Paradox about black hole evaporation

[Kostik]

TL;DR Summary

A paradox about two travelers, one of which crosses the event horizon of a black hole, while the other watches him and waits until the black hole completely evaporates.

Paradoxical scenario. Suppose Jack and Jill are sitting safely a kilometer above the event horizon (EH) of a large black hole. Now suppose:

1. Jack decides to head toward the center of the black hole, traveling at an easy pace (say 10 km per hour).
2. Jill sees Jack (with her ultrasensitive infrared camera) asymptotically approach the event horizon, but never reach it.
3. Assume that Jill is immortal and can go on watching him forever.
4. The event horizon slowly shrinks due to evaporation (Hawking radiation), but Jill still sees Jack infinitesimally close to the event horizon, his clock moving infinitesimally slowly. The radius of the event horizon shrinks extremely slowly as the black hole evaporates, but Jack has a constant speed of 10 km/h, so Jill always sees him "frozen" on the event horizon.
5. Assume that Jack is infinitesimally small (point-particle), so we can ignore any "temporal" tidal effects at the event horizon itself.
6. At some finite time T on Jill's clock, the black hole completely evaporates and exists no more. According to Jill, Jack never actually reached the event horizon. So in Jill's timeline, Jack is still alive and well. She thinks that he can return to her, and they can live happily ever after (although she is much, much older than he).

However, in Jack's timeline, he sailed through the event horizon and died billions of years ago. How is it possible that in Jill's timeline he was (in principle) able to return from the trip? The key is that the BH evaporates in finite time.

 

[PeroK]

However, in Jack's timeline, he sailed through the event horizon and died billions of years ago. How is it possible that in Jill's timeline he was (in principle) able to return from the trip? The key is that the BH evaporates in finite time.

First, you have to distinguish between events in spacetime and what signals reach Jill. Whether Jack reaches the event horizon or stops short and waits is a physical fact. You have to decide which physical scenario we are analysing. Jack may only return from the trip if he doesn't cross the horizon. If he does cross the horizon, he cannot come back (regardless of whether Jill has this information). The signals Jill eventually receives must be consistent with whether Jack crossed the horizon or not. How long it takes Jill to receive these signals is not relevant.

Second, the process of black hole evaporation will change the spacetime outside the horizon. It's no longer an eternal static spacetime. If we simply consider signals sent every second from Jack (his proper time), then every signal either eventually reaches Jill or does not. There is no immediate paradox or contradiction there. Jill will eventually have the full set of signals from Jack that she is ever going to receive. Although, now you are in the realms of the information paradox:

https://en.wikipedia.org/wiki/Black_hole_information_paradox

Finally, you have to avoid the assumption that there is a imperturable source of light continuously reflecting off Jack. If you want to pursue the scenario, you need to consider where the light originates that Jill sees. The simplest is to have Jack shine a torch back in Jill's direction. This is more complicated that sending a signal every second (which makes things more clear cut); but, in principle, every photon would eventually be accounted for and either escapes the hole or reaches the singularity.

 

[Tomas Vencl]

1. Jack decides to head toward the center of the black hole, traveling at an easy pace (say 10 km per hour).
2. Jill sees Jack (with her ultrasensitive infrared camera) asymptotically approach the event horizon, but never reach it.
3. Assume that Jill is immortal and can go on watching him forever.
4. The event horizon slowly shrinks due to evaporation (Hawking radiation), but Jill still sees Jack infinitesimally close to the event horizon, his clock moving infinitesimally slowly. The radius of the event horizon shrinks extremely slowly as the black hole evaporates, but Jack has a constant speed of 10 km/h, so Jill always sees him "frozen" on the event horizon.
5. Assume that Jack is infinitesimally small (point-particle), so we can ignore any "temporal" tidal effects at the event horizon itself.
6. At some finite time T on Jill's clock, the black hole completely evaporates and exists no more. According to Jill, Jack never actually reached the event horizon. So in Jill's timeline, Jack is still alive and well. She thinks that he can return to her, and they can live happily ever after (although she is much, much older than he).

