To what extent in general relativity do we get unique solutions to the Einstein field equations given the topology of space-time and a boundary condition? What if we're given only the boundary condition, but not the topology of space-time?(adsbygoogle = window.adsbygoogle || []).push({});

I know that symmetry under diffeomorphisms means that any solution to the EFE can be adjusted into a different solution by applying a diffeomorphism. However, these solutions are diffeomorphic to the original, meaning they aren't really any different from the original. (effectively it's just relabelling the points) But can there be multiple solutions that arenotdiffeomorphic to one another?

For concreteness, let's consider two specific cases, and another class of cases.

1. We have the situation in the original hole argument: a void in space-time with a topologically trivial topology. Let's say... the void has the shape of a 4-cube in some coordinate chart. Knowing the value of the metric on the boundary of the void, there are two questions:

1a. Is there an essentially unique solution for the value of the metric inside the void?

1b. If we are allowed to change the topology of the interior of the void, can we come up with new solutions?

2. Consider a Schwarzchild black hole. (Incidental question: can we add an additional point at [itex]r = +\infty[/itex]?) Take as the boundary condition the value of the Schwarzchild metric on the event horizon for all times. To what solutions can it be extended? Clearly we see two solutions: the interior and exterior parts of the Schwarzchild black hole, and these two solutions are compatable in the sense that if you put them on opposite sides of the boundary, they line up.

2a. Are there any other solutions (with any topology)?

2b. Is the exterior Schwarzchild solution compatable with any of those other solutions?

2c. What about the interior Schwarzchild solution?

2d. What about a solution where the boundary doesn't divide space-time into two regions? (i.e. the the boundary can be attached to the solution at two different places)

3. Same question as #2, but with a rotating black hole. (Or any other interesting black hole)

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# Uniqueness of solutions to EFE?

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