I Falling through the event horizon of an evaporating black hole

[PeterDonis]

What we want for an evaporating black hole, however, is an object that starts with a finite mass, emits outgoing null dust for a finite time, and ends up with zero mass. We also need the object to be a black hole, formed by collapsing matter, not a white hole that, as you can see from the figure I referenced, has to be "built in" to the spacetime in its infinite past. So we would need to take a finite range of $u$ from the figure, and start somewhere outside the "EH" line, and join that region to the other regions we need.

For an example that's not exactly what I described earlier, but is similar, see Fig. 5 in this paper:

https://arxiv.org/abs/gr-qc/0506126

This paper is actually discussing what it calls "regular black holes", of which the Bardeen black hole that I mentioned before is an example. These solutions have no actual event horizons or black hole regions; every event in them can send light signals to future null infinity. But they do give an example of joining an outgoing Vaidya region to other regions. In the figure, the region between the "pair creation surface" and future null infinity, bounded by the $u = v_d$ and $u = v_f$ lines, is an outgoing Vaidya region.

Note that the region at the right marked "static $m = m_0$" is a Schwarzchild vacuum region. The diagram is very distorted in terms of actual proper time: in an actual instance of this kind of model, an observer could remain in the "static" region for a time similar to the Hawking evaporation time, i.e., $10^{67}$ years for a one solar mass hole.

Note also that this figure shows the original "hole" forming by ingoing null radiation (an ingoing Vaidya region) instead of the collapse of a timelike object. If we did the latter instead, the beginning regions marked "flat" and "radiation - positive energy flux" ingoing, would instead be occupied by something like a collapsing FRW region as in the Oppenheimer-Snyder model.

 

[Dale]

Fig. 1 of this paper shows the maximal extension of the outgoing Vaidya metric:

https://arxiv.org/abs/2307.06139

Note that the "EH" line goes up and to the left, i.e., it is an ingoing null line.

Yes, to me, that diagram clearly shows that the horizon where radiation is emitted (labeled EH) is a white hole. Timelike paths only go out.

So we would need to take a finite range of u from the figure, and start somewhere outside the "EH" line, and join that region to the other regions we need

I don’t care about the other regions. The problem is EH itself. It is a white hole horizon. It doesn’t matter what you join it up to before or after. It doesn’t match what we want for Hawking radiation, which is a surface that both emits some dust (not necessarily null) and which timelike paths can only enter.

I don’t think that is the Vadiya metric and that diagram certainly supports my previous understanding.

 

[PeterDonis]

The problem is EH itself. It is a white hole horizon.

Yes, but...

It doesn’t matter what you join it up to before or after.

It matters if the portion of the diagram that you join up to something else does not include the white hole horizon. Which is what I have been describing. If you look at the figure I referenced in post #31, you will see that it includes an outgoing Vaidya region just like what I described in post #30: a finite range of $u$ values and no white hole horizon on the "inner" side (meaning that that region was "cut" from a portion of the maximal extension of outgoing Vaidya that does not include the "EH" line). In fact there is no event horizon, either ingoing (white hole) or outgoing (black hole) anywhere in that entire diagram.

It doesn’t match what we want for Hawking radiation, which is a surface that both emits some dust (not necessarily null) and which timelike paths can only enter.

No, that's not quite what we want for Hawking radiation.

An outgoing null surface, which is what a black hole horizon is, can't emit anything outward. It is already traveling outward itself at the speed of light. If it emits anything, it can only do so inward. So wherever the Hawking radiation is coming from in a model like Hawking's original one, where there is a black hole event horizon, it cannot be coming from that horizon. IIRC Hawking handwaved this in his original proposal by saying that the radiation was coming from a "layer" just outside the event horizon. But I don't know if this issue with models of evaporation that contain true event horizons has ever been fully addressed.

 

[Dale]

It matters if the portion of the diagram that you join up to something else does not include the white hole horizon. Which is what I have been describing.

Well, then that is not particularly relevant. The most interesting part is the horizon (and the singularity second).

The paper that you objected to was about general spherically symmetric spacetimes, and specifically investigated solutions where the horizon ends in finite time. That is the region of specific interest, and that is the specific region that I do not think is modeled by the outgoing Vaidya metric.

Both of your papers seem to support that. The first clearly showed the horizon of the outgoing Vaidya metric as a white hole horizon and the second appeared to maybe show the horizon evaporating in an ingoing Vaidya metric with negative energy.

