# Black hole horizon and Hawking evaporation

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Marilyn67
TL;DR Summary
Apparent contradiction between the formation of a horizon and the evaporation of black holes
Hello,

I take the example of two observers :

- A distant observer
- A falling observer

For the distant observer, the formation of the horizon is not part of his future cone of light, we agree.

For the falling observer, the consensus says it is crossing the horizon.

First question: the evaporation of the black hole occurs in a very long time, but finished for the distant observer.
How can the falling observer cross a horizon that has not happened, since the black hole has already evaporated? (for him the "film" is simply "accelerated").

Second question: The consensus is that the Hawking radiation occurs at the horizon. How can a distant observer perceive radiation coming from a horizon that does not yet exist (as if the radiation came from an event that occurs in the future?)

see you soon

Marilyn

Mentor
For the distant observer, the formation of the horizon is not part of his future cone of light, we agree.
I don’t agree. In the Oppenheimer Snyder metric there certainly are observers for whom the formation of the horizon is in their future light cone. This set of observers includes both observers that eventually cross and those that do not cross the horizon.

Personally, I do have issues with Hawking radiation. I do not know of a valid metric for an evaporating black hole. The outgoing Vadiya metric is the closest, but it is a white hole not a black hole. So I am unsure what metric to use to actually calculate the correct answer to questions like this.

vanhees71 and Ibix
2022 Award
For the distant observer, the formation of the horizon is not part of his future cone of light, we agree.
Horizon formation is never in an external observer's past light cone. It can be in the future light cone.
How can the falling observer cross a horizon that has not happened, since the black hole has already evaporated?
I don't think an evaporating black hole has an event horizon, strictly speaking. It has an apparent horizon, something that's indistinguishable from an event horizon on timescales much less than the evaporation time of the black hole, but not an actual event horizon.
The consensus is that the Hawking radiation occurs at the horizon.
I don't think that's correct. Hawking radiation comes from the region just above the horizon.

vanhees71 and Dale
Marilyn67
Hello Ibix,

Thank you for your quick and concise response.
I think we agree on all but one point.
You say :

Horizon formation is never in an external observer's past light cone. It can be in the future light cone.

When I describe a distant observer, I mean a distant observer who stays that way (ad infinitum).

In this case, isn't talking about a past light cone or a future light cone the same thing?
The point is, for a distant observer (and who stays that way), the horizon of the black hole never forms, right ?

From what I understand, the (real) horizon never forms for anyone (distant or falling observer).

The model of General Relativity (with horizon) becomes incompatible with the Hawking radiation which evaporates the collapsing star in a finite time.

We agree ? If not, where is the error ?

Thank you very much,
See you soon,

Marilyn,

Mentor
In this case, isn't talking about a past light cone or a future light cone the same thing?
The point is, for a distant observer (and who stays that way), the horizon of the black hole never forms, right ?
No. The formation of the event horizon is a specific event in the manifold. That event has a past light cone, and that past light cone goes out to infinity. For every event inside the past light cone, including those on the distant observers worldline, the formation of the event horizon is in the future light cone.

From what I understand, the (real) horizon never forms for anyone (distant or falling observer).
This is not correct. Perhaps you are thinking of the Schwarzschild spacetime instead of the Oppenheimer Snyder spacetime. In the Schwarzschild spacetime the horizon is eternal and static so there is no formation. So if you are talking about the formation of an event horizon then you cannot use a spacetime where it always existed.

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vanhees71 and Ibix
2022 Award
When I describe a distant observer, I mean a distant observer who stays that way (ad infinitum).

In this case, isn't talking about a past light cone or a future light cone the same thing?
For an eternal black hole like the Schwarzschild solution the direction of time doesn't make much difference, no. But you were talking about the formation and evaporation of a black hole. This is a different model, and there is clearly a unique sense of future and past light cones, because there's a star there before the black hole. The horizon formation event is part of the final collapse of the star, which can be in my causal future but not in my causal past until the hole evaporates.
The model of General Relativity (with horizon) becomes incompatible with the Hawking radiation which evaporates the collapsing star in a finite time.
A model of an eternal black hole is incompatible with a model of a black hole that forms and evaporates, certainly. But GR can certainly describe the former (for example the Oppenheimer-Snyder collapse model mentioned by @Dale), and can describe the latter with a semi-quantum extension without which you can't describe Hawking radiation at all. So I think it's wrong to say that GR is incompatible with finite-lifetime black holes. GR is more than just the Schwarzschild black hole.

vanhees71 and Dale
Tomas Vencl
Personally, I do have issues with Hawking radiation. I do not know of a valid metric for an evaporating black hole. The outgoing Vadiya metric is the closest, but it is a white hole not a black hole. So I am unsure what metric to use to actually calculate the correct answer to questions like this.

