Falling through the event horizon of an evaporating black hole

[brooknorton1]

I have heard that an person falling toward a black hole would 1) start to "freeze" from the viewpoint of an outside observer as the infalling person's (call him Bob) time dilation slowed Bob's time to a crawl, and 2) from Bob's viewpoint it would seem that he simply falls through the horizon unimpeded.

In contrast to assertion 2, it seems that actually falling through the horizon would be impossible for the following reason. The closer that Bob approached the horizon, the slower his time would progress and the more rapid would appear events of the outside universe. As Bob came arbitrarily close to the horizon, events of the outside universe would be appearing arbitrarily far in the future.

Through Hawking radiation, all black holes will eventually evaporate. As Bob fell close enough to the horizon, the needed passage of time (a very long time) would be achieved, the black hole would completely evaporate, and Bob would find himself floating in space with the black hole completely evaporated.

This scenario would unfold regardless of the size of the blackhole or the timescales involved. Therefore, in general, it is impossible for Bob, or anything else to fall into a black hole; or to be more precise, to fall through the event horizon.

For Bob, falling toward a super massive black hole, he would feel nothing unusual as tidal forces near the horizon of a supermassive black hole are small. He would see outside events speeding up. As the the black hole continued to evaporate and shrink (over astronomical time frames), Bob would feel increasing tidal forces. When the black hole finally completely evaporated, the tidal forces would vanish and Bob would find himself floating in free space with the black hole completely evaporated.

Has anyone heard of this line of reasoning for the impossibility of anything actually falling into a black hole? See anything wrong with it?

I suspect that after a black hole initially forms and an event horizon appears, that further things "falling into the black hole" actually just accumulate near the outer surface of the horizon, increasing the black hole system mass by increasing the mass of the shell around the horizon, without ever falling into the center of the black hole.

Thanks,
Brook Norton

 

[berkeman]

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I have heard that an person falling toward a black hole

Can you please link to the references you've been reading? That would help a lot in our responses to your question. Thanks.

 

[phinds]

The closer that Bob approached the horizon, the slower his time would progress

Not according to Bob, who would just fall through the EH as though there were nothing there (but that's only because there IS nothing there).

EDIT: by the way, don't feel bad. Your misunderstanding is so common that it is debunked here several times a year.

Also, you should still answer Berkeman's question since it would help us know WHY you have that misconception if we knew where you got it.

 

[Ibix]

I have heard that an person falling toward a black hole would 1) start to "freeze" from the viewpoint of an outside observer as the infalling person's (call him Bob) time dilation slowed Bob's time to a crawl, and 2) from Bob's viewpoint it would seem that he simply falls through the horizon unimpeded.

Pretty much correct, although "perspective" is unhelpful. Due to increasing gravitational redshift the distant observer never directly sees Bob fall in. Whether they interpret that as "Bob never fell in" or just "I haven't seen Bob fall in, but he has" depends on the coordinate choice. There isn't an unambiguous answer. Unfortunately, a lot of sources don't make that clear and treat "Bob never fell in" as if it were some fixed truth instead of a choice of interpretation of the data (a naive choice of interpretation developed by Schwarzschild in 1918, in fact, rather than more helpful interpretations by several others in the intervening century).

In contrast to assertion 2, it seems that actually falling through the horizon would be impossible for the following reason.

Your reasoning is incorrect because you are treating the distant observer's poor choice of coordinate scheme as physically significant. This is roughly the same mistake as looking at a Mercator map of the Earth and concluding that you can't cross the Pacific because there's an edge in the way. The distant observer just needs to pick a definition of "now" other than Schwarzschild's, just as the route planner needs to pick a map with its longitude zero elsewhere.

What actually happens to an infalling observer in a black hole with Hawking radiation isn't known because we don't have a proper model of quantum gravity. Depending on different assumptions you make the predicted final fate of an evaporating hole is different, but none of them lead to Bob seeing the whole history of the exterior universe so you don't get the contradiction you are thinking of.

