Redshift z

[coldcreation]

Modest,

 

Your objections have been addressed (see the links in post #729 above). In the process, a world-view consistent with observations has emerged that is a continuation from where Einstein left-off. The result is a four-dimensional static general relativistic universe that possesses intrinsic Gaussian curvature of the spacetime manifold.

 

If you feel that any one of your objections has not been fully explained, please state explicitly which of these is lacking.

 

 

CC

[modest]

I was hoping someone could offer a fresh perspective.

 

Just as in an expanding universe the global geometry may be either curved positively, negatively, of flat, in a non-expanding universe the geometry may be spherical, hyperbolic or flat. But again, the flat spacetime is ruled out on empirical grounds (no redshift z).

I'm sure that your first sentence refers to the geometry of space (ie spatial hyperslices) and the latter sentence is referring to the geometry of spacetime.

 

To me curved spacetime means, by definition, gravitational force. Flat spacetime, like Minkowski spacetime, means there is no gravitational force attracting objects to one another.

 

When you say that globally curved spacetime implies a static universe I have no idea how you got there, but it flies in the face of everything I know of GR.

 

~modest

[coldcreation]

 

I'm sure that your first sentence refers to the geometry of space (ie spatial hyperslices) and the latter sentence is referring to the geometry of spacetime.

 

To me curved spacetime means, by definition, gravitational force. Flat spacetime, like Minkowski spacetime, means there is no gravitational force attracting objects to one another.

 

When you say that globally curved spacetime implies a static universe I have no idea how you got there, but it flies in the face of everything I know of GR.

 

~modest

 

 

Ok, let's backtrack a little here.

 

Certainly, in an expanding universe, the geometry may be hyperbolic, spherical or flat.

 

In a non-expanding universe, where general relativity is operational, the geometry of the universe cannot be flat (and cannot be hyperbolic, according to Bruce, Fischer and possibly Einstein), it must be spherical. That is, the geometrical structure of the 4-dimensional spacetime continuum is curved. A flat spacetime a la Minkowski would indeed have no gravitational force acting anywhere. So, both the flat and hyperbolic models for a non-expanding universe are ruled out by both general relativity and observational evidence.

 

As long as general relativity is the correct theory with which we must describe the global properties of the universe (whether expanding or not), the general geometrical properties of the spacetime manifold depend upon the energy and pressure along with the gravitating mass-density of the universe. Albeit, the velocity of expansion can modify the geometry, whereas in the nonexpanding case such is nonexistent.

 

 

Yes, curved spacetime means, by definition, gravitational force. Globally curved spacetime implies that light from distant objects is redshifted in a static universe. This does not 'fly in the face of GR'. Quite the contrary. The fact that gravitation (spacetime curvature) is everywhere present does not imply that all objects will attract one another, ending up in a great ball of fire. This, amongst other misconceptions, is what I've tried to explain here and here.

 

What we have is a globally homogeneous gravitational field in accord with GR. The misconception arises when considering inhomogeneous local gravitational fields as synonymous with one that is global and homogeneous in nature. Locally, matter is impelled to move gravitationally towards a center of mass, or may remain in equilibrium when velocity is adequate, and-or when the combined gravitational pull of the two massive objects provides the centripetal force required to rotate with them. This latter phenomenon allows objects (such as superclusters) to remain in a "fixed" position in space rather than an orbit where its relative position changes continuously.

 

In a general relativistic homogeneous spacetime there is no center of gravity towards which objects will all gravitate. I gave the example above (following form the notion that a homogeneous cloud of gas and dust will collapse towards a center, locally) that differentiates between curved spacetime surrounding massive objects and a globally curved spacetime. The latter may indeed remain stable since the problem of diverging integrals does not arise in a homogeneous general relativistic universe of constant positive Gaussian curvature, unlike a Newtonian universe where the gravitational potential diverges with distance (see from the second link above: Homogeneous (background) gravitational field in general relativity, the problem of diverging integrals, and the inverse-square law of gravity).

 

 

It is quite remarkable that such a concept expressed here reconciles the homogeneous field of general relativity and a classical homogeneous gravitational field of Newtonian cosmology (would you not say?). Whereas before, the universe was though to be static and ended up unstable, now, the universe still thought to be unstable ends up static.

 

 

CC

[coldcreation]

Here are a few quick points in addition to the above post, and in order to better understand what we are dealing with here vis-à-vis cosmological redshift z and global stability:

 

 

1. What we have is a globally continuous isotropic and homogeneous gravitational field consistent with Einstein's general theory of relativity.

 

2. This is a 4-dimensional spacetime continuum. The geometry that describes this spacetime was founded by Gauss and Riemann.

 

3. We are talking about a non-degenerate pseudo-Riemannian manifold of constant positive mean curvature with Lorenzian structure. This manifold of constant curvature is defined everywhere in non-degenerated quadratic differential positive definite form. It has the property of convexity. All point on the manifold as the same.

 

4. Locally this spacetime is equivalent to Minkowski spacetime (or Euclidean space). For this reason there is no tendency, due to the global intrinsic Gaussian curvature, for all objects in the field to collapse towards one another.

 

5. The underlying space-time curvature is intrinsic to the manifold, and is thus physically significant. It can be measured empirically directly at the telescope, since the spectrum of light emitted by distant objects is affected (shifted toward the red).