I think that important in your scenario is that Jack (if I understand correctly) moves at some constant speed (question is what it means, dr/dtau, dr/dT, ...?). If really Jack is not free falling, but feels also gravitational forces, it can change the situation. First, he must use some "drive" (rocket engine, rope..), and IF he really cross the horizon, he is forced to accelerate (can not keep constant speed).
But if he is moving slowly enough, then can happen, that he never reach the horizon, because horizon shrinks faster (for him) then he moves (question is what is slow enough). In this case there is not any "paradox".
Imagine, that Jack decide near (but still outside) horizon to turn back and return to Jill. When they meet again, Jill is much older than Jack, but both now see the same BH that shrinked a little (the same shrink for both as both are at the same place when they meet again). So for Jack BH was shrinking faster during stay near horizon.
And if Jack is not moving slowly enough, I think this

The event horizon slowly shrinks due to evaporation (Hawking radiation), but Jill still sees Jack infinitesimally close to the event horizon

cannot happen. Because there are no light signals of Jack, which are emited below horizon (at r coordinates smaller than r at which Jack crossed the horizon) and which leads to Jill.
So correct approach (I think) is to use different metric. As said

Second, the process of black hole evaporation will change the spacetime outside the horizon. It's no longer an eternal static spacetime.

 

[Dale]

Summary:: A paradox about two travelers, one of which crosses the event horizon of a black hole, while the other watches him and waits until the black hole completely evaporates.

According to Jill, Jack never actually reached the event horizon. So in Jill's timeline, Jack is still alive and well.

There are several issues with this paradox, so multiple ways to resolve it. I will focus on this. Here A doesn’t imply B. All Jill can claim based on the evidence she has is that Jack was alive and well as he approached the EH. As it evaporates in finite time she receives no further information.

 

[anuttarasammyak]

The summary says Jack crosses the event horizon. So I think Jill will see the final moment of his entrance to BH soon after its evaporation.

 

[PeterDonis]

Jack decides to head toward the center of the black hole, traveling at an easy pace (say 10 km per hour).

10 km per hour relative to what?

Jack can't head toward the center of the hole at a constant speed relative to Jill. And "speed" doesn't even make sense if it's speed relative to the hole.

At some finite time T on Jill's clock, the black hole completely evaporates and exists no more. According to Jill, Jack never actually reached the event horizon.

Wrong. Jill sees Jack cross the horizon at the same time she sees the hole finally evaporate.

 

[PeterDonis]

All Jill can claim based on the evidence she has is that Jack was alive and well as he approached the EH. As it evaporates in finite time she receives no further information.

That's not correct. As I posted in response to the OP just now, Jill sees Jack cross the horizon at the same time that she sees the hole finally evaporate.

 

[Ibix]

That's not correct. As I posted in response to the OP just now, Jill sees Jack cross the horizon at the same time that she sees the hole finally evaporate.

Am I right that what she sees after that depends on what really happens instead of a singularity? The Penrose diagram for an evaporating black hole shows a region where all future-directed timelike and null paths end in the singularity. So GR says that the last Jill sees of him is the horizon crossing, and she sees that at the same time she sees the hole evaporate. But no one expects the singularity to be real so this is probably an incomplete picture.

 

[Dale]

That's not correct. As I posted in response to the OP just now, Jill sees Jack cross the horizon at the same time that she sees the hole finally evaporate.

What is incorrect about it? She does not receive any further information after evaporation, as I said.

 

[PeterDonis]

Am I right that what she sees after that depends on what really happens instead of a singularity?

I don't think so, at least not if we're talking about models that have a true event horizon. As far as I know, at the classical (or semi-classical) level, any model that has a true event horizon at all will have the property I described. Whether there is a singularity inside the horizon or something weirder will not change that.

There are proposed models in which there never is a true event horizon at all, only an apparent horizon. In those models, what happens to Jack, and what Jill sees, would depend on what is in the region at and inside the apparent horizon.

I discuss these kinds of issues in this Insights article:

https://www.physicsforums.com/insights/black-holes-really-exist/

 

[hmmm27]

If a BH is evaporating, doesn't the apparent radius of the EH shrink ? ie: Jill will eventually actually see (what's left of) Jack disappear into the (past) EH, and that before the complete evaporation ; what she will never see, of course, is anything that happened after the crossing.