I don’t think that your criticism of the paper I cited is well founded. Their solution seems a more reasonable candidate than the Vaidya metric. The ingoing Vaidya metric with negative energy is something I hadn’t seen before. That seems more promising than what I previously knew of the Vaidya metric. But I don’t see your objection to the paper.

 

[PeterDonis]

that is not particularly relevant. The most interesting part is the horizon (and the singularity second)

I disagree. A white hole is physically unreasonable because it would have to be built into the universe from the beginning. It's not something that can be formed by the collapse of a star, as a black hole can. So it does not seem to me to be a viable candidate for modeling evaporation of a black hole.

The paper that you objected to was about general spherically symmetric spacetimes, and specifically investigated solutions where the horizon ends in finite time.

Do you mean this paper?

https://arxiv.org/abs/1102.2609

If so, that paper's very title is "Black Hole - Never Forms or Never Evaporates". That means either no horizon ever forms, or if a horizon forms, it never goes away. The paper appears to claim that it is impossible to construct a solution in which a horizon forms and then evaporates--so to the extent that it investigates such solutions, it is only to (claim to) rule them out. That claim, for reasons I have already given, appears to me to be, at the very least, extremely implausible.

second appeared to maybe show the horizon evaporating in an ingoing Vaidya metric with negative energy

The solution described in the figure I referenced in the second paper, as I have already said, does not contain an event horizon anywhere. The only horizons are apparent horizons, i.e., marginally trapped surfaces. I posted it to give an example of a solution that joins a region of the outgoing Vaidya metric that does not contain the past event horizon or the white hole to other regions with different geometries, to show that such a thing is possible--in other words, that using the outgoing Vaidya metric as one region of a solution does not commit you to using the white hole region of the maximally extended outgoing Vaidya metric.

It also serves as an example of an alternative model to Hawking's original evaporating black hole that can look like that original Hawking proposal from the outside for a very long time, on the order of the Hawking evaporation time, without actually having an event horizon anywhere. As I have commented in other threads on this topic, if I had to put my money on one possibility for what we will eventually find to be the correct type of model for such objects, that is where I would put it.

I don’t think that your criticism of the paper I cited is well founded. Their solution seems a more reasonable candidate

If you mean the paper I referenced above, I don't see any specific solution being proposed. It claims to derive a "universal" spherically symmetric metric, and then uses it to make claims about black hole formation and evaporation that, as I stated above, I find at the very least highly implausible. Not to mention in contradiction to much other literature.

The ingoing Vaidya metric with negative energy is something I hadn’t seen before.

Mathematically it just means choosing the opposite sign for the mass function that appears in the metric. Physically it is indeed one aspect of the solution described in the paper I referenced for which I would want to see more justification. It is possible that it has something to do with the fact that the deep interior of that solution has to contain dark energy, but I don't see that really addressed in the paper I referenced.

I don’t see your objection to the paper.

I stated them above, but just to quickly summarize (and to add one more item I referred to in an earlier post):

The paper claims that either black holes (i.e., event horizons) never form, or if they form, they never evaporate away. Both claims contradict much other literature, from Hawking's original paper on (indeed, the "never form" part contradicts the original 1939 Oppenheimer-Snyder paper). They also seem highly implausible to me just based on the Penrose diagram of Hawking's original model.

The paper uses Schwarzschild coordinates in its treatment of the possibilities for evaporating black holes, which you yourself pointed out doesn't work. (I mentioned this in post #21.)

 

[Dale]

A white hole is physically unreasonable because it would have to be built into the universe from the beginning. It's not something that can be formed by the collapse of a star, as a black hole can. So it does not seem to me to be a viable candidate for modeling evaporation of a black hole.

I agree. This is why I do not think that the Vaidya metric is a viable candidate for representing Hawking radiation.

hat paper's very title is "Black Hole - Never Forms or Never Evaporates".

Yes, the title is a little click-bait-ish. And I am not sure if they refer to the singularity, the horizon, or both. So a more factual title would have been better.

I posted it to give an example of a solution that joins a region of the outgoing Vaidya metric that does not contain the past event horizon or the white hole to other regions with different geometries,

OK, so stipulated. You can join regions of spacetimes together. That was never disputed on my end.

Then it seems to me that the question of the metric for a black hole evaporating ala Hawking is still open. The only proposal I have seen here or elsewhere has been Vaidya, and I simply don't think that holds. Sure, maybe some part of the Vaidya metric is some part of a Hawking evaporating black hole, but what is the whole metric or at least the metric for the horizon?

can look like that original Hawking proposal from the outside for a very long time, on the order of the Hawking evaporation time, without actually having an event horizon anywhere

That sounds a lot like "never forms". Just saying. Are you sure that you actually disagree with the paper?