 

6. Whereas locally-curved space is quite complex, the curvature of a space which is globally isotropic and homogeneous is described by a single Gaussian curvature (as for a surface). All points and all directions are indistinguishable. All reference frames are equivalent. Geometrically these are strong conditions, and they correspond to reasonable physical assumptions.

 

7. When a signal is sent from point A (say a distant galaxy) to point B (the reference frame of any observer) the time it takes for the signal to arrive will appear to take longer than would be the case in a flat Minkowski spacetime. This is the general relativistic phenomenon of time dilation. Events (e.g., SN Type Ia explosions) will appear to occur more slowly the further removed the emitting object.

 

8. In a 4-dimensional manifold of constant positive Riemannian curvature, the volume of a given spherical shell differs from the area of a given spherical shell of the same radius in Minkowski spacetime. This difference, in a suitable limit, is measured by the scalar curvature. The difference in volume of a given spherical shell is measured by the Ricci curvature. Both the scalar curvature and Ricci curvature are particularly important in general relativity, where they appear in Einstein's field equations on the side that represents the geometry of spacetime. The other side represents the presence of matter and energy.

 

 

The beauty of such an attribution (of curved spacetime) to the phenomena of redshift z and global stability are too numerous to list here. What stands out, perhaps most, is that there is no instability due to the Riemannian manifold associated with galaxy superclusters, no origin or big bang event at some time t in the past, and no accelerated expansion. It follows that there is no so-called dark energy. It follows too that the universe is much older than currently assumed: in fact, it may have no "age" at all.

 

Certainly, a reduced dimension Gaussian manifold (the surface of a sphere) is finite but unbound. In 4-dimensions, however, there is no need to postulate a finite universe, since the size can be extended without limit. On a spherical surface light travels around great circle paths, curved geodesic trajectories that may continue until they reach their point of origin, but in in 4-D universe light travels in "straight" geodesic trajectories, never returning to the point of origin, eliminating the reduced-dimension problem of finiteness. Such a universe may be thus infinite and unbounded spatiotemporally.

 

While such a though might be pleasing philosophically, the idea can now be sustained with real physical arguments, as one of the properties of a 4-dimensional (pseudo)Riemannian manifold of constant positive curvature.

 

The 'fate' of this Einsteinian universe is an important question that depends on evolutionary trends; to be discussed shortly. One certainty is that a Big Crunch or Big Rip are non-options.

 

Interestingly enough, it was Einstein who first postulated this world-model, though he ended up adding a cosmological term to his equations to keep it from collapsing (or expanding), and latter removed it, adhering to the unstable model. What has been shown here is the the cosmological term was never required for the maintenance of equilibrium (see Coldcreation, 2011, Large-Scale Structures - Superclusters and Supervoids for more in depth coverage on the issue of stability and Curvature of a globally homogeneous isotropic general relativistic spacetime..., 2010, for a discussion about needlessness of the cosmological constant).

 

It may strike the reader as curious that it is general relativity itself that casts a pervasive shadow on the current standard model; just as Newtonian gravitation itself cast doubts on the cosmology that followed directly from it, with it's diverging integrals in a homogeneous universe. Yet in the case of general relativity the solution to the problem of stability in a pervasive homogeneous field was not so evident, in retrospect. It is understandable that the problem passed by Einstein, albeit not without notice.

 

Do I have my doubts about this scenario? I certainly do. But I also have doubts about the standard model. The main problem I see with this scenario is not how redshift z may be a result of curved spacetime, or even how stability is maintained, but if there is enough mass/energy in the universe to cause the global curvature of the manifold.

 

 

CC

[cruel2Bkind]

... the obvious (at least to you and me) truth derived from GR that redshift is a curvature effect,...

... both spherical and hyperbolic curvature induce redshift, ...

What is the mathematical justification for this claim?

Links to any papers that supply a rigorous mathematical rationale for this idea would be most welcome.

... it really is puzzling how cosmologist can say the expanding universe is based on GR and not use it to explain the single most important cosmological observation ever,...

Since no role is played by 'global curvature' in the geodesic equation for null-geodesics, but the scale factor of the Friedmann equations does play a role, then it seems that the above quote displays a lack of comprehension of the relevant physics.

May I be so bold as to suggest reading Misner, Thorne & Wheeler "Gravitation", where there is an adequate explanation of the occurrence of the cosmic redshift as a consequence of the field equations for a homogenous isotropic universe.

[coldcreation]

What is the mathematical justification for this claim?

Links to any papers that supply a rigorous mathematical rationale for this idea would be most welcome.

 

Since no role is played by 'global curvature' in the geodesic equation for null-geodesics, but the scale factor of the Friedmann equations does play a role, then it seems that the above quote displays a lack of comprehension of the relevant physics.

May I be so bold as to suggest reading Misner, Thorne & Wheeler "Gravitation", where there is an adequate explanation of the occurrence of the cosmic redshift as a consequence of the field equations for a homogenous isotropic universe.

 

First, cruel2Bkind, welcome to scienceforums. :)

 

There are several links that supply a rigorous mathematical rationale for this idea.

 

An Equilibrium Balance of the Universe by Ernst Fischer.

