Clarifying my position
Hi all,
pervect said:
there is apparently more to this question. See for instance
http://www.math.ucr.edu/home/baez/PUB/deser
I also believe that Jheriko was probably referring to the work by Deser et al. on an infinite sequence of corrections to the naive linearized theory, which eventually "yields general relativity". (This is one of the most difficult, but most intriguing, routes to "deriving gtr from first principle".)
pervect said:
If we assume (for the sake of argument) that there is a self-consistent theory that enforces a particular topology on space-time but is somehow locally equivalent to GR,
This is an essential point (unfortunately probably too sophisticated for PF, since it might take years of graduate level study to appreciate "local" versus "global" issues, and most readers here do not possesses this kind of background). There were one or two threads in sci.physics.research on this point long ago, where I and others pointed out that newcomers to the literature can easily misunderstand claims from string theory proponents, for example.
pervect said:
If I understand Chris Hillman's position on this issue correctly, he believes that there isn't even a self-consistent theory of this nature (?).
I insist that the "local versus global distinction" is absolutely critical when examining claims that some theory constitutes a "reinterpretation" or "reformulation" of gtr. In particular, I believe that the work of Deser et al. (which is solidly mainstream) need to be carefully interpreted in this light. That is, if I am not mistaken, Deser et al. show that under their assumptions, in
any sufficiently small neighborhood one must obtain something indistinguishable from gtr. I would add that it shouldn't be surprising that a classical field theory of gravitation, which is a metric theory, might have difficulty in unambiguously determining a unique topology, or that for many "initial values", solutions in such a theory might develop Cauchy horizons, so such difficulties appear to be common to a large class of theories.
As far as I tell, it is not yet known whether some well-defined theory of gravitation exists which is "locally equivalent" to gtr, but which in some sense excludes solutions which are spacetimes with nontrivial topology. Although there are many claims to this effect in the literature, as far as I can recall, I consider the ones I have studied unconvincing or even incorrect. And I think we must expect that obtaining the required "topological filter" in a convincing fashion might be very difficult. It appears to me that this would require exiting the domain of classical field theories.
One can also ask whether or not there is yet rock-solid evidence for nontrivial topological features of the universe in which we live. Or perhaps better put: one can ask whether or not there is rock-solid evidence that no model in gtr (Lorentzian four-manifold plus any additional mathematical structure required to describe nongravitational physics in the model) which fails to feature nontrivial topology can be consistent with all the available evidence. As far as I know, a reasonable answer would be "not yet, but astrophysics seems to be generally headed in that direction".
Note that nontrivial topological features could arise in many ways:
1. It might turn out that the "best-fit" FRW models are actually quotient manifolds of an FRW lambdadust model, having nontrivial topology (c.f. Cornish and Weeks),
2. Of those (lamentably rare!) known exact solutions in gtr which have clear and unobjectional physical interpretations, including many models of black holes, many do feature nontrivial topology. (For example, the Kerr vacuum is homotopic to the real line with circles attached to each integer, and the deSitter lambdavacuum is homeomorphic to {\bold R} \times S^3.) However, "idealized but realistic models" would presumably be (at best) nonlinear perturbations of exact solutions with nontrivial symmetries, so to tell whether or not gtr firmly predicts nontrivial topological features in realistic scenarios, one would have to characterize a local neighborhood (in the solution space) of one of these solutions. At present, the only rigorous results appear to concern models like Minkowski vacuum (small nonlinear perturbations of Minkowski vacuum are indeed homeomorphic to {\bold R}^4 and de Sitter lambdavacuum (small nonlinear perturbations of de Sitter lambdavacuum are indeed homeomorphic to {\bold R} \times S^3). Caution: these results are actually a bit weaker than we would really want, even in the case of these particular neighborhoods, which are unfortunately not the ones we really want.
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