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Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about this? Is there any experiment one can do to prove we live in a non-zero genus world?

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  • $\begingroup$ Have you considered a thin spherical shell asymptotically collapsing to its own Schwarzschild radius in the infinite coordinate time to form a black hole with nothing inside? $\endgroup$ – safesphere Nov 2 '18 at 17:02
  • $\begingroup$ Sounds interesting ... I wonder what it's corresponding stress energy tensor would look like? $\endgroup$ – More Anonymous Nov 2 '18 at 17:07
  • $\begingroup$ Check this out: physics.stackexchange.com/questions/414695/… $\endgroup$ – safesphere Nov 2 '18 at 17:19
  • $\begingroup$ As an example, you can introduce flat Lorentzian metric on torii. $\endgroup$ – Blazej Nov 2 '18 at 19:12
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Any manifold, regardless of its properties, is a solution to the Einstein field equations for some stress-energy tensor $T_{\mu\nu}$, providing it admits a metric, $g_{\mu\nu}$.

In theory, you are free to pick your favourite algebraic manifold of genus $g$, compute the metric and find its corresponding stress-energy tensor.

However, if we restrict the stress-energy, say by requiring that $T_{\mu\nu}$ be infinitely differentiable, then we also restrict the space of allowable metrics. Likewise, if we demand the manifold be globally hyperbolic, which is a frequent requirement due to the implication on causality, then we also restrict possible solutions.

Based on a cursory search, I cannot find any classification of genus $g$ spacetime solutions yet. However, in the paper here it is written that,

In the current paper we study that equation on closed 2-dimensional surfaces that have genus $>0$. We derive all the solutions assuming the embeddability in 4-dimensional spacetime that satisfies the vacuum Einstein equations...

It seems we can have sub-manifolds in spacetime which do have non-zero genus, and are sensible solutions to the vacuum Einstein field equations, meaning at least $T_{\mu\nu}$ is physically sensible, being zero.

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  • $\begingroup$ Thank you for answering! But what about the "Is there any experiment one can do to prove we live in a non-zero genus world?" Like surely having non-zero genus would affect the CMB, etc? $\endgroup$ – More Anonymous Nov 2 '18 at 17:02
  • $\begingroup$ @MoreAnonymous Well, it's the same for any solution you're considering. If you think the spacetime has a particular genus, compute the physical consequences of such a spacetime and then you can design experiments from that. $\endgroup$ – JamalS Nov 3 '18 at 5:59

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