 

Homogeneous cosmological solutions of the Einstein equation also by Ernst Fischer (2009)

 

 

Abstract Homogeneous solutions in the framework of general relativity form the basis to understand the properties of gravitation on global scale. Presently favoured models describe the evolution of the universe by an expansion of space, governed by a scale function, which depends on a global time parameter. Dropping the restriction that a global time parameter exists, and instead assuming that the time scale depends on spatial distance, leads to static solutions, which exhibit no singularities, need no unobserved dark energy and which can explain the cosmological red shift without expansion. In contrast to the expanding world model energy is globally conserved. Observations of high energy emission and absorption from the intergalactic medium, which can scarcely be understood in the ‘concordance model’, find a natural explanation.

 

 

This solution leads to an exact solution of the Einstein field equations consistent with a static Einstein universe.

 

This Bruce paper leads to the same conclusion, albeit by different means. [PM me if you would like more details on this work.]

 

 

Note: While cosmological redshift would also occur in a (non-expanding) hyperbolic universe, both of the above have come to the conclusion that the universe is spherically curved, not hyperbolically curved. A flat Minkowskian spacetime is ruled out by observation, since there would be no cosmological redshift in such a universe (unless of course it were expanding).

 

 

See too for an analytical and qualitative approach here and here.

 

 

Simply put, the concept of 'global curvature' can best be understood in terms of Einstein's static spherical model, 1916-17, developed when Einstein explored the cosmological implications of General Relativity.

 

A geodetic line on a sphere is a great circle arc (in reduced dimensions). A geodetic line in a 4-dimensional spacetime (of constant positive curvature) is represented by rays of light. The wavelength of light (throughout the entire spectrum) emitted by distant objects in such an Einstein universe is shifted toward the red end of the spectrum, due to the geodesic trajectory of photons as they pass through the curved spacetime manifold. The further the object, the greater the redshift, since the magnitude of curvature increases with distance (just as the curvature of the earth manifests itself with increasing distance from the point of view of an observer).

 

This universe is homogeneous and isotropic on the largest scales.

 

 

CC

[HydrogenBond]

There is an interesting coincidence about the cosmic microwave background radiation. The wavelength of this microwave radiation is roughly 160GHz. Devices such as the gyrotron, use this microwave wavelength to preheat materials into a plasma state as part of the fusion reactors. Is there a connection between stella plasma heating and the CBM, since the wavelength just so happens to line up well with what is needed for an induced plasma induction.

 

The gyrotron is a type of free electron laser (microwave amplification by stimulated emission of radiation). It has high power at millimeter wavelengths because its dimensions can be much larger than the wavelength, unlike conventional vacuum tubes, and it is not dependent on material properties, as are conventional masers. The bunching depends on a relativistic effect called the Cyclotron Resonance Maser instability. The electron speed in a gyrotron is slightly relativistic (comparable to but not close to the speed of light). This contrasts to the free electron laser (and xaser) that work on different principles and which electrons are highly relativistic.

[cruel2Bkind]

Thanks for the welcome, coldcreation.

 

Upon finding Barry Bruce's book via Google Books, I was able to read his first chapter. The idea that cosmic redshift has a gravitational cause, and in an homogeneous universe can be expressed as a function of distance alone, was an idea I rejected more than 15 years ago shortly after I commenced acquainting myself with General Relativity. Its disproof is easily grasped by assuming such a cause of redshift and constructing a thought experiment involving two far-separated observers, stationary with respect to each other and each seeing the radiation from the other as red-shifted, one of whom starts by sending a signal to the other and subsequently both send signals to each other immediately upon receiving signals from each other.

 

Since they each see the other's signals as red-shifted then they must see the other's time as dilated. If one sends out a signal at frequency f for a duration t and the other sees that signal as having frequency f/k (k>1) then that other will perceive that signal as having a duration of k*t because the number of wavelengths of the signal cannot change.

 

So let observer A send a signal to B and wait a time a₁ before receiving the return signal from B.

A immediately resends to B.

After having received the first signal B immediately sends a signal to A and waits for a time b₁ to receive the second signal from A.

Since B sees A as red-shifted then

 

b₁/a₁ > 1 .

 

Let k = b₁/a₁. So each will see the other's time as dilated by a factor of k.

Having received the second signal B immediately resends to A.

A's wait time between receiving the first signal from B and receiving the second signal from B is a₂. Since A sees B as red-shifted then

 

a₂ = kb₁ .

 

So a pair of sequences of durations is constructed:

 

(a₁, a₂, ... a_n, a_n+1, ...)

(b₁, b₂, ... b_n, b_n+1, ...) .

 

Now for each n in the natural numbers,

 

b_n = ka_n

 

and

 

a_n+1 = kb_n .

 

So

 

a_n = k^2n-2a₁

 

and

 

b_n = k^2n-1a₁ .

 

The sequences can be now expressed as:

 

(a₁, k²a₁, k⁴a₁, ... k^2n-2a₁, k²ⁿa₁, ...)

 

and

 

(ka₁, k³a₁, k⁵a₁, ... k^2n-1a₁, k²ⁿ⁺¹a₁, ...) .

 

For A the time elapsed from sending the first signal to sending the nth signal is

 

A_n = a₁[k²ⁿ - 1]/[k² - 1] .

 

Consequently the interval

 

a_n = ([k² - 1]/k²)A_n + a₁/[k²] .

