Uniqueness of solutions to EFE?

[Hurkyl]

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?


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 are not diffeomorphic 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 Schwarzschild black hole. (Incidental question: can we add an additional point at $r = +\infty$?) Take as the boundary condition the value of the Schwarzschild 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 Schwarzschild 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 Schwarzschild solution compatable with any of those other solutions?
2c. What about the interior Schwarzschild 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)

 

[cesiumfrog]

1. Let's say your cube contains a radially symmetric star. The metric inside depends on the density profile, the metric outside is Schwarzschild (no monopole grav. waves).

2. Just nitpicking, but the *Schwarzschild* metric coordinate values are singular at the horizon. Choose more appropriate coordinates; there is nothing that special about the horizon in particular.If you know the energy tensor (e.g., restricting to vacuum solutions), you can often get a degree of uniqueness (see maximal extensions of the various black hole solutions). Similarly, there is a Curzon space-time with tunnels that can be extended into either Minkowski space or something similar such as another Curzon iteration.

In general the EFE says nothing about topology. For example, there is nothing to stop you taking the flat space-time metric and pasting it on a torus (even if this produces closed time-like curves). You can express any space-time in cylindrical coordinates and (provided the metric has cylindrical symmetry) it is completely arbitrary whether, for the angular coordinate, you identify zero with 2 pi or with any other number.

 

[Stingray]

I think that a specified stress-energy tensor together with appropriate boundary conditions and topology do specify a unique solution (up to diffeos). I don't know the precise theorems, but I'm sure something appropriate has been worked out. Choquet-Bruhat probably did it, or maybe Marsden, Friedrich, or Rendall more recently.

One simple example is that the only vacuum asymptotically-flat spacetime with the normal topology of $\mathbb{R}^4$ is Minkowski. Changing that topology can allow Schwarzschild or any number of other interesting solutions.

 

[Hurkyl]

For example, there is nothing to stop you taking the flat space-time metric and pasting it on a torus (even if this produces closed time-like curves).

This source of variation is much closer to math I understand. R⁴ is the universal cover of the torus, making it in some sense a better topology for this particular solution. (Of course, the fact we can map it down to the torus is interesting...)

However, I am naïvely imagining that the boundary conditions for these problems would greatly constrain this sort of freedom... (in addition to there being less symmetry)