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General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:

  1. What topological restrictions do Einstein's equations put on the manifold. For instance, the existence of a Lorentz metric implies some topological things, like the euler characteristic vanishing.

  2. Are there any experiments being done or even any hypothetical experiments that can give information on the topology? E.g. is there a group of graduate students out there trying to contract loops to discover the fundamental group of the universe?

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6 Answers 6

That's a great question ! What you are asking about is one of the missing links between classical and quantum gravity.

On their own, the Einstein equations are local field equations:

$$ G_{\mu\nu} = 8 \pi G T_{\mu\nu} $$

and do not contain any topological information. At the level of the action principle:

$$ S_{eh} = \int_\mathcal{M} d^4 x \sqrt{-g} \mathbf{R} $$

the term we generally include is the Ricci scalar $ \mathbf{R} = Tr[ R_{\mu\nu} ] $, which depends only on the first and second derivatives of the metric and is, again, a local quantity. So the action does not tell us about topology either, unless you're in two dimensions, where the Euler characteristic is given by the integral of the ricci scalar:

$$ \int d^2 x \mathcal{R} = \chi $$

(modulo some numerical factors). So gravity in 2 dimensions is entirely topological. This is in contrast to the 4D case where the Einstein-Hilbert action appears to contain no topological information.

This should cover your first question.

All is not lost, however. One can add topological degrees of freedom to 4D gravity by the addition of terms corresponding to various topological invariants (Chern-Simons, Nieh-Yan and Pontryagin). For instance, the Chern-Simons contribution to the action looks like:

$$ S_{cs} = \int d^4 x {}^\star R \, R $$

where $ R \equiv R_{abcd} $ is the Riemann tensor and $ {}^\star R_{abcd} = 1/2 \epsilon_{ab}{}^{ij} R_{cd\,ij} $ is its dual. Here is a very nice paper by Jackiw and Pi for the details of this construction.

There's plenty more to be said about topology and general relativity. Your question only scratches the surface. But there's a goldmine underneath ! I'll let someone else tackle your second question. Short answer is "yes".

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Thanks for the answer. I do not see why EFEs cannot contain topological data since you need a global solution to them (you can solve it locally but they need to patch together to form a global metric). For example, if the EFEs implied something like positive scalar curvature then that would really limit the topology (being positive at a point is local, being positive everywhere is global). The adding of topological invariants looks very interesting-- I'll have to read more into it. –  Eric Dec 10 '10 at 15:46
    
I get what you're trying to say. The EFE's should encode some sort of topological information aside from the addition of topological terms to the action. Or perhaps that is because we consider the EFE's to be fundamental, when the Ricci term and the other topological terms can arise from something more general such as $BF$ theory Reference which is a topological theory. Anyhow, if you like the answer could you accept it as the answer. Thanks :-) –  user346 Dec 10 '10 at 19:43
    
@user346 "So gravity in 2 dimensions is entirely topological" Could you please expand on this in a less technical terms for me? –  Leos Ondra May 2 '13 at 11:15
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Just one additional point that I haven't seen mentioned above: if the space-time has non-trivial fundamental group, it won't be seen by an observer at infinity. This is the content of the Topological Censorship Theorem. The implication is that for an asymptotically flat space-time, any interesting topology will be hidden behind the event-horizon. The proof of the theorem is rather surprisingly simple: it is more or less a direct extension of Penrose's singularity theorem.

See:

Friedman, J. L.; Schleich, K. & Witt, D. M. Topological censorship Phys. Rev. Lett., American Physical Society, 1993, 71, 1486-1489

Schleich, K. & Witt, D. M. Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes http://arxiv.org/abs/1006.2890

Galloway, G. J.. On the topology of the domain of outer communication. Class. Quantum Grav. 12 No 10 (October 1995) L99 (3pp)

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You are a mathematician, correct? So please explain things at the level of a physicist to me :-) My question is, how does this conclusion change is the spacetime is asymptotically deSitter or anti-deSitter? Also what is your view on the dodecahedral universe hypothesis? –  user346 Dec 13 '10 at 23:41
    
