Is a Klein-bottle-like topology allowed for GR?

Question

As the spacetime of the universe seems to be quite flat, a torus topology comes mind easily. How about others?

Is it issue if manifold is non-orientable? I see challenges to find 3- or 4-Klein-bottle-like combinations though postulating Cauhcy property separately in in 2+2 dimensions i.e. two 2-Klein-bottles connected with non-Cauchy connections...

In the similar question it was remainded that strong causal condition may be the way to study GR topology. Is there any reasonable doubt to set that restriction?

Maybe I'm trying impossible. I didn't find direct list of topologies tried for GR. Can anyone help?

Answer 1

General relativity does not care about topology of the manifold. You could for example take a flat spacetime with cubical edges and identify the edges in such a way that you have a non-orientable manifold. This satisfies the vacuum equations of GR since the metric tensor is flat everywhere.

As noted in the answers to this similar question, it might be that the physics on the manifold does not allow non-orientability since it would transform particles looping around the universe into different chirality. But GR does not care.

Answer 2

Observations suggest that our Universe is spatially homogeneous and isotropic. This means that at a manifold level, the underlying spacetime should have 6 independent spatial Killing vector fields ($J_1, J_2, J_3, P_1, P_2, P_3$) associated with it such that

$$ \left[J_{a}, J_{b}\right]=\sum_{c=1}^{3} \epsilon_{a b c} J_{c}, \quad\left[P_{a}, P_{b}\right]=0, \quad \text { and } \quad\left[J_{a}, P_{b}\right]=\sum_{c=1}^{3} \epsilon_{a b c} J_{c} $$

Here the $J$ correspond to rotation symmetry and $P$ to translational ones and $\epsilon_{abc}$ is the Levi-Civita symbol.

As the spacetime of the universe seems to be quite flat, a torus topology comes mind easily. How about others?

So one way to filter out certain spacetime models in the above question in blockquote is: does a torus as a manifold satisfy such a condition? Intuitively my guess would be "no".

Is it issue if manifold is non-orientable?

Spacetime in GR is time-orientable. This means that there exist a vector field that allows us to distinguish the past lightcone from the future light cone. So once we pick one half of the light cone, the vector field then totally determines all futur choices. This is required as although the laws of nature appear time symmetric but the Universe typically evolve only into the future. As such non-orientable manifolds will lead to issues here.