 

I.e. the later the signal is sent the longer the wait before receiving the return signal.

 

But such a result is ridiculous, thus the initial idea is ridiculous. Therefore there can be no such redshift caused by distance alone.

 

Since 'global curvature' is not mathematically addressed in either of Fischer's papers, my first query still remains on the table: what is the mathematical justification for a redshift caused by a non-zero 'global curvature'?

[cruel2Bkind]

A geodetic line on a sphere is a great circle arc (in reduced dimensions). A geodetic line in a 4-dimensional spacetime (of constant positive curvature) is represented by rays of light.[1] The wavelength of light (throughout the entire spectrum) emitted by distant objects in such an Einstein universe is shifted toward the red end of the spectrum, due to the geodesic trajectory of photons as they pass through the curved spacetime manifold.[2] The further the object, the greater the redshift, since the magnitude of curvature increases with distance (just as the curvature of the earth manifests itself with increasing distance from the point of view of an observer).[3]

 

This universe is homogeneous and isotropic on the largest scales.[4]

 

[1] In space-time the path taken by any "free-falling" particle (not just light) is a geodesic. "Free-falling" particles with mass move along time-like geodesics. Particles with no mass (eg photons) move along null-geodesics.

 

[2] For homogeneous isotropic static universes of constant curvature, what are the equations that show how the redshift is functionally dependent on the curvature?

 

[3] How can the 'magnitude of curvature' increase with distance, in a space of constant curvature?

 

[4] The universe is not homogeneous and isotropic on the largest scales according to Labini & Pietronero: "The complex universe: recent observations and theoretical challenges".

[Sci]

If red shift is not from spacial expansion then it would be that the photons are gradually losing energy, and are expanding, not space, this energy loss having to be quantized and thus not occurring very often, but, over large distances it might add up. It's not that I go for this, but wish to keep something in reserve in case the Big Bang was not really so, but I don't go for this, either, at the moment. Just like to explore every avenue. Blue shift would then be the gaining of energy.

[cruel2Bkind]

If red shift is not from spacial expansion then it would be that the photons are gradually losing energy, and are expanding, not space, this energy loss having to be quantized and thus not occurring very often, but, over large distances it might add up. It's not that I go for this, but wish to keep something in reserve in case the Big Bang was not really so, but I don't go for this, either, at the moment. Just like to explore every avenue. Blue shift would then be the gaining of energy.

Years ago I chanced upon a web page with a refutation of the hypothesis that cosmic redshifts result from a quantum mechanical or quantum field theoretical decay of photons, but I can't recall its URL. I can only recall two of the reasons it gave.

1. Such a decay would necessarily be probabilistic, and hence result in a far greater smearing of spectral lines than is actualy observed.
   
2. There is a strong correlation between redshifts of galaxies and time dilations of supernove explosions of stars within them.
   

If the universe is indeed neither homogeneous nor isotropic but actually fractal at all scales then current cosmology theory is wrong and the Big Bang hypothesis may well need to be discarded.

[coldcreation]

Thanks for the welcome, coldcreation.

 

Upon finding Barry Bruce's book via Google Books, I was able to read his first chapter. The idea that cosmic redshift has a gravitational cause, and in an homogeneous universe can be expressed as a function of distance alone, was an idea I rejected more than 15 years ago shortly after I commenced acquainting myself with General Relativity. Its disproof is easily grasped by assuming such a cause of redshift and constructing a thought experiment involving two far-separated observers, stationary with respect to each other and each seeing the radiation from the other as red-shifted, one of whom starts by sending a signal to the other and subsequently both send signals to each other immediately upon receiving signals from each other.

 

Since they each see the other's signals as red-shifted then they must see the other's time as dilated. If one sends out a signal at frequency f for a duration t and the other sees that signal as having frequency f/k (k>1) then that other will perceive that signal as having a duration of k*t because the number of wavelengths of the signal cannot change.

 

[...]

I.e. the later the signal is sent the longer the wait before receiving the return signal.

 

But such a result is ridiculous, thus the initial idea is ridiculous. Therefore there can be no such redshift caused by distance alone.

 

Since 'global curvature' is not mathematically addressed in either of Fischer's papers, my first query still remains on the table: what is the mathematical justification for a redshift caused by a non-zero 'global curvature'?

 

Actually, the problem of Bruce is not that cosmic redshift has a gravitational cause, or that in a homogeneous universe redshift can be expressed as a function of distance alone. Redshift can indeed be expressed as a function of distance (and time) since Bruce is using spherical coordinates. Certainly, in a flat space such a redshift would not be observed. Time dilation follows too as a function of distance in a spherically curved manifold, in accord with general relativity.

 

In the following chapters Bruce goes on to derive the metrics for hyperbolic, flat and spherical space. Bruce uses a radially symmetric spherical coordinate system. The result is similar to that found by Ernst Fischer, though Fischer using local Cartesian coordinates.

 

The problem arises when Bruce describes the behavior of a large-scale system (i.e., when extending the locally defined metric tensor to large distances). The only physically relevant quantities are the derivatives of this functional at zero distance. In a homogeneous universe the derivatives that describe the spatiotemporal connection to neighboring points must be constant throughout spacetime. Because of this Bruce has great difficulty in explaining observations that show redshift greater than one (the maximum redshift according to his solution).

