Physics Asked on March 30, 2021
Consider a spherically closed Universe with spacial-topology $S_3$.
Put a black hole on both “poles” of this universe.
This seems like it should be in equilibrium. But then again maybe the Universe will collapse into a singularity.
Because of the symmetries of this Universe, I wonder if there would be an anayltic solution for the metric? Well if $r$ was the lattitude going from $-1..1$ the gravitational potential would be something like:
$$frac{m}{1+r} + frac{m}{1-r}$$
Is there any simple intuitive way to tell?
I’m not sure what use this solution would have, except one could imagine a test particle having orbits round both black holes at the poles.
Precisely this construction was attempted in the following paper:
Abstract:
In order to understand how locally static configurations around gravitationally bound bodies can be embedded in an expanding universe, we investigate the solutions of general relativity describing a space–time whose spatial sections have the topology of a 3-sphere with two identical masses at the poles. We show that Israel junction conditions imply that two spherically symmetric static regions around the masses cannot be glued together. If one is interested in an exterior solution, this prevents the geometry around the masses to be of the Schwarzschild type and leads to the introduction of a cosmological constant. The study of the extension of the Kottler space–time shows that there exists a non-static solution consisting of two static regions surrounding the masses that match a Kantowski–Sachs expanding region on the cosmological horizon. The comparison with a Swiss-Cheese construction is also discussed.
So, the configuration suggested by OP would not work as a static solution, but could be modified into time dependent solution.
Answered by A.V.S. on March 30, 2021
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