Clarification on a possible mathematical contradiction in general relativity This might be naive, but I have read that some solutions to the equations of general relativity allow black holes. But I have also read that black holes have infinite density at the center because of the mass divided by zero distance from the radius issue. But doesn't mathematics not allow division by zero? Perhaps someone can direct me to a mathematically rigorous text on general relativity?
 A: General Relativity predicts black holes and the singularities that they come with. However, note that most black hole metrics apply to any realistic situation only beyond a certain distance away from the singularity. Under this radius, the generic matter density resolves the singularity. For instance, in a spherically symmetric star system, the Schwarzschild metric accurately describes the geometry outside the star. Once inside, the metric is no longer that of a black hole the singularity is resolved
More generally, if you are not worried about realistic situations but want to understand theoretically how these singularities are handled, then you are in good company. This is still largely an open problem and has no good answer.
The best suggestion is that once we get down to small scales close enough to the singularity, quantum effects become significant and this may somehow resolve the singularity. Certainly to have a clear answer to this question requires a quantum theory of gravity of which the only example currently is string theory. In string theory this sort of resolution does indeed happen in some cases so that space-time still ends up being smooth.
A: Black holes can certainly exist in GR. I think it is still a matter of debate whether singularities can exist in our Universe.
Some well known textbooks on GR are those by Sean Caroll, Misner, Thorne and Wheeler or Steven Weinberg (searching their names followed by "GR textbook" should bring them up). 
