Schwarzchild and Reissner-Nordstrom singularities

[AlphaNumeric]

The Schwarzschild 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 continuous 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]

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.

 

[Entropia]

The concept of singularities in black holes is a fascinating and complex topic in physics. The Schwarzchild and Reissner-Nordstrom singularities are two types of singularities that occur in different types of black holes. The key difference between them is the presence of charge in the black hole, which leads to a change from a spacelike to a timelike singularity.

As you mentioned, the discrete nature of these singularities is a fundamental difference and can be seen in Penrose diagrams. This is because the nature of the singularity is not a continuous one, but rather a discrete one. In the case of a slightly charged black hole, the Schwarzchild metric is a good approximation because the ratio of charge to mass is small enough to be negligible. However, as the charge increases, the nature of the singularity changes and the Reissner-Nordstrom metric becomes a better approximation.

It is important to note that the singularity is a mathematical concept and does not necessarily reflect the physical reality of a black hole. It is a point where the equations describing the black hole break down, and we still do not fully understand what happens at the singularity. Therefore, using the SC metric can still provide valuable insights and approximations for certain aspects of the black hole, even though it may not fully capture the nature of the singularity.

In conclusion, the difference between the Schwarzchild and Reissner-Nordstrom singularities is a result of the presence of charge in the black hole. While the nature of the singularity is a discrete one, using the SC metric can still provide useful approximations in certain cases. The study of black holes and their singularities is an ongoing and complex field, and we continue to learn more about these mysterious objects.