# Schwarzchild and Reissner-Nordstrom singularities

The Schwarzchild metric has a spacelike singularity, while the R-N metric has a timelike one. The difference between the two physical systems is charge. Obviously you've a very slightly charged black hole, the SC metric is a good approximation because Q/M is too small to really be worried about. However, the change from spacelike to timelike (and vice versa) singularities is not a continous one, it's a discrete one.

The discrete nature of the singularity seems to be a fundamental difference (particularly when you're drawing Penrose diagrams, the staple diagram of my black hole course ATM) so I'm having a bit of trouble getting my head around how you could have a good approximation using the SC metric. It seems to me the nature of the singularity isn't a good approximation for that particular part of the system?

pervect
Staff Emeritus
I believe that actual physical collapse of a charged black-hole is expected to give a spacelike singularity.

I'm basing this statement on

http://lanl.arxiv.org/abs/gr-qc/9902008

We study the gravitational collapse of a self-gravitating charged scalar-field. Starting with a regular spacetime, we follow the evolution through the formation of an apparent horizon, a Cauchy horizon and a final central singularity. We find a null, weak, mass-inflation singularity along the Cauchy horizon, which is a precursor of a strong, spacelike singularity along the $r=0$ hypersurface. The inner black hole region is bounded (in the future) by singularities. This resembles the classical inner structure of a Schwarzschild black hole and it is remarkably different from the inner structure of a charged static Reissner-Nordstr"om or a stationary rotating Kerr black holes.

To try and clarify this a bit:

The Schwarzschild solution is a valid solution to Einstein's equation, and is stable in the exterior region. However, it is not expected to be stable in the interior region, and physically collapsing objects instead are expected to have a metric known as a BKL metric in the interior region, rather than the Schwarzschild metric.

(This is talked about in one of Thorne's excellent popular books on black holes, for a very terse online reference see

http://scienceworld.wolfram.com/physics/BKLSingularity.html.

The BKL metric is chaotic).

The R-N black hole is similar to the Schwarzschild solution. It is a mathematical solution to Einstein's equations, but it is not expected to be stable in the interior region (beyond the event horizon).

The expected physical solution for a charged collapse is not as well understood as the Schwarzschild case, but is felt to be likely (see the paper I quoted earlier) to have a significantly different interior structure than the R-N black hole. (In fact it is expected to be somewhat similar to the usual picture of the Schwarzschild black hole).

This is as much as I know - if anyone has any further information I would be interested in hearing about it.