I don't think an evaporating black hole has an event horizon, strictly speaking. It has an apparent horizon, something that's indistinguishable from an event horizon on timescales much less than the evaporation time of the black hole, but not an actual event horizon.

I must react. I am really happy to read such a reply, which I definitely agree. A few years ago standard answer for Marylins question was usually strong rejection with some mantra about obvious difference between freefalling and Schwarzschild observer (which by the way says nothing to this problem).
The question is pretty common and seems naive (and as naive is often answered) , but it has some logic.
Of course this naiv arguing breaks the (probably) fact that Hawking radiation do not occurs exactly at the horizon, but when you go deeper to problem of evaporating BH it seems that you really need different solution (metric). This is my personal belief. I will be not so surprised if finally this simple and first view approach has correct result, of course the result will have much more complex and deeper reasons.

Mentor
Summary:: Apparent contradiction between the formation of a horizon and the evaporation of black holes

See this Insights article:

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

It discusses the scenario you describe and why there is no actual contradiction.

For the distant observer, the formation of the horizon is not part of his future cone of light, we agree.

No, we don't. The horizon is in the future light cone of a distant observer.

In a spacetime with a black hole that does not evaporate, the horizon is not in the past light cone of distant observers, so they cannot see the horizon, but that does not prevent observers from falling into it.

In a spacetime with a black hole that does evaporate, the horizon (but not the region inside it) eventually does enter the past light cone of distant observers; but there is a very long period of time before that happens when it is still perfectly possible for those distant observers to fall into the hole.

Mentor
I do not know of a valid metric for an evaporating black hole. The outgoing Vadiya metric is the closest, but it is a white hole not a black hole.

I think a spacetime that includes a region similar to the Oppenheimer-Snyder collapse, joined to its future with an outgoing Vaidya region, in turn joined to its future with a Minkowski region inside the last "shell" of outgoing radiation, would work. The fact that the outgoing Vaidya region in this heuristic model is only a region, not the entire spacetime, avoids the white hole issue you refer to.

Mentor
I don't think an evaporating black hole has an event horizon, strictly speaking.

There are models of evaporating black holes that do have one; the simplest is the one Hawking originally used. But it's still an open question whether those models actually describe things in our real universe.

vanhees71
Mentor
The fact that the outgoing Vaidya region in this heuristic model is only a region, not the entire spacetime, avoids the white hole issue you refer to.
I am not convinced that it does, but I have never seen a formal derivation one way or another. I can’t see how the “stitching” you describe changes the fact that the outgoing Vaidya metric is a white hole. It produces an event horizon which allows matter to cross inward during the OS portion and then suddenly does not allow matter to cross inward during the outgoing V portion. I can’t visualize how that could work (which is of course not a proof, but just a limitation of mine)

I have wondered if the ingoing V metric would work using a negative radiation rate but this is beyond my capability to evaluate.

vanhees71
2022 Award
There are models of evaporating black holes that do have one; the simplest is the one Hawking originally used. But it's still an open question whether those models actually describe things in our real universe.
Ok - I misunderstood something. Having tracked down a Penrose diagram I see you are correct. There's a part of spacetime where you must hit the singularity before evaporation happens. So whether there's actually an event horizon or not depends on what actually happens where GR says there's a singularity?

Mentor
I can’t see how the “stitching” you describe changes the fact that the outgoing Vaidya metric is a white hole.

Because you're only using a portion of the full spacetime described by the outgoing Vaidya metric. The portion being used does not include the white hole region of the outgoing Vaidya metric. It's no different from the fact that the Oppenheimer-Snyder model you refer to does not have a white hole issue, even though the maximally extended Schwarzschild spacetime includes a white hole, because the portion of maximally extended Schwarzschild spacetime used in the Oppenheimer-Snyder model does not include the white hole region.

It produces an event horizon which allows matter to cross inward during the OS portion and then suddenly does not allow matter to cross inward during the outgoing V portion.

See Figure 2 of this paper:

https://www.groundai.com/project/a-linear-mass-vaidya-metric-at-the-end-of-black-hole-evaporation/1

This is the kind of model I was describing. The region marked "III" is the outgoing Vaidya region. Note that that region is only a portion of the full outgoing Vaidya spacetime, which is shown in Figure 1 (the region marked "I" in that figure, if I'm understanding it correctly, is the white hole region).