Very interesting! Thanks for this answer. –  Eric Dec 14 '10 at 0:26
    
@space_cadet: I don't know much about the dodecahedral universe hypothesis, but from what I know, isn't it an attempt to explain certain "features" of WMAP data? I don't think there's any a priori reason to rule it in or rule it out: only data will tell. As to topological censorship in dS or AdS spaces: the Penrose argument itself only uses the null energy condition, which is not affected by the cosmological constant. But the statement of topological censorship I think requires a time-like or null Scri to make sense. Indeed, in the AdS case, there is a 2001 paper by ... –  Willie Wong Dec 16 '10 at 1:38
    
... Galloway, Schleich, Witt, and Woolgar which shows that the same result (topological censorship) holds for asymptotically anti-de-sitter space-times. That is, defining the domain of outer communications as the intersection of the past and the future of Scri, they showed that for (n+1) dimensional (with n at least 3) asymptotically AdS space-times, the domain of outer communications is simply connected, in the sense that any time-like curve going from Scri to (the same connected piece of) Scri can be deformed continuously to a causal curve in Scri. –  Willie Wong Dec 16 '10 at 1:43
    
Interesting answer, but you may be interested in this: link.springer.com/article/10.1134%2FS0202289313010064. –  William Mar 15 '13 at 5:37
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I don't know the answer, but your intuition is right on -- the fact that the equations are local doesn't mean that there cannot be a constraint on the topology of a global solution. For example, in Euclidean signature, $R_{ij} = g_{ij}$ immediately implies that the scalar curvature is positive, which in turn leads to topological constraints. If the four-manifold is Einstein and complex, then it must be a del Pezzo surface (highly constrained). I don't know much about Lorentzian signature, but I know that the PDE's are a whole different beast. I have seen a few results about classification of possible holonomy groups of Lorentzian Einstein manifolds, but I don't know anything global (I actually don't know anything at all).

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  1. Einstein equations describe local structure of the space-time. They contain no global or topological information.

    While I heard that some restricitons on the scale of topology can be derived from curvature of the Universe if the curvature is negative. (Something like "scale = integer multiple of 1/curvature".)

  2. Well, if our space has non-trivial topology, then light rays will "wrap around" our universe multiple times and you'll be able to see the same (similar) copies of galaxies. I heard of people searching for such similarities without success.

    Also nontrivial topology must result in some correlation in CMB -- no such correlations were found (yet?) either.

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What do you mean by scale of the topology? But Einstein's equations need to be solved globally so couldn't they put some restrictions on the topology? For example if Einstein's equations implied positive scalar curvature, then that would limit the possible manifolds. Also, With there not being any classification of even simply connected 4-manifolds, it seems likely there are nontrivial ones which wouldn't have the "wrap around" property of light rays. –  Eric Dec 10 '10 at 15:35
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Simplest example -- consider flat space-time. You can imagine it "wrapping", so when you travel distance L in one direction you will get to the same place. As far as I understand that would be called the 3D torus (in simplest case). The distance L is the scale of topology. It can be arbitrary -- Einstein equations do not impose any restrictions on it. –  Kostya Dec 10 '10 at 17:03
    
Oh ok, so that would still be a geometric thing: scaling a cylinder doesn't change any topology. –  Eric Dec 10 '10 at 17:55
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These are two independent questions, one mathematical, and one about observations.

  1. What constraints do the Einstein equations imply about the global structure of space and/or spacetime? I don't know the general answer, my impression is that not as much as is known about Lorentzian manifolds as about Euclidean manifolds. Furthermore, there is no reason to suspect the space/spacetime is singularity-free (at the very least we know of many black holes in the universe), and I doubt much can be said about the global structure of any manifold if you allow for singularities.

  2. About observational physics: the only observable I can think of that is sensitive to global structure is the low multipoles of the CMB, and every now and again there are papers on the subject, to explain anomalies in such multipoles (e.g. stories about football-shaped universe). Alas, cosmic variance limits how seriously you can take such observations and models aimed to explain them.

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On the experiments and topology question, there is some work on the subject by Glenn Starkman et al. In their work, they search for structures in the CMB that would indicate some particular topology for the universe. There is a very nice lecture given in PI on the subject as well as other issues that have to do with CMB. To give you a spoiler on the lecture, they haven't found anything in large angle correlations.

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