 

[PeterDonis]

If a BH is evaporating, doesn't the apparent radius of the EH shrink ?

The surface area of the horizon does shrink when evaluated in a suitable way, yes. However, that doesn't mean quite what you think it means. See below.

Jill will eventually actually see (what's left of) Jack disappear into the (past) EH, and that before the complete evaporation

We have to be very careful here. The event horizon is an outgoing null surface. That means radially outgoing light rays stay there. So it's impossible to see any event that happens on the event horizon until the hole finally evaporates and the event horizon disappears completely. The radially outgoing light rays that stay at the horizon (until it disappears) are actually "staying" at a radial coordinate (surface area) that is shrinking, as above. That means that, viewed in terms of the radial coordinate, those light rays are actually moving inward until the hole finally evaporates!

However, there is another kind of horizon called an "apparent horizon", which is a surface at which radially outgoing light rays stay at the same radial coordinate. For an evaporating black hole, that means the apparent horizon is just outside the event horizon. It also means that the apparent horizon at any particular time does not stay an apparent horizon; as the hole shrinks, the apparent horizon shrinks too, so light that was trapped at the apparent horizon a little time ago can now escape.

So what Jill will actually see is Jack getting "stuck" briefly at apparent horizons that get smaller and smaller, until finally she sees Jack cross the actual event horizon at the same time she sees the hole finally evaporate. So Jill will indeed see Jack get initially "stuck", at wherever the apparent horizon was when he first fell in, before she sees the hole finally evaporate. And then she will continue to see Jack getting "stuck" at smaller and smaller radial coordinates. But none of those are the actual event horizon; they're all just apparent horizons.

 

[Ibix]

I don't think so, at least not if we're talking about models that have a true event horizon.

I was groping towards heuristic 2 in your article, I think. Looking at the Penrose diagram (e.g. the one on this page - please excuse the source, but the diagram matches my recollection of one you've posted previously) I think for an evaporating GR black hole the true event horizon exists because the singularity exists and "gets in the way" of the interior sending signals to future null infinity. We hope that there isn't a singularity in a quantum gravitational black hole, but then we don't know if there's a true horizon or not - it depends what actually goes on in there.

 

[Tomas Vencl]

We have to be very careful here. The event horizon is an outgoing null surface. That means radially outgoing light rays stay there. So it's impossible to see any event that happens on the event horizon until the hole finally evaporates and the event horizon disappears completely. The radially outgoing light rays that stay at the horizon (until it disappears) are actually "staying" at a radial coordinate (surface area) that is shrinking, as above. That means that, viewed in terms of the radial coordinate, those light rays are actually moving inward until the hole finally evaporates!

However, there is another kind of horizon called an "apparent horizon", which is a surface at which radially outgoing light rays stay at the same radial coordinate. For an evaporating black hole, that means the apparent horizon is just outside the event horizon. It also means that the apparent horizon at any particular time does not stay an apparent horizon; as the hole shrinks, the apparent horizon shrinks too, so light that was trapped at the apparent horizon a little time ago can now escape.

At which of those 2 horizons freefalling observer from infinity reach c ? Is it at apparent horizon ? (compare to situation, when freefalling to non evaporating BH reach c at event horizon). Thank you.

 

[otennert]

Yes, that's a classic and often laid out. And mostly inadequately discussed in my opinion, because the main point is actually often not understood. And in addition I think it is still not really resolved, due to the lack of understanding of the causality structure that quantum gravity (when known in some future theory) imposes upon spacetime.

One minor correction in your description of the paradox however: let Jack be a freely falling observer, not one casually traveling at "constant speed" because that doesn't make sense anyway. "Freely Falling" on the other hand is well-defined.

The main point in this paradox consists of the following takes upon the scenario:
1.) first take: Classical GRT, no BH evaporation, no semiclassical theory. The freely falling observer does not necessarily move radially towards the horizon, but may move in some nearly tangential orbit towards it. What then occurs is essentially a scattering process which eventually ejects Jack into future in the sense that: by the time he has somehow made it to some outer region away from the hozion to safer regions, the asymptotic observer will have measured say e.g. 10^50 years, whereas Jack (freely falling) measures only e.g. 25 mins between any two consecutive light signals. The BH is still there.