 

 

 

[1] In space-time the path taken by any "free-falling" particle (not just light) is a geodesic. "Free-falling" particles with mass move along time-like geodesics. Particles with no mass (eg photons) move along null-geodesics.

 

[2] For homogeneous isotropic static universes of constant curvature, what are the equations that show how the redshift is functionally dependent on the curvature?

 

[3] How can the 'magnitude of curvature' increase with distance, in a space of constant curvature?

 

[4] The universe is not homogeneous and isotropic on the largest scales according to Labini & Pietronero: "The complex universe: recent observations and theoretical challenges".

 

[1] Locally, both redshift and time dilation result when particles with no mass (eg photons) move along null-geodesics in a gravity field, from the point of view of an observer judiciously placed. These are the well-known phenomena of gravitational redshift and time dilation. The idea here is that both redshift and time dilation occur on the large-scale too, as long as the pseudo-Riemannian manifold is of constant positive Gaussian curvature.

 

[2] I will see if I can find the exact equations you seek that show how redshift is functionally dependent on the curvature. Certainly they are to be derived from the Einstein field equations. Recall, Hubble's program (1926-1934) to find space curvature from the galaxy count: the Gauss protocol (Hubble, Tolman, 1935, ApJ 82, 302). This was based on experimental geometrical arguments inherent in general relativity. Years later (1977), Ellis, G.F.R., Is the Universe Expanding?, General Relativity and Gravitation, Vol. 9, No. 2 (1978), showed that “spherically symmetric static general relativistic cosmological space-times can reproduce the same cosmological observations as the currently favored Friedmann-Robertson-Walker universes.” In this case the systematic redshifts are interpreted as “cosmological gravitational red shifts.” What was previously ascribed to a time variation in an expanding frame is now ascribed to a spatial variation in properties of a static universe, as we observe the past light cone (in the look-back time). Weyl had considered this possibility in 1921, when he still sought a middle ground between the Einstein and de Sitter models (Kerszberg 1989 p.392). Ellis urges “a closer investigation of the field equations and astrophysical aspects of these models” and considers that the interpretation of an expanding universe (an idea that was first put forth by de Sitter) “is based on the assumption of spatial homogeneity, which is made on philosophical rather than observational grounds.”

 

[3] The 'magnitude of curvature' increase with distance in a space of constant curvature from the reference frame of any observer, even though curvature is constant. Locally space is flat (excluding local deviations in spacetime due to ponderable massive bodies), just as the surface of the earth appears flat out to a few hundred kilometers. Curvature begins to manifest itself beyond a certain range (astronomically, beyond the conventional limits of the Local Group). At great distances, and increasingly the further the distance under consideration, the magnitude of curvature increases from the point of view of an observer. This analogous to comparing the curvature of the earth between say New York and Washington DC, with the curvature of the earth between NY and and Shanghai. Though the curvature of the earth is relatively constant (spherical), the magnitude of curvature that distinguishes between these locations is large. Likewise, when we look into the deep universe the magnitude of curvature appears large compared with the local region.

 

[4] Thanks for the link. I'll read it and get back to you. Note: whether the matter distribution of the universe at scales greater than 200 or 300 Mpc is perfectly homogeneous and isotropic or not make little difference for this model. It just means that instead of a perfectly spherical manifold (of constant positive Riemannian curvature) there are irregularities in the redshift-distance relation. The global curvature still remains essentially spherical, with 'bulges' and depressions, not like mountains on earth (which are more like deviations of stars and galaxies astronomically), but like the bulges cause by the moon and sun. The earth is not homogeneously curved but as a generalization a sphere works fine.

 

The bigger problem, as you mention above (reproduced below) is the concept of a fractal universe (formerly known as a hierarchical universe: see Charlier, 1908). This would lead to the possibility that galaxy superclusters may indeed be gravitationally bound objects, something which obviously poses problems for the standard model.

 

 

[...]

[*]There is a strong correlation between redshifts of galaxies and time dilations of supernove explosions of stars within them.

If the universe is indeed neither homogeneous nor isotropic but actually fractal at all scales then current cosmology theory is wrong and the Big Bang hypothesis may well need to be discarded.

 

I don't know who such a fractal model could be tested "at all scales." We can only observe a section of the universe (inside our light cone). The rest would simply be inferences from indirect observations (e.g., gravitational effects on matter near the horizon).

 

The big bang model may be in need of revision (or be discarded) even if the universe turns out to be both homogeneous and isotropic. This is only one of the fundamental assumptions of the model. As long as any other fundamental assumption is incorrect such a revision is required (if not a wholesale renunciation of the model).

 

Finally, a strong correlation between redshifts of galaxies and time dilations of Type Ia supernovae within them is not at all incompatible with the curved spacetime interpretation of redshift z. On the contrary, it would be expected. In the case of homogeneous, isotropic cosmologies where the mass distribution is non-expanding, symmetry considerations would require the vanishing of the net gravitational force on a test particle, regardless of its location. Seeliger (1895, 1896) noted that adding an attenuation factor to the inverse-square law of gravitation would be sufficient to solve the problem. The larger the distances, the faster the force of gravity would fall-off relative to the standard inverse-square law. Such an attenuation factor would be negligible effects on scales compatible with the solar system, making empirical verification very difficult (Norton 2002), but not impossible. Curved spacetime produces the same deviation from the inverse square law on the propagation of light. This deviation manifests itself in the form of redshift z and time dilation that increases with distance from the reference frame of the observer.

