What is known about the topological structure of spacetime?

Question

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?

Answer 1

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, $ G_{\mu\nu} = 8 \pi G T_{\mu\nu}$, are local field equations and do not contain any topological information. At the level of the action principle,

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

the term we generally include is the Ricci scalar $ \mathbf{R} = \mathrm{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:

$$ S_{cs} = \int d^4 x \frac{1}{2} \left(\epsilon_{ab} {}^{ij}R_{cdij}\right)R_{abcd} $$

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".

Answer 2

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)

Answer 3

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).

Answer 4

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

While I heard that some restrictions 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 non-trivial topology must result in some correlation in CMB -- no such correlations were found (yet?) either.

Answer 5

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.

Answer 6

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.