Attention: this is classical GRT! No Hawking effect etc. considered. It can all be exactly calculated within general relativity for different parameters.

2.) second take: Classical GRT, no BH evaporation, no semiclassical theory. The freely falling observer is now on a radial path. He will cross the horizon in a finite proper time. The asymptotic observer however will observe Jack approaching the horizon forever, increasingly slowly, and more and more redshifted and dimmed. Again, this is still classical general relativity, and can be exactly calculated.

However, takes #1 and #2 also -- if one wants to be consistent -- implies that horizon formation cannot be observed by an asymptotic observer to finish, as this also is observed to be an asymptotic process. Again, everything can be exactly calculated within GRT.

3.) third take: now the paradox comes into play when semiclassical theory is applied. It is important to understand that the semiclassical theory cannot explain any quantum effects, neither does it provide any hint towards the nature of a quantum theory of gravity.

What a semiclassical theory provides however, is a link between *any* quantum gravitation theory and classical GRT, similar to the correspondence principle in quantum mechanics. The consequence namely is that BH evaporation in a finite time implies that horizon formation for an asymptotic observer can never be observed before the BH itself has completely evaporated, neither will it be observed afterwards, of course. What remains is the question of what the remnant of the "BH" (now in quotes) actually is, and how it is to be defined in an obsever-independent manner. It is even completely unclear if the very notion of a horizon and a BH makes sense and can be defined in an observer-independent way, if at all.

In a strict sense, Jill will never see a horizon form, only motion slowing down for a finite albeit extremely long period of time. Jack will definitely be there after BH evaporation if his path had not been radial in the first place. If it has been only slightly tangential, he was then "surfing" on the horizon.

The important point here is to take as a premise that Jack's path can only be just tangential enough to *not* cross the "receding" (in his coordinate system) horizon. Then the argument given in this thread already breaks down, and still we end up with Jack being 10^50 years into the future, a Jill long deceased, and the BH gone (maybe also the whole universe after such a long time period), and maybe a naked singularity left.

The real unknowns are:
1.) What is the nature of the remnant? Is it a spacelike naked singularity? Is there one at all?
2.) What is the causal structure of spacetime, taking the stance that BH formation itself is not happening, as the whole concept of a horizon is clearly defined and observer-independent in classical GRT, but unclear in a semiclassical theory, and definitely unknown in any quantum gravity theory (which does not exist so far).
3.) What happens with a radially freely infalling obsever?

All of the above has a meaning with regards of the BH information paradox, which in my opinion, also is not understood and resolved so far at all.

My summary therefore: the paradox is not really resolved so far.

 

[Ibix]

At which of those 2 horizons freefalling observer from infinity reach c ?

Reach $c$ relative to what? And in what sense do you think a massive object can reach $c$?

 

[Ibix]

What then occurs is essentially a scattering process which eventually ejects Jack into future in the sense that: by the time he has somehow made it to some outer region away from the hozion to safer regions, the asymptotic observer will have measured say e.g. 10^50 years, whereas Jack (freely falling) measures only e.g. 25 mins between any two consecutive light signals. The BH is still there.

I don't believe this is possible with such an extreme time dilation factor. A free-falling observer can't cross the photon sphere and return, so $dt/d\tau$ never becomes more than 3 (for an observer dropped from rest at infinity). An accelerated observer, however, could reach arbitrarily close to the horizon given sufficient thrust.

 

[Tomas Vencl]

Reach $c$ relative to what? And in what sense do you think a massive object can reach $c$?

In sense dr/dtau=sqrt(2GM/r)

 

[otennert]

I don't believe this is possible with such an extreme time dilation factor. A free-falling observer can't cross the photon sphere and return, so $dt/d\tau$ never becomes more than 3 (for an observer dropped from rest at infinity). An accelerated observer, however, could reach arbitrarily close to the horizon given sufficient thrust.

Agreed, you are right for a freely falling observer in classical GRT, he/she needs to be accelerated. I was incorrect here. Nevertheless it is inessential for the main point in the paradox.

The whole paradox here affects the principal statements and the internal consistency of the semiclassical theory, not whether the scenario under discussion may be set up in realistic circumstances or not. In this sense, it is a gedanken experiment like all the classics in QM back a long time ago.

