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Let Einstein's equations satisfy $ R_{\mu \nu } = 0 $. Suppose we solve it numerically with the aid of a computer. Can we know from the numerical solution if there is a black HOle in the solutions? For example, how can you know when you solve Einstein's equation if your solution will be a black hole or other particular non-smooth solution?

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Did you mean $R_{\mu\nu} = 0$? That would normally mean the Ricci tensor is zero i.e. the space is Ricci flat, so there couldn't be any horizons or singularities. – John Rennie Sep 3 '12 at 14:14
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@JohnRennie: the Kerr solution is Ricci flat. All $R_{\mu \nu}=0$ tells you is that you don't have any matter. – Jerry Schirmer Sep 3 '12 at 14:56
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Just an addition to Jerry's comment; John, you may have confused Ricci flatness and Riemann flatness. Riemann flatness - literally (piecewise) flat Minkowski spacetime - requires $R_{\kappa\lambda\mu\nu}=0$, not just $R_{\mu\nu}=0$. The former condition is stronger and implies the latter but not vice versa. – Luboš Motl Sep 3 '12 at 15:00
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@JerrySchirmer and Luboš: thanks, this site teaches me something new every day :-) – John Rennie Sep 3 '12 at 17:06

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It isn't clear if you're asking how to identify horizons, singularities or both. Singularities are easy because the curvature becomes infinite, but horizons are harder. Usually to find horizons you study the null geodesics i.e. the paths taken by light rays, but you have to be careful about your choice of co-ordinates. As it happens there's a Living Reviews article on just this subject and this would be a good place to start.

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The way to do this is to look for a closed trapped surface in the solution. This is a spherical surface such that all the null geodesics, both going out and going in, have area that is locally going down per unit affine parameter. When you find such a surface, you know that it is inside a black hole, and you can stop simulating the interior of this region, since no influence from the interior will reach infinity.

The union of all closed trapped surfaces at any one time is called the apparent horizon, and finding the apparent horizon is done in simulations to find regions which can be excluded, since they are black hole interiors. This is useful, because the curvature will blow up inside the black hole for sure, and you don't want to have to simulate that, and you don't need to.

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