Not to mention in contradiction to much other literature.

Are you sure it is in contradiction to other literature? I am not seeing it. I am not seeing the other literature which actually calculates the metric of an evaporating black hole. Vaidya is an evaporating white hole. The second paper wasn't any hole. So what literature is it contradicting? Hawking didn't write down a contradictory metric for an evaporating black hole. Who did?

Both claims contradict much other literature, from Hawking's original paper on (indeed, the "never form" part contradicts the original 1939 Oppenheimer-Snyder paper).

You are missing the "or" in there, particularly wrt Oppenheimer-Snyder. Never forms or never evaporates. The OS black hole does not evaporate. It forms, but it does not evaporate. This paper does not contradict OS at all.

And again, Hawking did not propose a metric that could be contradicted here.

 

[PeterDonis]

I don’t see your objection to the paper.

Comparing that paper with two of its references might help to show the issue I see with it. The papers linked to below are listed as [Vac07] and [VSK07] in the paper you referenced earlier.

https://arxiv.org/abs/0706.1203v1

https://arxiv.org/abs/gr-qc/0609024

The key difference I see in both of these papers, as compared to the Yi Sun paper, is that they acknowledge that there is no problem classically with models like the original Oppenheimer-Snyder model or the original Hawking proposal. (Note that this thread is in the relativity forum, so classical GR is the appropriate framework for discussion.) By contrast, the Yi Sun paper, at least as I read it, is claiming to cast doubt on those models at a classical level, by claiming to derive a classical "universal" metric for spherically symmetric spacetimes and claiming that it rules out classical models where event horizons form and then evaporate.

The question the papers I linked to above investigate is whether quantum corrections might make a significant difference. The first paper looks at possible quantum corrections to a collapse with a Schwarzschild exterior that would effectively stop collapse prior to an event horizon forming (for example, by changing the effective stress-energy tensor to something more like dark energy, which would exert "gravitational repulsion"). The second paper looks at quantum field effects in a collapsing domain wall spacetime that would classically form an event horizon and a black hole. Both papers conclude that yes, quantum corrections might make a signficant difference.

 

[PeterDonis]

That sounds a lot like "never forms". Just saying. Are you sure that you actually disagree with the paper?

The paper I referenced shows, if you agree that its model is self-consistent, that it is possible to have solutions that look from the outside like black holes but don't actually have event horizons.

The paper you referenced claims that it is impossible for any valid solution to have an event horizon that forms and then evaporates. The paper I referenced is irrelevant to that claim.

Are you sure it is in contradiction to other literature?

It's in contradiction to all of the literature from Hawking's original paper on that treated Hawking's model as a valid solution--valid in the sense of being self-consistent--and then investigated its properties. The paper you referenced is claiming that Hawking's model is not even self-consistent--that there is no solution of the Einstein Field Equation that corresponds to it.

Hawking did not propose a metric that could be contradicted here.

The paper you referenced claims that it is impossible to have a spherically symmetric solution whose Penrose diagram looks like Hawking's. Fig. 8 of the second paper I referenced shows that Penrose diagram. You are quite correct that Hawking did not explicitly write down a metric corresponding to that Penrose diagram, but that does not mean it is impossible to do so. Impossibility is a very strong claim, which IMO the paper you referenced does not even come close to justifying.

(The comments in the VSK paper in Section VII on the diagram show in Fig. 8 of that paper are relevant, btw. In contrast to the paper you referenced, they never claim it is impossible to have a solution with that Penrose diagram. They only give reasons for thinking that that diagram is not physically plausible as a model that would actually be realized.)

 

[Dale]

The paper you referenced is claiming that Hawking's model is not even self-consistent--that there is no solution of the Einstein Field Equation that corresponds to it.

Hawking never made a model that showed that claimed that a black hole could both form and evaporate. Hawking only proposed a model showing that it could evaporate. This paper does not contradict that.

In fact, Hawking’s paper assumes an already existing black hole, and shows local arguments for evaporation. He never produced a global solution to be contradicted. As you yourself indicated, we can divide spacetimes up into different regions and patch them together. Hawking only proposed a local patch and this paper confirmed that there exist global solutions that have such local patches.

You are quite correct that Hawking did not explicitly write down a metric corresponding to that Penrose diagram, but that does not mean it is impossible to do so.

Has anyone else written such a metric then? Showing that it is possible? I have asked before and only ever been pointed to the Vaidya metric, which doesn’t have that Penrose diagram.

