Equivalence of the 'expanding space' & 'kinematic' paradigms

[nutgeb]

The Equivalence Principle suggests the possibility that a 'kinematic' paradigm might produce exactly the same cosmic observations as the standard 'expanding space' paradigm. In my view, functional equivalence demands that the 'kinematic' paradigm make the same observational predictions as the other paradigm, comply with the cosmological principle, and not make predictions contrary to existing physics.

Recall that in the 'kinematic' paradigm (sometimes called the 'ballistic' paradigm), galaxies move through pre-existing space, while in the 'expanding space' paradigm, the underlying hypersphere of spatial geometry is what expands, carrying galaxies along with it and causing new space to come into existence between the galaxies.

I've kicked the idea of equivalence around quite a bit and have flip-flopped between optimism and pessimism. Now I think that the two paradigms can be reconciled and are functionally equivalent, at least for a spatially flat model at critical density.

The FRW metric is the accepted mathematical model for the 'expanding space' paradigm with an idealized homogeneous and isotropic matter distribution. To represent the 'kinematic' paradigm, I have selected the Schwarzschild metric, coupled with SR when applicable. Both FRW and Schwarzschild are exact GR solutions. My goal is to examine whether the Schwarzschild solution, along with SR, can be applied in a way that matches the predictions of FRW.

'Fundamental comovers' in the FRW metric are objects (e.g. galaxies) which are comoving exactly with their local Hubble recession flow, and therefore have peculiar velocities of zero. However, their radial proper velocities (proper distance as a function of time) relative to each other are not zero, they are equal to the Hubble Velocity HD, where H is the Hubble rate and D is the proper distance between them.

The FRW metric. At the heart of the FRW metric is the RW line element, which varies depending on whether the universe is overdense, flat (i.e., at critical density), or underdense:

$$ds^{2} = -dt^{2} + a^{2}(t) \left[ dx^{2} + \left\{sin^{2} \chi, \chi^{2}, sinh^{2} \chi \right\} \left( d\theta^{2} + sin^{2 \theta} d\phi ^{2} \right) \right] \\ \left\{ overdense, flat, underdense \right\}$$

An important characteristic of the FRW metric is that no relativistic time dilation exists as between fundamental comovers: comoving coordinate time equals proper time. Fundamental comovers all share the same cosmological proper time since the Big Bang, and the same simultaneity. In theory they can manually synchronize their local clocks (e.g. by exchanging electromagnetic signals) and they will remain synchronized thereafter. The absence of time dilation can be seen by examining the RW line element (above), the time part of which is linear. Therefore, no 'net' SR time dilation and/or gravitational time dilation can occur as between fundamental comovers.

Also, in a 'flat' FRW model at critical density (which observations indicate our universe closely resembles), no spatial curvature exists as between fundamental comovers: comoving coordinate distance equals proper distance. Again, the space part of the flat RW line element is linear. Therefore, for example, no 'net' SR Lorentz contraction can occur as between fundamental comovers.

The metric for the flat FRW model is such that if a sphere of arbitrary radius and origin location is defined in homogeneous space, then fundamental comovers located at that radius will always have recession velocities, relative to the origin, exactly equal to the Newtonian escape velocity of the total mass contained within the sphere (the mass parameter).

The Schwarzschild metric. This metric can model only the 'static' gravitational field of a spherically symmetrical central mass embedded in flat Minkowski space. It cannot directly model the dynamic expansion of the cosmic scale factor, and the changing cosmic density, as FRW can. However, the Schwarzschild metric can be used with manually updated density parameters to produce a series of 'snapshots' (time foliations) each of which provides a static representation of an FRW model at a particular instant in its expansion. Birkhoff's Theorem says that Schwarzschild is the correct metric for any spherically symmetrical mass distribution embedded in flat space. We can carve a sphere of arbitrary radius in a homogeneous area of space, and treat it mathematically as if all the matter were compressed to a central point mass, surrounded by vacuum. Shells of matter outside the sphere have no effect and can be ignored.

