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.

Answer 2

As a sort of objection to @Anders Sandberg I should say:

The theory by no means is only local. The global degrees of freedom are determined by the curvature invariants integrated over all space. In other words if the local observables are determined everywhere, then topology can be discussed formally in terms of physical observables or better said: it's predetermined although inaccessible!

Take a look at the Chern-Gauss-Bonnet theorem for even dimensional Riemannian manifolds.

For example the Euler characteristic (a global topological feature) can be explicitly computed in terms of observables in case of Riemannian manifolds.

Another example is Sphere Theorem where enough knowledge about the curvature map(to be defined below) of the manifold fixes the topology.(although there's a mild topological assumption of simply connectedness here, but the topological information you are given at the end is way nontrivial in comparison to the input!)

Keep in mind that Riemann curvature tensor is the only physical observable that can be measured by means of geodesic deviation at every point of space.

But in case of Lorentzian manifolds, which is our case in relativity things become complicated.

There exist two topologies, one is that of the manifold topology and the other is the Alexandrov topology that is defined in terms of the Lorenzian metric. I like to call such topology the observable topology!

These two topologies do not necessarily coincide unless the spacetime is at least strongly causal on the causal ladder.

So in case the topologies coincide the global topology can be fixed in terms of what I call the curvature map: the map of curvature at all spacetime points.

But in case the two topologies do not coincide, the manifold topology is an independent degree of freedom that should be treated separately.

Nonetheless there are Pseudo-Riemannian variants of the Chern-Gauss-Bonnet theorem that determine the manifold topology partially, in terms of the observables: Chern-Gauss-Bonnet for Pseudo-Riemannian manifolds

So maybe the real question is: to what extent the manifold topology is restricted by the curvature map in the Lorentzian manifold that is not strongly causal?

PS: I always assumed that spacetime points are defined operationally by means of spacetime coincidences argument. Given such definition of spacetime, then a curvature map generally demands multiple compatible such observers.

PS: For those who might object that, the topological information can be localized by a singular gauge transformation(under the influence of the topological defects in gauge theory), one should note that such singularity in case of gravity is hidden behind a horizon(since this singularity is physical), and the information remains inaccessible to the external observer!