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Suppose a black hole is formed at time $t_0$, and after that even more energy falls in, in what we are calling mass shells. I'm inclined to believe that the initial black hole starts radiating before it continues growing, and that each shell of mass falling does change that pre-Hawking radiation. And I say pre-hawking because it's not needed to assume thermal radiation is coming out. However, it's not clear to me if there is a way to locate the source of the radiation, i.e. how much energy is drawn from each mass shell.

I may rephase the question by asking this: Is there a way to know how much of the radiated energy comes from each of the different mass shells as sources of pre-Hawking radiation?

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    $\begingroup$ I’m not aware of “mass shells” in black holes. Do you have a reference for this? $\endgroup$ Commented Apr 10, 2018 at 1:52
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    $\begingroup$ Mass shells is the way one describes spherical stars, one shell after the other make up the whole of it. To find a solution with a Vaidya spacetime you could use a null shell like in arxiv.org/abs/1705.05722 and also study the solutions to the case of many infalling shells by simply pasting black hole spacetime with the right Mass one after the other $\endgroup$ Commented Apr 10, 2018 at 2:00
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/22498/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented Apr 10, 2018 at 3:12
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    $\begingroup$ Hawking radiation comes from the black hole horizon, this arxiv.org/abs/1604.01181 and this arxiv.org/abs/1601.00921 may be relevant , it seems there are horizon cells only $\endgroup$
    – anna v
    Commented Apr 10, 2018 at 5:40

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The Schwarzschild solution is a vacuum solution of Einstein's equations. So a black holes consists of vacuum; its mass M is in the singularity. Therefor there are no "mass shells". Some people talk about shell observer, who are stationary at a constant r-coordinate outside the event horizon but such shells are definitively not a source of the Hawking radiation. The Hawking radiation is a black body radiation which is proportional to 1/M and can be thought to be emitted at or very close the event horizon.

I am not sure what you mean saying "different mass shells as sources of pre-Hawking radiation." The Hawking radiation does not depend on the history of a black hole, it depends only on its actual mass M.

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    $\begingroup$ 1 Schwartzschild solution is an eternal black hole, I'm talking about a black hole being formed at $t_0$ $\endgroup$ Commented Apr 14, 2018 at 11:28
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    $\begingroup$ 2 I speak of infalling Mass shells, not Mass shells that describe the BH $\endgroup$ Commented Apr 14, 2018 at 11:29
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    $\begingroup$ 3 a Vaidya spacetime describes one such dynamical spacetime. Thus, radiation can be computed... In particular, there is radiation I before the BH is formed and there is a somewhat continuous transformation of I into Hawking radiation [arxiv:1705.05722] $\endgroup$ Commented Apr 14, 2018 at 11:38
  • $\begingroup$ I am not familiar with the Vaidya metric. The formation of an astrophysical black hole through the gravitational collapse of a cloud of dust was described by Oppenheimer & Snyder and happens in finite time for an observer comoving with the dust. I think the newly formed BH radiates from this time on regardless that there is still some dust inside the event horizon which has not yet reached the singularity. $\endgroup$
    – timm
    Commented Apr 14, 2018 at 15:14

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