Mentor
The portion being used does not include the white hole region of the outgoing Vaidya metric.
I think you may be misunderstanding the outgoing V spacetime (or I am). This has noting to do with the maximally extended Schwarzschild spacetime where there are two horizons. To my knowledge the outgoing V spacetime has only one horizon and it is a white hole horizon.

As you say, this is similar to the OS spacetime which has only one horizon and it is a black hole horizon.

I need to read that paper.

Mentor
To my knowledge the outgoing V spacetime has only one horizon and it is a white hole horizon.

I know. But that doesn't mean you can't put a portion of the outgoing Vaidya spacetime, one that doesn't include the white hole region, into a model that includes portions of other spacetimes as well. The model I described, and which is shown in Figure 2 of the paper I referenced, does that: it uses only the portion of outgoing Vaidya spacetime that is outside the white hole horizon of that spacetime, and joins it to a portion of Schwarzschild spacetime that includes the Schwarzschild exterior region and the Schwarzschild black hole region. The black hole horizon in that model is in the Schwarzschild region, not the outgoing Vaidya region.

Mentor
it uses only the portion of outgoing Vaidya spacetime that is outside the white hole horizon of that spacetime
Yes, I see that in the paper, but that then does not represent exactly the portion of primary interest: the horizon during evaporation.

In this paper it looks like the V spacetime is used to transition from Schwarzschild to Minkowski, but not for the event horizon except possibly right at the end in Figure 6.

I think the portion of primary interest is what they describe as “adiabatically evaporating Schwarzschild” which I have never seen before.

Mentor
In this paper it looks like the V spacetime is used to transition from Schwarzschild to Minkowski, but not for the event horizon except possibly right at the end in Figure 6.

Yes, that's my understanding.

I think the portion of primary interest is what they describe as “adiabatically evaporating Schwarzschild” which I have never seen before.

I'm not entirely sure what that means. It might be that that portion is also technically an outgoing Vaidya region, just with a different mass function.

I think part of the problem might be that the model in that figure is leaving out the collapse part--the Oppenheimer-Snyder type region where the black hole is initially formed. In the figure it looks like the Stretched Horizon (SH) line goes all the way back to past timelike infinity, which obviously can't be the case for an actual collapsing object.

Dale
Mentor
I'm not entirely sure what that means. It might be that that portion is also technically an outgoing Vaidya region, just with a different mass function.
Me neither, and a cursory reading of the text didn’t provide details. I will need to read it more in depth to see if there are additional hints

Marilyn67
Thanks @PeterDonis for your response, as well as for the very extensive article link you wrote.
I understood that my question is simplistic and that intuition, the most "rational" explanation is not necessarily the right one.
Your knowledge is superior to my knowledge, and it's completely beyond me.
I can't get a clear picture of the chronology of events.
I understood that the answer is not yet clear and that it depends on the models.

Cordially
Marilyn

Mentor
I understood that the answer is not yet clear

The answer to whether black holes actually evaporate in our actual universe (and even whether the objects we call "black holes" are actually black holes in the sense of having true event horizons) is not yet clear, yes.

But, as I said in the article, if it turns out that our current ideas about black holes and black hole evaporation need modification, it won't be because there is any inherent contradiction between black hole formation and black hole evaporation. We have consistent models that contain both; the open question is whether those models are actually realized in our universe.

Dale and andrew s 1905
Marilyn67
it won't be because there is any inherent contradiction between black hole formation and black hole evaporation. We have consistent models that contain both; the open question is whether those models are actually realized in our universe.

Yes, I understood the meaning of your article correctly.
It's these consistent models that I don't yet understand very well.

Cordially
Marilyn

Personally, I do have issues with Hawking radiation. I do not know of a valid metric for an evaporating black hole. The outgoing Vadiya metric is the closest, but it is a white hole not a black hole. So I am unsure what metric to use to actually calculate the correct answer to questions like this.
Strictly speaking you are saying that there isn't a known explicit solution. Not that there are no solutions. Most solutions are not possible to write in closed form, but they do exist.

