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.(adsbygoogle = window.adsbygoogle || []).push({});

Recall that in the 'kinematic' paradigm (sometimes called the 'ballistic' paradigm), galaxies movethroughpre-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 havepeculiar velocitiesof zero. However, their radialproper velocities(proper distance as a function of time) relative to each other are not zero, they are equal to the Hubble VelocityHD, whereHis the Hubble rate andDis 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:

[tex] 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\} [/tex]

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 theinteriorof a body of constant density. That's because we want to compare the time dilation at thecenterwith the dilation at thesurfaceof 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 thatnogravitational 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 bytwonon-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 cancelled 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 outboththe 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 incompatibleat some levelwith 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.

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

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