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I have read this question:

Null Energy Condition Violation in QFT and Area Theorem

Can black holes shrink in classical general relativity, violating area theorem?

And this one:

When one considers effects from quantum field theory such as Hawking Radiation, the area theorem fails to apply.

Evaporation of BH and area theorem

And this one:

Indeed if you consider general relativity plus quantum field theory the black hole can evaporate. Since the radiation stress energy tensor doesn’t satisfy the null energy condition the area law is violated and the black hole loses mass and it shrinks.

That said, black holes can loose mass via Hawking Radiation. (or gravitational waves during a "collision/merging"). Consequently the event horizon area may reduce if either of these effects dominate.

This argument doesn't work. The Schwarzschild spacetime is just one example of a spacetime that has an event horizon. Also, a region of spacetime that contains one or more event horizons can lose mass; for example, this happens in black hole mergers.

How to show that the area of the event horizon never decreases?

There are currently these two arguments on this site, and I cannot easily reconcile them:

  1. Hawking radiation manifests in the form of photons (amongst other forms) and this means energy is leaving the black hole, causing a decrease in mass (ADM etc.) thus violating the area theorem, and based on the quotes from this site, this could mean that the area of the event horizon should shrink

  2. The area theorem has been experimentally verified, and, we do know that Hawking radiation originates from outside the event horizon, taking energy from the black hole's gravitational field (which we know extends outside the EH), and this could make one think that the energy content "inside" the event horizon does not decrease, leading to the (maybe misleading) thought that the area theorem is still valid, and the event horizon should not shrink

One of the answers says that yes the event horizon may decrease if we consider QFT and black holes, the other one says the cause is Hawking radiation, but the comment clearly states this argument does not work. Why?

It adds to the contradiction that the area theorem has been experimentally verified.

https://news.mit.edu/2021/hawkings-black-hole-theorem-confirm-0701

Question:

  1. Can Hawking radiation decrease the area of the event horizon or not?
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  • $\begingroup$ the energy content ‘inside’ the event horizon does not decrease, leading to…” - It cannot “lead” to anything outside, because the “inside” if in the future relative to “outside”. Nothing that happens tomorrow can lead to anything today. Everything in #2 is wrong +1 $\endgroup$
    – safesphere
    Apr 29 at 9:21
  • $\begingroup$ @safesphere thank you so much! $\endgroup$ Apr 29 at 15:34

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Whenever one talks about experimentally verifying a result, one always has to remember that there is always experimental uncertainty. So an experimental result is only ever of the form "$X$ statement is consistent (or inconsistent) with our measurements," never "$X$ is absolutely true (or false)."

The area theorem has only ever been tested experimentally by studying mergers of binary black holes. These tests are nowhere near sensitive enough to detect Hawking radiation, and therefore are not able to study whether there are small deviations of the area theorem due to Hawking radiation. They do not even study the right process; what the area theorem tests show is that the final black hole of the merger has a larger area than the two incoming black holes, but to test the decrease in area from Hawking radiation you would want to watch one black hole in isolation shrink as it emitted radiation. Note that I'm not saying that these tests are unimportant, just that we should appreciate them for what they are: a consistency test of classical GR for astrophysical black holes, not a measurement with any implications for Hawking radiation.

If Hawking radiation is emitted and escapes to asymptotic infinity, it is pretty clear that the first argument you cite is correct, since the radiation will carry away energy, decreasing the mass of the black hole, and therefore its area will decrease.

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  • $\begingroup$ Thank you so much! $\endgroup$ Apr 28 at 2:53

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