# If Hawking radiation is observer dependent then is the mass of a black hole too?

I have read this question (in the comments):

Late response, but the energy really comes from the gravitational field, which exists outside the event horiz0n and in space. This field is what lets us measure the "mass" of the black hole. Loosely speaking, the field, not the mass, "is" the gravity of the black hole. Since there's energy in the field permeating a region near the horizon, pair-creation is possible by "borrowing" that energy to form particles. But if some of that energy escapes (the photons of the Hawking radiation), that energy cannot be returned to the field, so the field loses energy.

How does the Hawking Radiation mechanism cause a black hole to lose its mass?

And this one:

This paper discusses these issues in a fairly comprehensible way. Faraway observers (like your observer A) see thermal Hawking radiation with an effective temperature given by the Hawking temperature TH:=ℏc38πGMkB, where M is the black hole's mass. If an observer on a string is very slowly lowered toward the black hole (so that her dr/dτ is very small), then the effective temperature increases without bound and diverges at the horizon, so your observer B inevitably gets burned up.

Intensity of Hawking radiation for different observers relative to a black hole

So naively saying, the black hole loses mass (energy) by Hawking radiation in the form of photons that originates from the gravitational field of the black hole, and this radiation is observer dependent, ergo, the mass lost is observer dependent? So basically, different observers see the mass (energy) of the black hole decrease by different amounts (the amount of Hawking radiation they observe) so they calculate different amounts for the mass (energy) of the same black hole.

Question:

1. If Hawking radiation is observer dependent then is the mass of a black hole too?
• Disregarding Hawking radiation, mass in GR can be tricky to covariantly define because mass=energy, and it's a very subtle question about how to disentangles the energy of matter/radiation from the energy within the gravitational field itself. See the wiki on the Landau–Lifshitz psuedo-tensor for more. But about black hole mass in particular, one usually needs an asymptotic boundary like those in Minksowki or AdS to rigorously define it. See ADM mass and Bondi mass. Coincidentally, the same is needed for Hawking temperature. Commented Jan 25 at 5:56