Why is the natural singularity $r=0$ in Schwarzschild geometry a spacelike one?
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asked Dec 22, 2016 at 14:26
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$\begingroup$ What research have you done? What is your definition of "spacelike singularity"? $\endgroup$– ACuriousMind ♦Commented Dec 23, 2016 at 14:15
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1 Answer
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Nice question. Topologically, a singularity isn't a point or set of points. It's treated as a hole in the manifold. Therefore it doesn't have its own topology or geometry. We can't even say what its dimensionality is. So if we want to define what is a spacelike or timelike singularity, we need to define it in terms of the nearby spacetime, which is a point-set and does have a geometry.
A timelike singularity is one such that there exists an observer (i.e., a timelike world-line) who has it both in his past and in his future light cones.
Given that definition, I think it should be pretty clear why a black hole singularity is not timelike. It's in your future light cone, because you can fall into it. It's not in your past light cone, because we don't observe things popping out of it.
Black hole singularities can form by gravitational collapse. If timelike singularities could form by gravitational collapse, it would be shocking, because the laws of physics can't predict what could pop out of such a singularity, and therefore the laws of physics would lose their power to predict what happens in our universe.
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$\begingroup$ The ring singularity in the Kerr metric is timelike, and so is the Reissner Nordström one. $\endgroup$– YukterezCommented Mar 9 at 9:33
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$\begingroup$ @Yukterez While they are technically timelike, their timelike nature has no relation to our physical time, so they are not problematic for us. They “exist” inside the Cauchy horizon where past doesn’t determine the future anyway with no predictability even without the timelike singularity. A troublesome timelike singularity in our physical time would be Schwarzschild with a negative mass. I agree though that Ben Crowell’s answer above is overall incorrect. $\endgroup$ Commented Mar 10 at 5:27