# Can solutions of GR have non-zero genus?

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?

• 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? – safesphere Nov 2 '18 at 17:02
• Sounds interesting ... I wonder what it's corresponding stress energy tensor would look like? – More Anonymous Nov 2 '18 at 17:07
• Check this out: physics.stackexchange.com/questions/414695/… – safesphere Nov 2 '18 at 17:19
• As an example, you can introduce flat Lorentzian metric on torii. – Blazej Nov 2 '18 at 19:12

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.

• 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? – More Anonymous Nov 2 '18 at 17:02
• @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. – JamalS Nov 3 '18 at 5:59