter universe. 

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. (Source). 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. (Source)

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. (Source)

Figure 7Ab

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.

Events at great distances appear to take longer than in the rest-frame of the observer. Time dilation reaches a maximum at the horizon.

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. 

Distances in a hyperbolic universe can be measured in terms of a unit of length  (see here). This is analogous to the radius of a sphere in spherical geometry.

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. 

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.

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). 

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.

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.   

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.

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. (Source).

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.

Figure SA

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 (or pseudospherical) curved spacetime continuum.

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. Over 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.

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.

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.

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.

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.

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. 

Cosmological Redshift z

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.

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.

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).

In an static regime, the redshift and distance relation of every supernova records not 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). 

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:

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. Source: The Deep Universe, M.S. Longair, page 369

The point is, as you may have guessed, there is a direct empirical correlation between redshift z as interpreted by Longair and the interpretation of redshift z as a curved spacetime phenomenon (with both minor and major differences in physical outcome).

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 uswhen interpreted as a general relativistic phenomenonis 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.

Figure PLEM

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.

Figure PLSM

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. 

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 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.

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.

Figure PLHM

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.

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 deviation from linearity 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.

Just for fun, let's see what happens when a hyperbolic grid is superimposed on the infamous Sloan Digital Sky Survey (SSDS) image.


See SDSS for the original, unadulterated, image.


A global Pseudo-Riemannian manifold with local Loentzian submanifolds

Recall, it is mathematically in the language of differential geometry that general relativity describes the universe of events. 

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.

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. 

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.

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.

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. 

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.

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.

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. 

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.

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.

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).

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).

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.

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.

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.

Figure PRM-LSM 

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)

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.

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.

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.

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.


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.

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.

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).

It was written in the first few paragraphs above that this work is a qualitative and conceptual 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. 

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:

  • Ellis, G.F.R. 1977, Is the Universe Expanding?, General Relativity and Gravitation, Vol. 9, No. 2 (1978), pp. 87-94

  • 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

  • Hubble, E. 1936, The Realm of the Nebula 108-201

  • Kerszberg, P. 1989, The Invented Universe, The Einstein-De Sitter Controversy (1916-17) and the Rise of Relativistic Cosmology

  • Kragh, H. 1996, Cosmology and Controversy, The Historical Development of Two Theories of the Universe

  • Daigneault, A. 2003, Standard Cosmology and Other Possible Universes, (SCOPU)

  • Segal, I.E. 1976, Mathematical Cosmology and Extragalactic Astronomy

  • 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

  • Segal, I.E., Zhou, Z. Oct. 1995, Maxwell’s Equations in the Einstein Universe and Chronometric Cosmology, Astrophys. J. Suppl. Ser. 100, 307-324

  • Segal, I.E., 1997, Cosmic time dilation, Ap. J. 482:L115-17

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. 

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 quasi-steady state cosmology(QSSC), Arp's hypothesisZwicky's tired light scenarios, plasma cosmology 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 is in line with GR (i.e., in line with non-Euclidean geometry). 

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 have 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. 

Infinity and Stability

Clearly, the notion that astronomical objects are not 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).

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. (E. R. Harrison, 2000, Cosmology: the science of the universe, p. 334).

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.

The subsequent 'natural' generalization of Einstein's field equations—that allow the radius of curvature R(t) 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 The Arrow of Time in Cosmology, M. Castagnino, O. Lombardi and L. Lara).

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. 

On equilibrium and the cosmological constant: Λ

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 ideal gas law). 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.

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.

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 mechanical equilibrium. 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).

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 equations of state 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.

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.

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 lambda 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: “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.” 

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).

“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) 

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.

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.

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.

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. 

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.

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).

Concluding remarks

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.

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.

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, entre autre

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.

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.

Falsifiability or refutability: there exists the logical possibility that the assertions, regarding cosmological redshift z and global stability presented here, could be shown false. If the contentions discussed in this work are false, then the falsehood can be demonstrated, simply by showing (1) that electromagnetic radiation is not 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.   

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.  

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. 

A continued discussion on this topic may be found here: Redshift z at

Alex Mittelmann

June-August 2010, Barcelona, Spain