For this very reason, all replies that go along the lines of "the very last photon that ever gets emitted before the horizon is crossed" or "come 10^50 years, the universe will be gone by then anyway" do really miss the point.

 

[Ibix]

Of course it can, because the math allows for it. What limit should there be for $dt/d\tau$? 3? 3.5? 10? 5,000?

If you are free falling around a Schwarzschild black hole, $dt/d\tau=E/(1-R_S/r)$, where $E$ turns out to be the Lorentz gamma factor you have at infinity. If your freefalling observer wishes to leave without accelerating, $r>3R_S/2$. Thus $dt/d\tau<3E$. Assuming rest at infinity $E=1$ and the limit is as I said.

 

[otennert]

If you are free falling around a Schwarzschild black hole, $dt/d\tau=E/(1-R_S/r)$, where $E$ turns out to be the Lorentz gamma factor you have at infinity. If your freefalling observer wishes to leave without accelerating, $r>3R_S/2$. Thus $dt/d\tau<3E$. Assuming rest at infinity $E=1$ and the limit is as I said.

I have already corrected myself here, and you are correct in your observation. But this point is actually inessential for the paradox per se, because it can be be circumvented by acceleration.

 

[PeterDonis]

I think for an evaporating GR black hole the true event horizon exists because the singularity exists and "gets in the way" of the interior sending signals to future null infinity.

Yes.

We hope that there isn't a singularity in a quantum gravitational black hole, but then we don't know if there's a true horizon or not - it depends what actually goes on in there.

I'm not aware of any quantum gravity model (at least not one that has a reasonably precise mathematical formulation--there's a lot of vague speculation out there) that has a true event horizon but no singularity. All of the quantum gravity models that remove the singularity that I'm aware of also end up not having a true event horizon at all. Of course that's not a proof that no model with a true event horizon can be without a singularity; but it's hard for me to see what the geometry of such a model would look like at the semiclassical level, because of the heuristic you mention: what would "get in the way" of any event sending light signals to future null infinity, if there isn't a singularity inside the horizon?

 

[PeterDonis]

She does not receive any further information after evaporation, as I said.

But she does receive the information that Jack crossed the horizon, at the same time she receives the information that the hole evaporated; and once she receives that information, she knows Jack is not "alive and well"; he fell into the hole and got destroyed in the singularity. Your post did not appear to me to recognize that Jill gains that information when she sees the hole evaporate away.

 

[PeterDonis]

At which of those 2 horizons freefalling observer from infinity reach c ?

Neither. "Reaching c" is not well-defined.

 

[Tomas Vencl]

Neither. "Reaching c" is not well-defined.

Even in the sense of dr/dtau (for freefalling observer) ?

 

[Dale]

once she receives that information, she knows Jack is not "alive and well"; he fell into the hole and got destroyed in the singularity. Your post did not appear to me to recognize that Jill gains that information when she sees the hole evaporate away

By information I meant signals from Jack. The last signal she actually gets is him crossing, which she receives with the light from the moment of evaporation. Anything else she has to infer.

Specifically, the last actual observation she has of Jack is that he was alive and well as he crossed the horizon. She does not observe but only infers that he “got destroyed in the singularity” based on the fact that there is no Jack left after evaporation.

 

[PeterDonis]

The consequence namely is that BH evaporation in a finite time implies that horizon formation for an asymptotic observer can never be observed before the BH itself has completely evaporated, neither will it be observed afterwards, of course.

This is not correct. The asymptotic observer observes all events that happen on the horizon at the same time he observes the hole's final evaporation. Light signals from all of those events reach the asymptotic observer at the same event on that observer's worldline.

It is even completely unclear if the very notion of a horizon and a BH makes sense and can be defined in an observer-independent way, if at all.

This is not correct. Please read the Insights article I linked to earlier in this thread. The term "event horizon" has a perfectly well-defined definition (the boundary of a region of spacetime that cannot send light signals to future null infinity) that is observer-dependent.

Jack will definitely be there after BH evaporation if his path had not been radial in the first place. If it has been only slightly tangential, he was then "surfing" on the horizon.