Dale
Marilyn67
Hello @PeterDonis,

I just found a link to an article published by Stephen Hawking in the journal "Nature" in 2014.
His conclusions are the same as yours :

https://www.sciencesetavenir.fr/fondamental/stephen-hawking-les-trous-noirs-n-existent-pas_23069

D'une manière générale, précise Hawking, on ne peut rendre compte parfaitement de ces astres curieux tant que nous n'avons pas élaboré une théorie de la gravité unifiée. C'est à dire capable de concilier les lois de la physique quantique des échelles subatomiques avec la relativité générale qui rend compte de l'astronomie…

It's in French (sorry, I am of French-speaking origin) but if we translate :

Generally speaking, Hawking specifies, we cannot fully account for these curious stars until we have developed a unified theory of gravity. That is to say capable of reconciling the laws of quantum physics of subatomic scales with general relativity which accounts for astronomy…

Consistent models are a real problem for me !

As a distant observer, and who stays that way indefinitely, I don't "see" the horizon forming before the black hole evaporates, it's impossible.

In my opinion, these "models" are highly speculated:

S.H himself says:

En effet, la description des trous noirs par la physique classique, telle qu'on l'enseigne aux cours des premières années universitaires aux étudiants, n'est pas exacte.

If we translate :

Indeed, the description of black holes by classical physics, as it is taught during the first years of university to students, is not exact.

Cordially
Marilyn

Mentor
Strictly speaking you are saying that there isn't a known explicit solution. Not that there are no solutions. Most solutions are not possible to write in closed form, but they do exist.
If you have a published numerical solution I would certainly accept that also. I don’t know anywhere that has such a solution either. Do you?

Mentor
As a distant observer, and who stays that way indefinitely, I don't "see" the horizon forming before the black hole evaporates, it's impossible.
I don’t believe that claim can be asserted either. Without a valid metric you cannot assert this.

With the OS metric the formation of the EH is a specific event. I am skeptical that feature would change with an evaporating model. And without such a model you have no basis to make this claim one way or the other

Mentor
In my opinion, these "models" are highly speculated

Hawking is not saying what you think he is saying. He is not saying the models that are actually used in black hole physics, for example to study quasars, or to model the black hole at the center of our galaxy, or to model black hole mergers in order to analyze data from LIGO, are speculative or inexact.

He is only saying that the highly idealized and simplified models that are taught in introductory university courses in GR are not exact (he is not saying they are speculative, just that they are highly idealized and simplified and so are not exact).

Big difference.

Dale
Mentor
He is only saying that the highly idealized and simplified models that are taught in introductory university courses in GR are not exact (he is not saying they are speculative
And that is true of nearly everything taught in introductory courses. So it is not a big criticism at all

Marilyn67
Hello,

He is only saying that the highly idealized and simplified models that are taught in introductory university courses in GR are not exact (he is not saying they are speculative, just that they are highly idealized and simplified and so are not exact).

Yes, I know, I was the one who said that these idealized models were speculative. The term is abusive.
I don't understand these models, and I don't have your level, I recognize it.

My problem with these models is that I don't understand, as a distant observer (and who stays that way indefinitely), how we can witness the formation of a real horizon, with an infinite redshift (we observes a star undergoing asymptotic collapse) before a finite time corresponding to the evaporation of the black hole.

How can a model remove this contradiction ? Where is the error in this current description?

And that is true of nearly everything taught in introductory courses. So it is not a big criticism at all

Yes I agree

Mentor
My problem with these models is that I don't understand, as a distant observer (and who stays that way indefinitely), how we can witness the formation of a real horizon

In a spacetime with an evaporating black hole, the distant observer sees the formation of the horizon at the same time as he sees the final evaporation of the black hole. That finite time (which will be very, very long--something like ##10^{67}## years for a black hole of roughly the mass of the Sun, and the time increases as the cube of the mass) takes the place of the "infinite time" for the distant observer in the non-evaporating black hole model.

Marilyn67
Hello @PeterDonis,

In a spacetime with an evaporating black hole, the distant observer sees the formation of the horizon at the same time as he sees the final evaporation of the black hole. That finite time (which will be very, very long--something like ##10^{67}## years for a black hole of roughly the mass of the Sun, and the time increases as the cube of the mass) takes the place of the "infinite time" for the distant observer in the non-evaporating black hole model.

I thank you for your message.
This question has kept me awake for years and you are the first to give me a relevant answer! Thank you !

Most of the speakers with a sufficient culture on the French forums bypass the question and never answer it.
You are the first to give me a relevant answer.
Thank you very much.

You have both the level of study and the pedagogy to explain what escapes us.

I will finally be able to better understand these models with this contradiction that you have just resolved and which seems to me to be completely consistent.

Thank you very much,

Marilyn

timmdeeg, phinds and berkeman
Mentor
Thank you very much

You're welcome! Glad I could help.