I said:
nutgeb said:
In the Milne model, the degree of negative curvature (i.e., the radius of curvature) must be finely tuned in the initial conditions, such that the amount that space "bends" as a function of distance is proportional to the increasing velocity as a function of distance. ... Is it reasonable to expect empty space to accommodate such infinite flexibility in the scale factor, while spontaneously generating the corresponding needed amount of negative curvature?
I realize that my posts on this subject have wandered around, but I think I’ve hashed through this enough to come to some conclusions. I had started with the assumption (or hope) that Birkhoff’s Theorem could be combined with SR to find equivalence between a snapshot of the FRW model and the Schwarzschild static model. I will describe why I no longer think that’s true. The best way to sort this problem out is to contrast the FRW metric under the ‘expanding space’ paradigm with the Schwarzschild metric under the “kinematic” paradigm, with Lambda=0. First I will set out the RW line element (which is the relevant part of the FRW metric):
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\{ closed, flat, open \right\}
FRW metric with 'expanding space' paradigm: In this paradigm, empty, static space must start out with an intrinsic, maximal negative curvature. Then the 'expansion of space' causes space itself to automatically stretch exactly in proportion with the growing scale factor, causing the negative curvature to progressively flatten out, i.e. become less curved per unit of proper distance. In this respect the model does not require fine tuning of the initial conditions.
Comoving observers remain "stationary" while the space between them expands. Since they are stationary, there are no SR velocity-related effects at all. Instead, in FRW there is negative spatial curvature. In mathematical terms, the FRW metric simply inserts the hyperbolic
sinh function in the space part of the metric, as can be seen in the RW line element. The metric seems to require that the underlying distribution of comovers would be hyperbolically contracted if empty space were flat. Such that when this hyperbolic distribution of comovers is viewed through hyperbolically curved space, it actually looks flat, in the sense that the distribution of comovers looks homogenous. This is nicely pictured in Figure 4.1 of Tamara Davis’ http://arxiv.org/abs/astro-ph/0402278v1" .
The intrinsic hyperbolic distribution of comovers required by the metric seems oddly out of place, because nothing is (or ever was) moving in the ‘expanding space’ model. What effect other than velocity would cause this exactly Lorentz-equivalent hyperbolic distribution? It is equally curious that FRW doesn’t apply the same requirement of intrinsic SR time dilation, which normally would accompany Lorentz-contraction as a velocity-induced effect. The FRW metric includes no time dilation element; the time part of the FRW metric can be seen to be linear. This allows comovers to share the same cosmological time even in an empty model.
Adding a critical density of matter in the FRW ‘expanding space’ paradigm introduces positive spatial curvature which exactly offsets the intrinsic negative curvature of empty space, so space becomes flat. The RW line element accomplishes this by simply deleting the
sinh function from the space part of the metric. That’s fine, but one then wonders, what became of the original the distribution of comovers that would have been hyperbolically non-homogeneous if the space were intrinsically flat? In other words, how can the same distribution of comovers be homogeneous in negatively curved space and yet remain homogeneous when space becomes flat? I suppose one answer is that the expansion of space over time could (almost) accomplish that feat if it (almost) infinitely stretched out the scale factor. Inflation theory uses a similar explanation. I’ll adopt that answer for now, although it suggests that the FRW metric for an empty, expanding model permits a homogeneous distribution of comovers only at physical scales that are infinitesimal compared to the scale at which a homogeneous distribution is observed in flat space.
The FRW metric does not introduce any time contraction on account of the added matter, so comovers continue to share the same cosmological time and homogeneity.
Schwarzschild metric with ‘kinematic’ paradigm: In this paradigm, empty, static space starts out intrinsically flat. When recession velocity is introduced, comoving observers really are in motion relative to each other. This motion causes Lorentz-contraction and SR time dilation as between comovers. There is no underlying curvature to offset the Lorentz effects. The distribution of comovers (as viewed by each other) is hyperbolically non-homogeneous due to the Lorentz-contraction. Comovers do not share a common cosmological time, because time is dilated.
Adding a critical density of matter in this paradigm again introduces positive spatial curvature which exactly offsets the Lorentz-contraction of comovers. The Lorentz SR time dilation at escape velocity is:
d\tau = \gamma dt = \frac{dt}{\sqrt{1 - v^{2} /c^{2}}} = \frac{dt}{ \left( 1 - \frac{2M}{r} \right) ^{1/2} }
Although the spatial curvature theoretically has become locally positive everywhere (as measured by an observer at zero-density infinity,) it is measured to be locally flat as between all comovers, who see each other to be homogeneously distributed.
One could argue that adding matter introduces no gravitational time dilation or contraction in the Schwarzschild ‘kinematic’ model, but I now think the better answer is that it introduces the amount of gravitational time contraction specified by the Schwarzschild
interior metric. In the spatially flat model expanding exactly at escape velocity, this amount of gravitational time contraction is never enough to mathematically entirely offset the SR time dilation. So an element of time dilation remains, and comovers still do not share a common cosmological time. (Note in passing that in a very
over dense model in which expansion velocity is less than escape velocity, there is a balance point at which the Schwarzschild interior time contraction
does match the SR time dilation, and comovers can share a common cosmological time, but this does not coincide with spatial flatness.)
. . . . . . . . . .
This comparison of the FRW ‘expanding space’ and Schwarzschild ‘kinematic’ paradigms shows that their treatment of the time element for comovers is different and irreconcilable. There is no direct Schwarzschild kinematic analogy for the relativistic time observed by FRW comovers. One must sadly conclude that while Birkhoff’s Theorem can model the spatial aspects of comovers in the FRW metric, it cannot be combined with SR Lorentz effects to model the time relationship between comovers.
This analysis reinforces the limitations on how SR Lorentz effects can be used within an FRW model. Since FRW comovers are not actually moving, SR doesn’t apply to their recession "motions", even by analogy. However, I see no reason why both Lorentz space and time effects can’t be applied validly to peculiar motions (i.e. the difference between proper motion and the local Hubble rate). That could explain why we can clearly observe many kinds of SR effects across distances far greater than the infinitesimal extent of a true "local" inertial reference frame, despite the fact that we are always immersed in the cosmic gravitational background.
This analysis also supports the proposition that SR and gravitational time dilation are not contributing causes of the cosmological redshift, because that would require net time dilation as between FRW comovers.
I don’t want to express any preference here for the FRW ‘expanding space’ paradigm or the Schwarzschild ‘kinematic’ paradigm. I’m just explaining why they seem to predict observational differences, due specifically to the incompatible ways in which they treat time dilation.