This seems too optimistic. Jack cannot get arbitrarily close to the horizon even in the evaporating case; there are no stable orbits (even hyperbolic orbits that come in from infinity and escape to infinity) arbitrarily close to the horizon even if it is receding. So if Jack gets too close, he will fall into the hole even if he is not moving purely radially.

The real unknowns are

While these are unknowns in the general case, they are not unknowns in the specific case being discussed in this thread. That case assumes a particular spacetime geometry (basically the original evaporating black hole geometry that Hawking used), and in that geometry, all of your questions have definite answers:

1.) What is the nature of the remnant? Is it a spacelike naked singularity? Is there one at all?

There is no remnant after the hole finally evaporates; but there is a spacelike singularity inside the event horizon that objects that fall inside the horizon will hit and be destroyed.

2.) What is the causal structure of spacetime

The one in, for example, https://www.researchgate.net/figure/Maudlins-version-of-an-evaporating-black-hole-Penrose-diagram_fig1_320223267.

3.) What happens with a radially freely infalling obsever?

If he falls in early enough, he falls through the horizon and gets destroyed in the singularity. If he falls in late enough, he passes through the outgoing spherical light wave front produced by the hole's final evaporation and finds himself in a flat Minkowski region of spacetime.

 

[PeterDonis]

Even in the sense of dr/dtau (for freefalling observer) ?

That quantity is not limited to $c$; it increases without bound as $r = 0$ is approached. So it cannot be interpreted as a "speed".

If you are fine with that, then I think the answer is that $dr / d\tau = c$ when the true event horizon is crossed, but I have not done the math to confirm that.

 

[PeterDonis]

She does not observe but only infers that he “got destroyed in the singularity” based on the fact that there is no Jack left after evaporation.

Hm, ok, your requirement for using the word "knows" is much more strict than mine. To me, since Jill knows the laws of physics and the spacetime geometry, she knows that Jack is doomed once he has crossed the horizon. But it's true that she never actually sees any light signals showing him getting destroyed (we are assuming that the hole is large enough that Jack can withstand the tidal gravity at the horizon).

 

[Dale]

Hm, ok, your requirement for using the word "knows" is much more strict than mine. To me, since Jill knows the laws of physics and the spacetime geometry, she knows that Jack is doomed once he has crossed the horizon. But it's true that she never actually sees any light signals showing him getting destroyed (we are assuming that the hole is large enough that Jack can withstand the tidal gravity at the horizon).

Agreed, I was intending to be strict in response to the OP's statement "So in Jill's timeline, Jack is still alive and well".

Jill does not know that because all Jill can observe is that he WAS alive and well as he crossed the horizon. If Jill applies the laws of physics she will indeed agree with your assessment, but it appeared to me that the OP was not applying the laws of physics but simply relying on the raw observation. The raw observation only tells about Jack's state up to the crossing and cannot be used to infer that he IS alive and well.

 

[Tomas Vencl]

That quantity is not limited to $c$; it increases without bound as $r = 0$ is approached. So it cannot be interpreted as a "speed".

If you are fine with that, then I think the answer is that $dr / d\tau = c$ when the true event horizon is crossed, but I have not done the math to confirm that.

Thank you. (I used "speed" because it corresponds to Newtonian escape velocity, But I do not need to interpret it as "speed",just $dr / d\tau$ is ok.)

 

[otennert]

Found the paper by Ashtekar and Bojowald 2005. It is not very technical but has some references that surely need to be read in addition, but summarizes a lot of issues with the "traditional" semi-classical approach instigated by Hawking on BH evaporation.

 

[PeterDonis]

Found the paper by Ashtekar and Bojowald 2005.

Arxiv link is here since the iopscience paper is behind a paywall:

https://arxiv.org/abs/gr-qc/0504029

 

[PeterDonis]

this is based on the assumption that the final naked singularity is spacelike

The singularity isn't naked since it's behind a horizon, but it is spacelike, yes, like the one for an eternal Schwarzschild black hole.

to me it is not really clear that this is the case. It is not even clear to me that a final singularity exists at all.

Yes, that's an open question. But this particular thread is discussing that model, since that's the setting for the OP's scenario.

 

[PeterDonis]

The discussion has evolved

You're the only one who is trying to do that. If you want to discuss the more general questions you are raising, please start a separate thread.