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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.

Does every black hole have its mass within its Schwarzschild radius?

No, not all of the BH's mass needs to be inside rS.

Does every black hole have its mass within its Schwarzschild radius?

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.

Does every black hole have its mass within its Schwarzschild radius?

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).

Question:

  1. 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?
Árpád Szendrei
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  • @ChiralAnomaly I understand, though, the difference is, whatever is inside the EH, is causally disconnected from the outside world. In the case of a planet in your example it is not the case. – Árpád Szendrei Sep 07 '21 at 00:24
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    Does this answer your question? How does gravity escape a black hole? – StephenG - Help Ukraine Sep 07 '21 at 01:36
  • I VTC as a duplicate as I think you are thinking of the event horizon as a barrier to the gravitational field. – StephenG - Help Ukraine Sep 07 '21 at 01:37
  • @StephenG no, that is not what I am asking. I am asking about the fact that we restrict the mass to be inside the EH, although it is sourced from the gravitational field, which extends to infinity. – Árpád Szendrei Sep 07 '21 at 03:35
  • If a black hole's mass is derived from its gravitational field Which is not really the physics of it - the mass is causing the field (a curvature of spacetime). But if you took the view you state, that does not require the mass to be spread outside the event horizon for the curvature outside to be what it is. That's just the way the math works out. – StephenG - Help Ukraine Sep 07 '21 at 05:33
  • Gravity doesn't have it's own $T_{\mu\nu}$, so unlike EM field or any local fields, you can't simply integrate components from $T_{\mu\nu}$ to get local mass density. Gravitational energy is non-local...in this sense it is different from Newtonian gravitational potential energy. Not sure why the downvote. Also, the linked post doesn't answer this question. – KP99 Sep 07 '21 at 06:14
  • "the mass is entirely within the Schwarzschild radius" - It is not. That answer is incorrect. A.V.S. is a true GR expert here. See his response: https://physics.stackexchange.com/questions/630816/does-every-black-hole-have-its-mass-within-its-schwarzschild-radius/630895#comment1420251_630895 – safesphere Sep 07 '21 at 06:41
  • @StephenG correct, but my point is, that even though when you say "the mass is causing the field (a curvature of spacetime)", there are no matter particles we can address the mass to, and instead for a black hole, the mass is determined by the spacetime geometry (energy of the gravitational field), which extends to infinity, thus we should not restrict the mass to be located inside the EH. – Árpád Szendrei Sep 07 '21 at 16:10
  • Surely this is addressed by Birkhoff's Theorem and it's consequences (and generalizations). – StephenG - Help Ukraine Sep 07 '21 at 16:31
  • Upvoted. " Shouldn't the mass be located at the same as the gravitational field, and extend wherever the gravitational field extends to?" - No. Consider any vacuum solution, (where $T_{ab}=0$), it doesn't mean that there is no gravitational field, there is still non-zero conformal curvature. So mass is localised inside planet/BH. You can define mass density $\rho$ but there is nothing like gravitational energy density in GR. – KP99 Sep 07 '21 at 16:37

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