Space is not locally flat in Schwarzschild near the central mass. The mass' gravity causes positive spatial curvature, which results in radial stretching of space relative to the central mass. The closer a test particle is to the central mass, the more spatial stretching occurs as between the test particle and the central mass.

If we follow the flat FRW example by giving the test particle a recession velocity radially away the central mass equal to the latter's escape velocity, we might expect SR Lorentz contraction to occur. Lorentz contraction causes radial compression of space that increases with velocity. In the Schwarzschild configuration with a single central mass, the closer the test particle is to the central mass, the greater the escape velocity, and therefore the greater the Lorentz contraction.

It can be readily calculated that at escape velocity, the SR Lorentz contraction exactly offsets the Schwarzschild positive spatial curvature, at any radius. (I'll demonstrate the math on request.) As a result, space is measured to be flat as between the central mass and the test particle. Thus we have achieved equivalence with the space part of the flat FRW metric.

Now let's see if we can match the undilated time element of the FRW metric for fundamental comovers. Our test particle, moving radially outward at escape velocity, might be expected to experience both gravitational and SR time dilation relative to the central mass.

If there is gravitational time dilation, it should be calculated using the version of the Schwarzschild metric for the interior of a body of constant density. That's because we want to compare the time dilation at the center with the dilation at the surface of the sphere. (By comparison, the exterior Schwarzschild metric measures the time dilation at the surface, as compared to an infinitely distant observer who experiences no effects of gravity from the central mass.)

However, I think there is a strong argument to be made that no gravitational time dilation occurs, due to the symmetry of the cosmological matter configuration. Unlike the asymmetrical Schwarzschild configuration with a single central mass, we are really trying to model what happens in a space uniformly filled with homogeneously dispersed matter. That scenario is more accurately modeled by two non-overlapping spheres of matter which touch at one point on their surfaces. When we compress the matter in each identical sphere into a central mass, we see that the two resulting central masses are identical. No gravitational time dilation can occur as between two identical point masses.

Next let's consider whether our test particle, moving at escape velocity, experiences SR time dilation relative to the central mass. I submit that no SR time dilation occurs, due to the gravity arising from the presence of a critical density of mass. The explanation is as follows.

SR time dilation and Lorentz contraction are flip sides of the same coin. The two always occur together, never separately. That is implicit in SR spacetime diagrams. This duality requirement is necessary so that light always travels at the speed of c in every local inertial frame. In our scenario, the Lorentz contraction of the distance between the point mass and the test particle has been exactly offset (or canceled out) by the positive spatial curvature caused by the central mass, resulting in spatial flatness. If we allowed SR time dilation in this scenario, we would find that the locally measured speed of light departs from c. That cannot be allowed to happen, so SR time dilation must be excluded. (And we've already concluded that gravitational time dilation does not enter into this scenario.)

I think it's reasonable to ask, if SR time dilation and Lorentz contraction always occur together, then how can we say no time dilation occurred, while at the same time saying that the Lorentz contraction must be present in order to cancel out the positive spatial curvature? It does seem contradictory. I think the contradiction is resolved by reversing the causation, and saying that the positive spatial curvature is what cancels out both the Lorentz contraction and the time dilation. It makes sense that the positive spatial curvature cannot cancel out the Lorentz contraction while leaving the time dilation intact. But we must also acknowledge that if the SR effects were entirely omitted, there would be nothing to cancel out the positive spatial curvature. So SR's role in the calculation is mandatory, but circumscribed.

(I acknowledge that the interior Schwarzschild metric does not supply its own element of time dilation that directly offsets the SR time dilation. That would make things too simple.)

