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Bob falls into a black hole, watched by Alice who is far away.

Alice sees Bob asymptotically approach the event horizon while his clock asymptotically approaches 12:00 noon as his image redshifts (illustrated by a recent Veritasium video on black holes.)

From Bob's perspective, he crosses the event horizon at 12:00 according to his clock, not noticing anything special at that time, and proceeds onward.

Let's now take into consideration that at some point in the future, the black hole evaporates due to Hawking radiation.

From Alice's point of view, Bob never passes the event horizon. Bob's clock shows a time prior to 12:00 while the black hole evaporates. The matter/energy he and his ship is composed of is eventually emitted back out to the external environment without ever crossing the black hole's event horizon.

Question

What does Bob see?

Specifically, what occurs first for Bob: 12:00, or completed black hole evaporation?

One way to reconcile these two perspectives is that from Bob's point of view, as 12:00 approaches, he "watches" a sped-up version of the black hole's lifetime as it evaporates due to Hawking radiation, as he gets transformed into radiation along with it, all finishing prior to (what would have been) 12:00 on his clock.

It seems that the usual non-quantum GR description of Bob not noticing anything special happening at 12:00 when crossing the event horizon is no longer accurate when Hawking radiation + black hole evaporation are taken into account.

Other posts have come close to addressing this question, such as Doesn't the Schwarzschild metric combined with Hawking radiation imply that nothing ever gets past the event horizon of a black hole? however it seems more can be said about reconciling Alice's and Bob's perspectives when taking black hole evaporation into consideration.

Matt
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1 Answers1

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Let's now take into consideration that at some point in the future, the black hole evaporates due to Hawking radiation.

From Alice's point of view, Bob never passes the event horizon.

This last statement is not correct. The issue is that you are taking reasoning based on the eternal black hole Schwarzschild coordinates and trying to apply that to an evaporating black hole. An evaporating black hole and an eternal black hole are completely different spacetimes, and both Schwarzschild coordinates and the behavior of geodesics are different in the different spacetimes.

See “Black Hole - Never Forms, or Never Evaporates”. In particular, Figure 3 shows that, in an evaporating spacetime that satisfies the usual energy conditions, null geodesics do not cross the horizon, but become outgoing null geodesics after the evaporation.

So if Bob is continually emitting a signal of his proper time then his signals will escape and he, staying within his own light cone, must also escape. His 12:00 signal will arrive to Alice in finite time, as will all subsequent signals.

This is physically different from the eternal black hole where Alice would continue to receive signals from Bob forever, all of which would show finite proper time from Bob.

Note that the above will not constrain spacetimes that violate one or more of the usual energy conditions. Such might have timelike paths that enter the horizon and meet the singularity before it evaporates.

Specifically, what occurs first for Bob: 12:00, or completed black hole evaporation?

If Bob crosses the event horizon at 12:00 proper time then he will hit the singularity before it evaporates. If he escapes, then light could also escape, so he did not cross the horizon.

Dale
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