Surely the best way to refute this claim is to explicitly show a counterexample.

1) I think you are seeing contradictions with literature that it is not actually contradicting

2) where it does in fact contradict the literature, the literature is incomplete, containing proposals but not solutions.

This paper is not the be-all and end-all, but I think you are dismissing it far too abruptly. The conflict with the literature is not as great as you say, and the conflict is limited to a field where the literature is incomplete. Is that not how science should progress?

 

[PeterDonis]

Hawking never made a model that showed that claimed that a black hole could both form and evaporate.

This is not correct. Hawking's model does require the black hole to form. Hawking just didn't emphasize that part.

A black hole that does not form from collapse of an object must have a white hole in its past: that is what the maximal extension of Schwarzschild spacetime tells us. Hawking's model does not have that. Just look at its Penrose diagram and compare it with the Penrose diagram of maximally extended Schwarzschild spacetime.

Has anyone else written such a metric then? Showing that it is possible?

I don't know that anyone has written down an explicit metric, but any Penrose diagram implicitly specifies one in the Penrose coordinates used to construct the diagram.

I think you are dismissing it far too abruptly

I have given a reason for being skeptical of it that has nothing to do with the question of conflict with the literature. That reason is one you yourself gave weight to in post #10. If that reason is valid, it invalidates the paper's entire analysis quite apart from anything else.

 

[PeroK]

Does it not take many orders of magnitude less time to fall into a black hole than for the black hole to evaporate? It's a bit like worrying about getting your shopping home in a biodegradable bag!

 

[Dale]

I don't know that anyone has written down an explicit metric, but any Penrose diagram implicitly specifies one in the Penrose coordinates used to construct the diagram.

Not any Penrose diagram specifies a metric that satisfies the energy conditions considered in this paper. If the metric for the standard Penrose diagram were written down and shown to satisfy the energy conditions then indeed there would be a contradiction between this paper and the literature (and this paper would be shown to be wrong). Otherwise, I think the most you can say is that you suspect there is a contradiction.

Personally, I doubt it. I suspect that this paper is correct in that any black hole metric which both forms and evaporates must violate at least one of the listed energy conditions. I do not think this claim is inconsistent with the literature.

it invalidates the paper's entire analysis quite apart from anything else

Not really. They define what they mean by Schwarzschild coordinates. It is not the standard coordinates, but it is something that I have seen elsewhere in the literature.

 

[PeterDonis]

that satisfies the energy conditions considered in this paper

We already know that anything that allows a black hole to evaporate violates the energy conditions, since black hole evaporation violates the area theorem that says that the area of an event horizon can never decrease, and the area theorem is based on the energy conditions being satisfied. (Similar remarks apply to any solution, such as the Bardeen black hole, that contains trapped surfaces but does not contain any incomplete geodesics--this violates the singularity theorems, which also are based on the energy conditions being satisfied. The Bardeen black hole violates the energy conditions by having dark energy, which violates them, in its deep interior.) So assuming the energy conditions in any treatment of evaporating black holes can't be right.

 

[PeterDonis]

I suspect that this paper is correct in that any black hole metric which both forms and evaporates must violate at least one of the listed energy conditions

Forming a black hole does not require violating any energy conditions; for example, the 1939 Oppenheimer-Snyder model satisfies them.

Black hole evaporation, as I said in post #43 just now, does require violating at least one of the energy conditions, and, as I noted, this is already known and has been since the 1970s. Furthermore, even Hawking's original analysis showed that Hawking radiation has to violate at least one energy condition. And we know that dark energy violates at least one (I think it violates several), and we have evidence for the existence of dark energy, i.e., that violation of energy conditions is physically possible. So if violation of the energy conditions is the basis for the paper's claim that it is impossible to have a black hole that forms and then evaporates, then that is an even better reason to be skeptical of it.

 

[PeterDonis]

I do not think this claim is inconsistent with the literature.

The claim that black hole evaporation must violate at least one of the energy conditions is of course consistent with the literature.

However, the claim that this constitutes a proof of impossibility of black hole evaporation, on the grounds that violation of an energy condition is not physically possible, is not consistent with the literature.

 

[PeterDonis]

They define what they mean by Schwarzschild coordinates. It is not the standard coordinates

I don't see the coordinates explicitly defined anywhere in the paper. The paper gives what it calls "the standard metric form under Schwarzschild coordinates" at the top of p. 5 and references Weinberg 1972, which, AFAIK, uses the standard definition for Schwarzschild coordinates. Unfortunately I don't have a copy handy to check.