It is widely recognized that SR is incompatible at some level with the presence of gravity. This is explained by the fact that SR requires a single 'global' inertial reference frame that encompasses both parties, but such an extended frame cannot exist in a gravitational field. Yet cosmology often ignores this supposed incompatibility, even across distances far beyond any local inertial frame. For example, the dipole in the CMB is conventionally interpreted as an SR redshift caused by Earth's (and the Milky Way's) peculiar velocity, superimposed on the normal cosmological redshift that would occur if we were exactly comoving with our local Hubble flow (which we are not).

This suggests that in the flat FRW model, SR is uniquely incompatible with respect to the recession velocity of fundamental comovers, but it is entirely compatible with respect to the peculiar velocities of non-fundamental comovers. This is a very important distinction, which underlies my explanation of the equivalence of the two paradigms.

Conclusion. We have analyzed how to apply the Schwarzschild and SR metrics to an expanding, homogeneous matter distribution, and have described how we can model the spatial flatness and absence of time dilation between fundamental comovers that the flat FRW metric requires. Thus it seems reasonable to conclude that the 'expanding space' and 'kinematic' paradigms are functionally equivalent.

 

[Chalnoth]

I would go one simpler. Just take Einstein's equations (neglecting the cosmological constant for simplicity):

$$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

On the left hand side we have space-time. On the right we have the matter. Once we have the distribution of matter, therefore, we have already fully-specified the behavior of space-time (up to the possibility of some gravitational waves). The inverse is also true: once we have the space-time, we have fully-specified the energy/pressure/stress distribution of matter. So if we formulate everything with respect to how the right hand side changes with time, we'll get the same exact result as if we formulate everything with respect to how the left hand side changes with time. Of course, to get the right answer you actually have to use both sides of the equation, but whether we're talking about "expanding space" or "kinematic" as descriptions, we're just talking about what side of the equation to formulate things with respect to.

 

[nutgeb]

Once we have the distribution of matter, therefore, we have already fully-specified the behavior of space-time (up to the possibility of some gravitational waves). The inverse is also true: once we have the space-time, we have fully-specified the energy/pressure/stress distribution of matter. So if we formulate everything with respect to how the right hand side changes with time, we'll get the same exact result as if we formulate everything with respect to how the left hand side changes with time. Of course, to get the right answer you actually have to use both sides of the equation, but whether we're talking about "expanding space" or "kinematic" as descriptions, we're just talking about what side of the equation to formulate things with respect to.

I think that's a helpful point Chalnoth. It is encouraging to know what the ultimate answer is before you starting start trying to work out the details! I think your conclusion is correct that the EFE require that the 'expanding space' and 'kinematic' paradigms be functionally equivalent. So any analysis that contradicts that conclusion must be flawed.

Both the FRW and Schwarzschild metrics are exact solutions to the EFE. The FRW metric includes a velocity parameter that presumably allows any SR effects to be dealt with. The Schwarzschild metric alone does not include a velocity parameter, so it needs to be manually combined with the SR gamma metric. I find that to be the difficult part, because if one tries to separately combine the space and time components of Schwarzschild and gamma respectively, the resulting time calculations are incorrect for a homogeneous matter distribution. So heuristic adjustments, like those I described for the time metric, are needed to make the combined metrics fit together to enable correct calculations. In this case, the whole is not the sum of its parts. This is not surprising, since the Schwarzschild metric alone does not inherently model a homogeneous and expanding matter distribution throughout space.

 

[nutgeb]

Here's a description of the equivalence of the 'expanding space' and 'kinematic' paradigms in an empty Milne model and a vanishingly empty FRW model:

Milne: Consider first an empty Milne model cosmology in Minkowski space. There is no gravity, so this is a straight SR model. Test particles all depart the origin at t = 0 and move away from it in uniform quantities at all speeds up to but not including c. At any point in time the distribution of test particles (as observed from the origin) is inhomogeneous, because the farther a particle is from the origin, the faster its recession velocity, and therefore the more Lorentz contracted the radial distance is between the origin and the particle. Thus the farther a test particle is from the origin, the greater the radial contraction of the space is between it and the origin.

