Allowed Topologies for General Relativity

Question

Studying the ADM formulation of General Relativity the ADM splitting comes out from the assumption that the spacetime is globally hyperbolic.

From that assumption thanks to Geroch's theorem, it is proved that:

$$\mathcal{M} = \mathbb{R} \times \Sigma $$

with $\Sigma $ a spacelike manifold with arbitrary but fixed topology.

My question:

Removing the assumption of globally hyperbolicity, what are all the other allowed topologies that are compatible with the theory of General Relativity? In the sense, does the theory, or other principles, constraint the allowed topologies for the spacetime manifold?

If there are too many, a brief description of the exclusive topological properties compatible with the theory, instead of the list of topologies, is also an acceptable answer.

Answer 1

The theory only deals with the local curvatures, not the global topology. Hence any manifold with an allowed metric is allowed. These can be infinitely many, especially for negative curvature space-times.

The most obvious infinite family would be flat space-times, where one can make 3-toruses, chimney space-times, and slab space-times of any period (and for the chimney case, one can use all the tiling groups to get extra versions); there are 17 flat 3D space families. Positive constant curvature space-times are more constrained with just a few cases, while negative constant curvature space-times can be split by an infinite number of discrete groups. Non-constant curvatures adds a lot more options.

More technically, the Killing–Hopf theorem states that a complete connected Riemannian manifold of constant curvature is isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously. However, if you loosen assumptions a bit like allowing it to be geodesically incomplete many more become possible.

That was the 3D case. The 3+1 dimensional case obviously is far more complex. But more importantly, you can take a non-trivially connected base manifold $\Sigma$, do the Cartesian product with $ \mathbb{R}$, and then just rotate the coordinates (assuming the manifold obeys the Einstein equations) to get a non-hyperbolic spacetime with nontrivial topology. That you now can get closed timelike curves that make causality break doesn't bother relativity: some other constraint is needed to rule out such manifolds.