 

See here: The important point to make is that the gravitational force between each particle on our spherical surface is not the same as the gravitational force between particles located on a flat, Euclidean, Minkowskian, or Newtonian manifold. Just as the propagation of light is affected by the Gaussian curvature, the gravitational force between particles diminishes with distance at a greater rate than predicted by the inverse-square law (ISL). In another way, the gravitational potential exerted on a test particle located in a manifold of constant positive Gaussian curvature with a homogeneous distribution of material particles converges with increasing volume. Integrals, rather than diverging to some arbitrarily large value (or even to infinity) as would possibly occur on a flat manifold, converge due to the geodetic propagation of force on the background manifold of constant positive intrinsic Gaussian curvature. There is a diminution of gravitational force with increasing distance, greater than inversely proportional to the square of the distance.

 

Just as a curved spacetime manifold solves Olbers' paradox by modifying the inverse-square law for the propagation of light, so too does a curved spacetime manifold solve the problem of diverging integrals by modifying the inverse-square law of gravitation. It is no miracle that the quantity by which the inverse-square law for the propagation of light is affected is exactly identical the quantity by which the inverse-square law of gravitation is affected. The deviation from the inverse-square law (for light and gravity) is directly proportional to the Gaussian curvature.

 

In another way, there is a divergence from the inverse-square law of gravitation (1/r^2) directly proportional to the magnitude of curvature of the manifold, and thus with increasing distance from the origin. The problem of diverging integrals inherent in a geometrically flat Newtonian manifold differs thus from the situation in an Einsteinian manifold of constant positive Gaussian curvature in such a way that the entire problem vanishes naturally.

 

[...] Einstein had shown that a slight modification of the inverse-square law of gravitation permitted a cosmology consistent with staticity and homogeneity (dissimilar in form but with the same result as Seeliger 1895, 1896, 1909, and Neumann 1896). The fact too that Newtonian gravitation could be adjusted to readmit cosmology by other means was immediately apparent (Norton 1999, pp. 297-303).

 

 

CC

[cruel2Bkind]

...

both redshift and time dilation occur on the large-scale too, as long as the pseudo-Riemannian manifold is of constant positive Gaussian curvature.[1]

...

Ellis, G.F.R., Is the Universe Expanding?, General Relativity and Gravitation, Vol. 9, No. 2 (1978), showed that “spherically symmetric static general relativistic cosmological space-times can reproduce the same cosmological observations as the currently favored Friedmann-Robertson-Walker universes.” In this case the systematic redshifts are interpreted as “cosmological gravitational red shifts.” What was previously ascribed to a time variation in an expanding frame is now ascribed to a spatial variation in properties of a static universe, as we observe the past light cone (in the look-back time)... Ellis urges “a closer investigation of the field equations and astrophysical aspects of these models” and considers that the interpretation of an expanding universe (an idea that was first put forth by de Sitter) “is based on the assumption of spatial homogeneity, which is made on philosophical rather than observational grounds.” [1]

...

Locally space is flat (excluding local deviations in spacetime due to ponderable massive bodies), just as the surface of the earth appears flat out to a few hundred kilometers.[2]

...

Curved spacetime produces the same deviation from the inverse square law on the propagation of light. This deviation manifests itself in the form of redshift z and time dilation that increases with distance from the reference frame of the observer. [3]

...

 

1. Irrespective of who wrote what, the above mentioned thought experiment in post #739 reveals that for a scenario of two observers, stationary with respect to each other and both observing the radiation from the other to be red-shifted, in any positively curved, negatively curved or flat space-time, an unsatisfactory result prevails.
    
   The duration experienced by either observer between sending a signal to the other observer and receiving the prompt return signal from the other observer depends on the total time elapsed since the commencement of the repeated signaling process; it does not have a singled value; it is multi-valued; it cannot be well defined.
    
   Hence, irrespective of the curvature of the space, it is impossible for each of a pair of mutually stationary observers to see light from the other as red-shifted. If any cosmological theory permits such an impossible scenario then it is a bad theory; it cannot be used to model the real world. If the addition of extra terms to the field equations permits that aforementioned impossible scenario then that addition is a blatant mistake; a theory that predicts nonsense is a nonsense theory!
    
   The only resolution is to unconditionally reject all models involving static universes in which both of a pair of mutually stationary observers can see each other as red-shifted. Either a pair of observers who do see each other as red-shifted cannot be mutually stationary, or a pair of observers who are mutually stationary cannot see each other as red-shifted, but not both mutually stationary and mutually red-shifted!
    
   
2. For the average person standing on a large "flat" plain on Earth the distance to the horizon is approximately 5 kilometres. The ancient Greeks knew that the earth was spherical because, from a vantage point on the shoreline, the tops of masts of approaching ships were visible before the hulls could be seen.
    
