I have read this question:
Let's consider the simplest kind of black hole, which is one with no rotation or electric charge. The quick answer to the question is "yes, the mass is entirely within the Schwarzschild radius", but I would like to elaborate a little to explain what we are saying.
No, not all of the BH's mass needs to be inside rS.
Now if you take Komar, ADM, or Bondi mass, as far as I understand, all of them use the calculation with the source being the gravitational field itself.
So the gravitational field (which by definition should extend to infinity) is the source of the mass, then how can we restrict the mass to be inside the EH?
Shouldn't the mass be located at the same as the gravitational field, and extend wherever the gravitational field extends to?
“Mass” in GR is not a local quantity, it is determined by the geometry itself and not by “something” in geometry, moreover there is no single definition of quasilocal mass.
Just for clarification, the difference between a planet and a black hole is that in the case of the planet, the mass comes from the matter particles it is made up of. In the case of a black hole, this is not true, we cannot use the matter particles as a source of the mass (vacuum solution), but the source is the energy of the gravitational field itself (the geometry itself). The comments specifically say that the answers are not satisfactory in the linked question. Thus, in the case of the black hole, the mass cannot directly be addressed through matter particles, but stems from the spacetime geometry (energy of the gravitational field), which extends to infinity, so we should not restrict the mass to be located inside the EH.
So one answer says yes, others say no, some say its undefined. Based on this, I believe it needs clarification why we should restrict the mass of the black hole inside the horizon (when we know that we cannot address it through matter particles but it stems from spacetime geometry, the energy of the gravitational field which extends to infinity).
- If a black hole's mass is derived from its gravitational field, then why would mass be restricted inside the EH, while the field extends outside?