Clocks attached to the test particles are time dilated (relative to the origin clock) in the same proportion as the Lorentz contraction of the space between the test particle and the origin.

Vanishingly Empty FRW: I use the description "vanishingly empty" because technically the FRW metric does not apply to a completely empty model. Regardless, the matter densitiy can be set so vanishingly small that any effect it has in the metric is entirely insignificant.

The physical description of this model is the same as the Milne model, except that the vanishingly empty FRW metric says that empty space is not flat Minkowski space, it actually has negative (hyperbolic) spatial curvature. This can be seen in the RW line element in the OP, as the hyperbolic sinh function is inserted in the underdense version of the metric.

Hyperbolic negative spatial curvature causes radial expansion which increases with distance from origin. Thus the further a test particle is from the origin, the greater the radial expansion of the space is between it and the origin.

The SR Lorentz contraction is the same in this model as it is in the Milne model, causing radial contraction that increases with distance from the origin. This hyperbolic contraction exactly offsets the hyperbolic stretching of space due to the FRW negative spatial curvature. The result is that, unlike in the Milne model, the test particles are now seen to be homogeneously distributed across all radial distances. Has space now actually become flat in the vanishingly empty FRW model, or does it just appear to be flat? Arguably it is a distinction without a difference. In any event, this outcome is unavoidable, because the FRW metric requires a homogeneous distribution of particles regardless of whether the matter density is underdense, at critical density, or overdense.

How about the time metric? As we've discussed, the FRW metric always prohibits time dilation between fundamental comovers, regardless of the matter density. The absence of such time dilation is hard-wired into the RW line element. So we must conclude that the negative spatial curvature of the vanishingly empty FRW model is accompanied by some element that either offsets or cancels out the SR time dilation caused by the recession velocities of the test particles.

There is no offsetting time contraction component apparent in the scenario. So I submit that, like in the flat FRW model, the SR time dilation is in fact cancelled out because the Lorentz contraction was canceled out (by the negative spatial curvature). As discussed in the OP, SR time dilation cannot occur without SR Lorentz contraction, because if it did, the locally measured speed of light would vary from c. So we can conclude that gravity itself is not the direct cause of the SR time dilation being eliminated, because the effect of gravity is insignificant in the vanishingly empty FRW model. We must conclude that the SR time dilation is eliminated directly by the presence of any spatial curvature that exactly offsets the SR Lorentz contraction.

The Schwarzschild metric revisited: The reader may note what appears to be a contradiction of the analysis here with the OP. In the OP analysis of the Schwarzschild metric, it was the introduction of positive spatial curvature that offset the Lorentz contraction arising from the recession velocity, while here it is the introduction of negative spatial curvature that performs the same role. How can positive and negative spatial curvature equally offset Lorentz contraction?

This is a potentially confusing point, but there is a subtly which resolves it. Recall that in the Schwarzschild metric, the radial expansion of space increases the closer one is to the point mass at the origin. That's because with a single mass, gravity increases toward the center. However, in the vanishingly empty FRW model, the negative spatial curvature increases hyperbolically with increasing distance from the origin. Therefore, rather unexpectedly, positive spatial curvature in Schwarzschild has the same functional characteristic as negative spatial curvature in FRW.

Also note that in the Schwarzschild model, escape velocity increases the closer the test particle is to the central mass -- meaning that in the spatially flat model, SR Lorentz contraction increases toward the center. In the FRW model with homogeneously dispersed matter, escape velocity increases with increasing distance from the origin -- meaning that SR Lorentz contraction increases away from the center. Thus in both the Schwarzschild and FRW, the 'gradient' of SR effects increases in the same radial direction that the 'gradient' of the gravitational effects increases. But that radial direction is opposite in Schwarzschild (inward) compared to FRW (outward).