   
3. The occurrence of deviations from inverse square laws are not in dispute. Deviations from the inverse square law for EMR affect only the intensity of the radiation, they have absolutely no bearing on whether the frequency of the EMR is red-shifted, blue shifted or unchanged; a deviation from the inverse square law does not "manifest itself in the form of redshift z and time dilation that increases with distance from the reference frame of the observer". Intensity and frequency are independent concepts.
   (see Misner, Thorne & Wheeler "Gravitation", pp 570-583, §22.6 'GEOMETRIC OPTICS OF CURVED SPACETIME')
   

[HydrogenBond]

Since 'global curvature' is not mathematically addressed in either of Fischer's papers, my first query still remains on the table: what is the mathematical justification for a redshift caused by a non-zero 'global curvature'?

 

This can be done with entropy. If we expand a gas, the volume will cool, since the heat energy will be absorbed into entropy. What we get is a red shift, but at the level of IR. What we need is an entropy expansion that cools visible energy so it appears to have a red shift to the eyes. How about dark matter entropy.

 

Let me give an example of the opticial effects. Say we start with a spherical tank of gas. I tell someone this is far away and they need to look through the telescope to see it. I then open the valve and the tank red shifts in the IR as it gets very cold. It looks like it just doppler shifted near the speed of light. But if stays stationary. If I leave this trick unanswered but let it run its course, all types of explanations will appear such as curved space time wells.

[cruel2Bkind]

^^^^^^^^^^^^^^^^^^^^^^^^^^ Goin' fishin' H-H ?

---

In support of the idea that this universe can be modeled by a spacetime with a space of constant curvature and constant volume, the book by Bruce and the papers by Fischer have been given as references. The irresolvable problem in Bruce's model has already been dealt with, so attention is now focused on Fisher. His model also fails because of that aforementioned irresolvable problem dealt with in a previous post. However there is more criticism that can be directed at Fischer. In his paper "Homogeneous cosmological solutions of the Einstein equation" he displays a perspective of GR that is somewhat naive; his 'grasp' of the subject is questionable.

 

• He fails to mention that if the g₀₀ is only a function of distance then a time coordinate can be chosen to ensure that g₀₀ = -1 , thus simplifying the equations considerably without any loss of generality because the original case, where g₀₀ is a function of distance, can be determined by transforming back to the original time coordinate.
   
  
• At the start of his section "4 Properties of static solutions" he makes the statement:

  In a homogeneously curved space of curvature radius a, the relation between the distance s_i and the coordinate x_i near x_i = 0 in a local orthonormal coordinate system is given by
  

  s_i = a*actan[x_i /sqrt{a²-x_i²)]

  without presenting any justification or reference for it. This formula has the same structure as the formula that relates the distance across the surface of a sphere, between two points on the sphere, to the length of the perpendicular dropped from one of the points to the radius connecting the other point to the centre of the sphere. Thus Fischer's use of x_i herein is as a Schwarzschild radial coordinate r ( 2(pi)r is the "proper circumference" of a circle whose points have a radial coordinate r in a Schwarzschid spherical coordinate system centered on the centre of the circle; the actual length of the radius of such a circle in a constant curvature space is a*actan[r /sqrt{a²-r²)] ). There are other local orthonormal coordinate systems for which the proper distance is a different function of the coordinates. Fischer seems to be aware of only the system he employed.
   
  
• In the first paragraph of his section "5 Energy considerations" he has the statement:

  The term 2/a, which is balanced by a negative pressure in Einsteins static solution, is simply compensated by the purely geometrical change of time scale. The negative pressure appears as a property related to the curvature of space, not as an intrinsic property of some matter field.

  which displays a misunderstanding of the physics involved. The pressure can only come from some matter/radiation field because it is only there by virtue of it being a component of the energy momentum tensor. Thus his model necessitates an all pervading matter/radiation field that has a negative pressure. So his model has nothing to contribute to the understanding of this universe.
  

[Sci]

Years ago I chanced upon a web page with a refutation of the hypothesis that cosmic redshifts result from a quantum mechanical or quantum field theoretical decay of photons, but I can't recall its URL. I can only recall two of the reasons it gave.

1. Such a decay would necessarily be probabilistic, and hence result in a far greater smearing of spectral lines than is actualy observed.
   
2. There is a strong correlation between redshifts of galaxies and time dilations of supernove explosions of stars within them.
   

If the universe is indeed neither homogeneous nor isotropic but actually fractal at all scales then current cosmology theory is wrong and the Big Bang hypothesis may well need to be discarded.

 

Continuing with this alternate theory, which may turn out to be absurd, it would be that the actual decay emittance only happens about every two million years, and in the microwave range, which would form the CMBR.

 

As photons are asymmetrical, as noted since they cannot all pass through the crystal lattice of a polarizing substance, their length may be acted on unevenly by gravitation, causing a differential velocity gradient.

[coldcreation]

Excuse the time delay in answering this post:

 

Irrespective of who wrote what, the above mentioned thought experiment in post #739 reveals that for a scenario of two observers, stationary with respect to each other and both observing the radiation from the other to be red-shifted, in any positively curved, negatively curved or flat space-time, an unsatisfactory result prevails.

 

The duration experienced by either observer between sending a signal to the other observer and receiving the prompt return signal from the other observer depends on the total time elapsed since the commencement of the repeated signaling process; it does not have a singled value; it is multi-valued; it cannot be well defined.

 

Hence, irrespective of the curvature of the space, it is impossible for each of a pair of mutually stationary observers to see light from the other as red-shifted. [...]