 

[Dale]

However, the claim that this constitutes a proof of impossibility of black hole evaporation, on the grounds that violation of an energy condition is not physically possible, is not consistent with the literature.

I am fine with that criticism. IMO, that goes along with the click-bait style of the title, which I don’t like. To me, that is not a substantive criticism, just a style choice I don’t prefer.

I actually have a different criticism now. I had misunderstood their figure 3. It seems to me that this is a white hole horizon, meaning that matter cannot go from the outside to the inside. So that puts it on the same footing as the Vaiyda metric, with my same objection as that one. And thus I still don’t know of anyone that has published an explicit form for a Hawking black hole metric.

Black hole evaporation, as I said in post #43 just now, does require violating at least one of the energy conditions, and, as I noted, this is already known and has been since the 1970s. Furthermore, even Hawking's original analysis showed that Hawking radiation has to violate at least one energy condition.

So I see no substantive disagreement with the literature.

So if violation of the energy conditions is the basis for the paper's claim that it is impossible to have a black hole that forms and then evaporates, then that is an even better reason to be skeptical of it.

I disagree that is a good reason, let alone a better reason. As with all derivations you start with some assumptions and derive some conclusions. When the assumptions are violated the conclusion doesn’t follow. There is nothing unique (nor even extreme) about this paper in that regard. But you are free to have your opinion on this topic. I don’t share it.

 

[PeterDonis]

I had misunderstood their figure 3. It seems to me that this is a white hole horizon

I am not sure, but I think Fig. 3 is supposed to represent a portion of maximally extended Schwarzschild spacetime, i.e., it would include both black hole and white hole horizons. I agree that the white hole portion is not relevant to an analysis of black hole evaporation, for reasons already discussed.

 

[PeterDonis]

To me, that is not a substantive criticism, just a style choice I don’t prefer.

I don't think a claim of impossibility is a "style choice".

 

[PeterDonis]

As with all derivations you start with some assumptions and derive some conclusions. When the assumptions are violated the conclusion doesn’t follow. There is nothing unique (nor even extreme) about this paper in that regard.

In that regard, taken in isolation, no, there isn't.

But if the assumptions are ones that have been known to be violated by evaporating black holes for almost four decades at the time a paper using those assumptions but purporting to be about evaporating black holes is written, then the paper would seem pointless. Certainly it would not seem to me to be a good paper to use in a discussion of evaporating black holes.

 

[PeterDonis]

As with all derivations you start with some assumptions and derive some conclusions. When the assumptions are violated the conclusion doesn’t follow.

To expand on my comment in post #50 just now, let's go back to your original post referencing the paper:

In an eternal black hole, all maximally extended geodesics that cross the event horizon reach the center in finite proper time. In an evaporating black hole there are geodesics that reach the center in finite proper time before it evaporates and these are what form the interior of the horizon.

https://arxiv.org/abs/1102.2609

As far as I can tell, Section II.B of the paper, titled "The forbidden region of light cone", is saying that there are no geodesics at all that reach $r = 0$ during the period after the horizon of an evaporating black hole forms but before it evaporates. This appears to me to be the basis for the paper's claim that it is impossible to have an event horizon that forms and evaporates. In other words, the paper is claiming the opposite of what you say in the bolded portion of the quote above.

The paper appears to be basing this on the assumption that the energy conditions are satisfied. But, as I have said, it has been known since the 1970s that black hole evaporation must violate the energy conditions. So the paper is just rediscovering, in a roundabout way, what has been known since the 1970s. (But it's not clear to me that the author of the paper actually recognizes that.)

 

[Dale]

As far as I can tell, Section II.B of the paper, titled "The forbidden region of light cone", is saying that there are no geodesics at all that reach r=0 during the period after the horizon of an evaporating black hole forms but before it evaporates. This appears to me to be the basis for the paper's claim that it is impossible to have an event horizon that forms and evaporates. In other words, the paper is claiming the opposite of what you say in the bolded portion of the quote above.

Yes, I misunderstood that. The paper doesn’t describe the metric I thought it did. It has the same limitations as the Viadya metric.

if the assumptions are ones that have been known to be violated by evaporating black holes for almost four decades at the time a paper using those assumptions but purporting to be about evaporating black holes is written, then the paper would seem pointless

This is an opinion I don’t share. Lots of proofs of impossibility have some assumptions that are known to be violated and yet are not pointless, IMO. The 2nd law of thermo and Earnshaws theorem come to mind. If you want to single this paper out as uniquely pointless or if you want to broadly paint all such proofs as pointless, that is your choice.