all this talk of GR and invariants, not to confuse a mathematical singularity with a gravitational one, at which point GR itself breaks down per the need for a quantum mechanical description. the truth is we don't know what happens within a black hole, and probably never will
> GR itself breaks down per the need for a quantum mechanical description
This is the current predominant belief among physicists, yes, but that doesn't mean it's established fact.
> the truth is we don't know what happens within a black hole, and probably never will
This is way too pessimistic. The predictions of GR for this regime are precise and unequivocal. Even if you believe GR breaks down when spacetime curvatures reach the Planck scale (which is the current predominant belief, as I said above), that still leaves most of the black hole's interior within GR's domain of validity.
> that still leaves most of the black hole's interior within GR's domain of validity.
Thanks, that's what I (as a layman) supposed: that inside the event horizon isn't some space that you can't think about; it's qualitatively not distinguishable from outide.
Upthread, someone mentioned that the singularity wasn't a place in space, it was a location in time. That made me think; the mass of a BH isn't the mass of a volume of space, it's the mass of the "singularity", which (presumably?) is located at the center of the space described by the event horizon. So there must be a place where the singularity is located, and there must be a direction towards it.
Someone else suggested there are two things: a mathematical singularity and a gravitational singularity, and that they aren't the same. That hadn't crossed my mind.
> someone mentioned that the singularity wasn't a place in space, it was a location in time.
That was me.
> That made me think; the mass of a BH isn't the mass of a volume of space, it's the mass of the "singularity"
This is not correct. The mass of the hole is not somehow "concentrated" at the singularity. There is no matter anywhere inside a black hole. The "mass" of the hole is a global property of its spacetime geometry.
> "singularity", which (presumably?) is located at the center of the space described by the event horizon.
Your presumption is wrong. There is no "center" of the black hole in the ordinary sense. Its spacetime geometry simply doesn't work that way; it is not like the spacetime geometry inside an ordinary spherical object.
> So there must be a place where the singularity is located
Wrong. See above.
> and there must be a direction towards it.
There is such a spacetime direction, but it's a timelike direction--towards the future--not a spacelike direction.
> Someone else suggested there are two things: a mathematical singularity and a gravitational singularity, and that they aren't the same.