 

The only resolution is to unconditionally reject all models involving static universes in which both of a pair of mutually stationary observers can see each other as red-shifted. Either a pair of observers who do see each other as red-shifted cannot be mutually stationary, or a pair of observers who are mutually stationary cannot see each other as red-shifted, but not both mutually stationary and mutually red-shifted!

 

The thought experiment argument is not substantiated, nor is the conclusion derived. The situation of A and B is completely symmetric. Therefore the constant k = b₁/a₁ must be equal to 1. Your arguments are based on the erroneous proposition k>1. All the further calculations are nonsense.

 

 

One of the basic concepts of differential geometry is that it is strictly local, describing the connection of points to neighboring points. Applying this method to the spacetime continuum forbids the definition of a global time-scale, just as it forbids the extension of a local Euclidean spatial coordinate system to extended systems, when space is curved. As by Lorentz invariance, time and space form an invariant connection. In curved space the time-scale can only be defined locally (there is no cosmic time). This is in accord with Ernst Fischer's model.

 

 

A further point to make according to general relativity, there is no unique method by which vectors at points separated by great distances can be compared in a curved spacetime, i.e., a definition of curved spacetime is the inability to compare vectors at different points. Therefor, in cosmology, there is an inability to distinguish between a Doppler effect, the expansion of space, and gravitational redshift. The interpretation of cosmological redshift z remains open.

 

Generally, in a curved spacetime manifold, the observed shift in frequency of a photon can be interpreted as a kinematic effect or a gravitational frequency shift (or even both together, superimposed), depending on the choice of coordinates.

 

In the case of the cosmological gravitational redshift interpretation, observer A and observer B (separated by a large distance) see each other's signal as if emitted at a lower elevation in a gravity field. Events at great distances appear to take longer than in the frame of the observer (clocks run slower when further removed). Observer A detects the phenomenon of time dilation and redshift from signal emitted by observer B. Observer B sees frequencies shifted and time dilation in the signal emitted by A. The relationship between the two stationary observers in entirely symmetric.

 

Any shift in frequency can be described as gravitational or Doppler. There is no "fact" about the cause of redshift z. The conclusion chosen is a function of the coordinate system or calculation method. (See here for example).

 

While it is generally believed that the kinematic origin of redshift z is the most natural interpretation, it is instructive to highlight that there exists another viable alternative that is just as natural (if not more so), based entirely on general relativity.

 

 

For the average person standing on a large "flat" plain on Earth the distance to the horizon is approximately 5 kilometres. The ancient Greeks knew that the earth was spherical because, from a vantage point on the shoreline, the tops of masts of approaching ships were visible before the hulls could be seen.

 

The point is that the earth appears flat locally. The further the distance, the more curvature will become apparent. So too, the universe appears flat locally (excluding local humps and bumps due to planets, stars and galaxies), and the further the distance, the more curvature will become apparent, even though the curvature is constant. The deviation from linearity manifests itself increasingly with distance (from the reference frame of the observer), whether it be on a reduced dimension spherical surface, or on a 4-dimensional manifold of constant positive Gaussian curvature.

 

In another way, curvature manifests itself increasingly with distance, i.e., it becomes more and more apparent, the further one looks (even though curvature is constant).

 

Thus redshift increases with distance, along with the associated time dilation factor.

 

 

The occurrence of deviations from inverse square laws are not in dispute. Deviations from the inverse square law for EMR affect only the intensity of the radiation, they have absolutely no bearing on whether the frequency of the EMR is red-shifted, blue shifted or unchanged; a deviation from the inverse square law does not "manifest itself in the form of redshift z and time dilation that increases with distance from the reference frame of the observer". Intensity and frequency are independent concepts.

(see Misner, Thorne & Wheeler "Gravitation", pp 570-583, §22.6 'GEOMETRIC OPTICS OF CURVED SPACETIME')

 

The deviation from the inverse square law of gravitation was mentioned earlier not to explain the cause of redshift or time delay. These result from the constant Gaussian curvature itself. Such a deviation of the inverse-square force law would explain why the gravitational potential (a la Newton) would not diverge toward infinity in an homogeneous universe, thus opening the route to static solutions. A perturbed gravitational inverse-square law modifies the Poisson equation, which affects the growth of over-dense regions and thus affects the large-scale structures (e.g., superclusters) here claimed to be gravitationally bound systems, without the need of dark matter (CDM) or dark energy (DE). This is a deviation from Newtonian gravity at large scales.

 

Certainly, such a deviation of the inverse square law of gravitation would affect the propagation of photons. What would manifest itself is nonlinearity. Redshift z and time dilation increase with distance from the reference frame of the observer regardless of the deviation. What changes is that the regime is now nonlinear. There is no longer a one to one relation between distance and redshift. High-z objects (e.g., SNe Ia) will appear further than their standard redshift-distance would indicate, and light curve rises times will appear slower or longer than would in an otherwise linear regime.

 

Indeed, the deviation in linearity observed is essentially a precision test of the inverse-square force law on scales up to that of the horizon. It is by virtue of the Gaussian curvature itself, induced by the nonzero mass-energy density, that the deviation arises.

 

In general relativity the mass-energy density affects the electromagnetic field itself.

 

Olber's paradox is solved in a static universe, since there is a modification of the inverse-square law for EM in a curved spacetime, regardless of the modification of the inverse-square law for gravity.

 

CC