# [Physics] Is the mass of a black hole somewhere at all

definitiongeneral-relativitymass-energysingularitiesvacuum

The Schwarzschild vacuum solution describes the gravitational field and thus the spacetime curvature of a black hole. The mass of a black hole is sometimes associated with its singularity as pointed out in an answer here:
Black holes: where is its mass? In a singularity or on the horizon?

An answer to the question If black holes are just empty vacuum of space inside, then what causes the curvature?
explains

"So realistically, the cause of the curvature would be stress-energy that is outside of the vacuum solution, in the part of the universe not described by the Schwarzschild metric."

The cause of the stress-energy is the mass $$M$$ of the black hole. If $$M$$ is associated with the singularity it is not a part of the manifold (corresponding to "outside of the vacuum solution") which seems to mean that the mass is not at a locality describable by coordinates.

Is the mass $$M$$ nowhere? Is it merely represented by spacetime curvature?

In general relativity, in general mass and energy are not well-defined globally since they are not conserved. So the question is in a sense meaningless in general, but black holes may be an exception... except that the location of the mass still remains iffy.

Locally everything is fine: the local flow of mass and energy is described by the stress-energy tensor, and there is energy conservation locally. Globally there is no good way of defining mass globally so it is conserved, in general. The fundamental problem is that energy gets its conservation from time translation invariance, and general spacetimes does not have that symmetry.

But the Schwarzschild spacetimes does! That means one can define the Komar mass. Indeed, it can be calculated by integrating a volume integral, and it does produce the "right" answer. But the volume stretches out to spatial infinity. Making "quasi-local" measures appears tricky.

Note that a collapsing material object turning into a Schwarzschild black hole seems to put all the mass into $$r=0$$. But this is a dynamic spacetime so the Komar mass is not defined until after the end of the collapse, not during it.

So where does this leave us? If you say "amount of mass-energy" is a measure of something conserved related to the stress-energy tensor, then it is clearly just zero in the Schwarzschild metric. The relativistic mass measures are all global, and do not correspond to any "where" of the mass. I have no idea of all the quasi-local measures.

In the end, maybe one should turn the question around: does it matter?