Global Properties of Spacetime Manifolds

Question

When solving the Einstein field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi GT_{\mu\nu}$$

for a particular stress-energy tensor, we obtain the metric of the spacetime manifold, $g_{\mu\nu}$ which endows the manifold with some geometric structure. However, how can we deduce global properties of a spacetime manifold with the limited knowledge we usually have (i.e. simply the metric)? For example, how may we deduce:

• Whether the manifold is closed or exact
• Homology and de Rham cohomology
• Compactness

I know if we can establish compactness, one can easily arrive at the Euler characteristic, and hence the genus of the manifold, using the Gauss-Bonnet-Chern theorem,

$$\int_M \mathrm{Pf}[\mathcal{R}] = (2\pi)^n \chi(M)$$

where $\chi$ is the Euler characteristic and $n$ half the dimension of the manifold $M$. In addition, the Chern classes of the tangent bundle computed using the metric give some information regarding the cohomology. Note this question is really not limited to spacetime manifolds. There are many scenarios in physics wherein we may only know limited information up to the metric, e.g. moduli spaces. It would be interesting to see how one can deduce global properties.

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This question is inspired by brief discussions on the Physics S.E. with user Robin Ekman, and I would like to thank Danu for placing a bounty; a pleasant surprise!

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Resources, especially journal papers, which focus on addressing global properties of spacetimes (or more exotic spacetimes, e.g. orbifolds) are appreciated.

Answer 1

Well, the simplest case is that some topologies of spacetime may only allow a particular class of metrics. But unfortunately, it usually requires the knowledge of the metric at every point to be quite certain.

Here's a few thing we can probably assume about the spacetime manifold :

• All the usual jazz about manifolds in general relativity (paracompactness, Hausdorff, etc). This is not necessarily the case, as some theories may allow weirder versions of it, or allow more general topologies like manifolds with boundaries and conifolds, but that is what is usually assumed to get a Lorentzian metric. The fact that spacetime is a connected manifold isn't really a physical constraint but more of a metaphysical one : you can't really say much about any disconnected piece of spacetime since it cannot affect our own. By the way, if your manifold is indeed compact, only spacetimes with Euler characteristics 0 admit a Lorentzian metric, since you need to have a line element.

• It is also usually assumed to have some causality conditions. It may not necessarily be true, but it seems like a rather remote possibility. If the spacetime is causal (no causal loops), it cannot be compact. If you also want it to be globally hyperbolic (No loops or naked singularities), it will fix the topology as $\mathbb{R} \times \Sigma$, $\Sigma$ some 3-manifold, per Geroch's theorem.

• To get the topology from the metric, another important constraint is geodesic completeness : no geodesic should have a finite range in its affine parameter. You can put de Sitter space in $\mathbb{R}^4$, but much like the stereographic projection of a sphere, you will reach the "edge" when a geodesic tries to go through the other pole but finds none.

• Because of the dependance of particle physics on time reversal and space reversal, it is assumed that spacetime is both time-orientable and space-orientable. If it was not, there would be no $SO^+(3,1)$ group and as such no spin groups, but only the Pin groups, with different properties for fermions.

Those are some rather generic things you can say about spacetime from some, we hope, rather reasonable assumptions. Experimental evidence of topology is harder though.

• The Topology Censorship Theorem is a (classical) theorem regarding the ability to gauge the topology of spacetime. From Visser :

"In any asymptotically flat, globally hyperbolic spacetime such that every inextendible null geodesic satisfies the averaged null energy condition, every causal curve from past null infinity to future null infinity is deformable to the trivial causal curve".

Which rules out, if all conditions are met, the ability to send a particle along any trajectory along a topological handle (or wormhole, in the science).

• You can try to check the topology of the spacelike hypersurface of the spacetime by simply observing any repeating patterns, but so far this has not met with any success. The PLANCK space observatory had, among other missions, looking for any correlations in the CMB that might indicate some compactified dimension of space.

• Edit : Oh, and by the way, space being compact in some dimension will also affect the modes of any field on it (the so called topological Casimir effect). Non-orientable compactness also has some different effects.

Answer 2

The initial value boundary problem in general relativity only gives you the metric on a patch of the spacetime. Other methods must be used to find the true global extension of that spacetime. Therefore, Einstein's equation alone cannot tell you the topology of the spacetime.