Qs re Hawking Radiation – Part I

[Buzz Bloom]

I am starting this thread to continue asking questions regarding the Hawking Radiation phenomenon which were discussed in the Comments thread about the Insights article “Misconceptions about Virtual Particles”. This discussion has been mostly a dialogue between A. Neumaier and myself starting with posts #4-6 on page 1 and continuing with posts #174-180 on pages 9-10. At this point the questions I want to ask are far removed from those issues about virtual particles that began my first post in the above cited thread.

The following is summary of what I think I have learned so far from the previous discussion cited above about Hawking Radiation.

1. Hawking Radiation (HR) involves multiple events, each involving the creation of a particle pair (PP), a particle and its anti-particle, outside but close to the event horizon (EH) of a black hole (BH). For the purpose of this discussion it is assumed that the BH is non-rotating, and it and its gravitational field (GF) are spherically symmetrical.

2. The EH of the black hole is a sphere of radius

R_c = GM/c².​

a) M is the mass equivalent of all the mass and energy inside the EH. This mass-energy is assumed to include a point singularity at the center of the black hole, and this point singularity contains all of the mass-energy M except the energy of the GF which occupies the volume within EH.
b) The amount M_i of M corresponding to the GF energy inside the BH is functionally related to the amount M_s of M at the point singularity.
c) M_s + M_i = M
d) M_s is functionally related to the total GF energy both inside and outside the EH.​

3. The PP creation event takes place over a very brief period of time in which several changes in M and the GF occur. For each such PP creation:

a) Some of the energy in the GF outside, but near, the EH is converted into the PP.
b) There are logically three possible scenarios regarding this PP.

(i) The most likely is that one of the pair escapes entirely from the BH as it begins its travels towards infinity. The other particle crosses into the volume inside the EH.
(ii) Both of the two particles escape the BH.
(iii) Both of the two particles cross into the volume inside the EH.​

c) An amount of mass-energy inside the EH, equal to the GF energy outside of the EH that was converted into the PP, “vanishes”. Most of this mass-energy “moves” outside the EH so that the GF there almost equals what it had been before the PP creation. The net reduction in the GF energy outside the EH is due to the weakening of the GF due the reduction of M inside the event horizon.
d) The reduction of energy inside the EH depends on which of the three above scenarios is involved.

(i) The reduction is slightly less than one-half of the total energy used to create the PP, since one of the pair replaces half of the energy that vanished inside the EH.
(ii) The reduction is slightly less than the total energy used to create the PP.
(iii) The reduction is zero since both particles replace the vanished inside energy.​

e) The reduction of M includes: (i) a reduction in the mass at the point singularity, and (ii) a reduction in the energy of the GF inside the EH.​

4. When the event has completed, the GF is again spherically symmetrical. The details of what is physically happening during the event that causes the energy to move around to accomplish the changes described above is unknown at the present state of understanding of Quantum Gravity (QG) theory.

My first question is: Have I made any mistakes? That is:

Does the above summary description of the Hawking Radiation phenomenon contradict anything that is known with respect to the current state of knowledge about this phenomenon?​

If I have made any mistakes, I would be most grateful if someone will post corrections.

Regards,
Buzz

 

[PeterDonis]

Hawking Radiation (HR) involves multiple events, each involving the creation of a particle pair (PP), a particle and its anti-particle, outside but close to the event horizon (EH) of a black hole (BH).

This is a heuristic description only and does not really correspond to anything in the actual underlying math. It is unfortunate that many pop science sources, even when written by experts in the field (including Hawking himself), use this heuristic language and don't tell you that it's heuristic only. But that is in fact the case.

The EH of the black hole is a sphere of radius
R_c = GM/c².

This is not correct. The correct statement is that the EH is a 2-sphere with area $4 \pi R_c^2$. The EH does not have a physical radius; the singularity at $r = 0$ is to the future of the EH, it is not any spatial distance away from it.

M is the mass equivalent of all the mass and energy inside the EH.

No, is isn't. M is a global property of the black hole spacetime. The black hole is vacuum everywhere and does not have any mass or energy inside it.

This mass-energy is assumed to include a point singularity at the center of the black hole, and this point singularity contains all of the mass-energy M except the energy of the GF which occupies the volume within EH.

Wrong. See above. Also, there is no such thing as a locally measurable "energy of the GF" in GR, and certainly no such thing that corresponds to the mass M of the spacetime.

Since all of the rest of your post is based on incorrect premises, per the above, it is incorrect as well.

 

[Buzz Bloom]

Hi @PeterDonis:

Thank you very much for your post. It has now become very clear to me that it is impossible for me to understand physics at an intuitive level.

Regards,
Buzz

 

[Nugatory]

It has now become very clear to me that it is impossible for me to understand physics at an intuitive level.

You may be going about it backwards. The intuition comes after you've developed some comfort with the mathematical description - intuitive understanding isn't a level below the full mathematical understanding, it's the result of achieving that mathematical understanding.

 

[Buzz Bloom]

The intuition comes after you've developed some comfort with the mathematical description

Hi @Nugatory:

Thank you for your post.

That is the conclusion that I have reluctantly reached. As an octogenarian whose math skills have significantly declined since my younger years, I do not believe I have the time to struggle with the math sufficiently to ever achieve the necessary comfort for intuition to follow.

Regards,
Buzz