TLDR: Confusing Schwarzschild's time coordinate (which goes wrong at the event horizon) with physical reality is a common mistake that people make. It's a "confusing the map with the territory" mistake. Once that confusion is cleared up, there is no problem with crossing the horizon.

 

[brooknorton1]

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Can you please link to the references you've been reading? That would help a lot in our responses to your question. Thanks.

I haven't read an references recently; mostly watching videos lately.

 

[PeterDonis]

Moderator's note: Thread moved to relativity forum.

 

[PeterDonis]

I haven't read an references recently; mostly watching videos lately.

Pop science videos are not a good way to learn actual science. Nor are they a good basis for discussion here.

it seems that actually falling through the horizon would be impossible

No, it isn't. @Ibix's response is a good one, but if you want more details, I suggest this Insights article:

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

 

[Demystifier]

I suspect that after a black hole initially forms and an event horizon appears, that further things "falling into the black hole" actually just accumulate near the outer surface of the horizon, increasing the black hole system mass by increasing the mass of the shell around the horizon, without ever falling into the center of the black hole.

This looks like a black hole firewall scenario, on which you may google for more info.

 

[PeterDonis]

This looks like a black hole firewall scenario

Not really. In one version of the firewall scenario, things still fall through the event horizon, but in doing so they encounter a high-temperature substance of some sort instead of vacuum. In another version, there is no event horizon at all; the "firewall" is an impenetrable barrier. (How to square this with any actual valid solution of the Einstein Field Equation is not explained). Neither of these are what your quote from the OP is describing--a black hole with an event horizon but which has things "pile up" against the horizon instead of falling through it. That is simply physically impossible.

 

[Dale]

One other problem is that the Schwarzschild coordinates apply to an eternal black hole, not one where the black hole evaporates due to Hawking radiation. So making conclusions about an evaporating black hole based on coordinates defined on an eternal black hole is almost guaranteed to be wrong.

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

 

[brooknorton1]

Pretty much correct, although "perspective" is unhelpful. Due to increasing gravitational redshift the distant observer never directly sees Bob fall in. Whether they interpret that as "Bob never fell in" or just "I haven't seen Bob fall in, but he has" depends on the coordinate choice. There isn't an unambiguous answer. Unfortunately, a lot of sources don't make that clear and treat "Bob never fell in" as if it were some fixed truth instead of a choice of interpretation of the data (a naive choice of interpretation developed by Schwarzschild in 1918, in fact, rather than more helpful interpretations by several others in the intervening century).

Your reasoning is incorrect because you are treating the distant observer's poor choice of coordinate scheme as physically significant. This is roughly the same mistake as looking at a Mercator map of the Earth and concluding that you can't cross the Pacific because there's an edge in the way. The distant observer just needs to pick a definition of "now" other than Schwarzschild's, just as the route planner needs to pick a map with its longitude zero elsewhere.

What actually happens to an infalling observer in a black hole with Hawking radiation isn't known because we don't have a proper model of quantum gravity. Depending on different assumptions you make the predicted final fate of an evaporating hole is different, but none of them lead to Bob seeing the whole history of the exterior universe so you don't get the contradiction you are thinking of.

TLDR: Confusing Schwarzschild's time coordinate (which goes wrong at the event horizon) with physical reality is a common mistake that people make. It's a "confusing the map with the territory" mistake. Once that confusion is cleared up, there is no problem with crossing the horizon.

I wish I were more knowledgeable on the physics to more directly reply. But this comes to mind - surely, time dilation in a gravitational field is real. Doesn't that make the outside observer's view a consistent way of seeing things? It sounds like you are saying the outside observer does not see what really happens which is that Bob freely falls through.

Pop science videos are not a good way to learn actual science. Nor are they a good basis for discussion here.


No, it isn't. @Ibix's response is a good one, but if you want more details, I suggest this Insights article:

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

Thanks for the relevant article. I'll need to read it a few more times.

 

[Ibix]

I wish I were more knowledgeable on the physics to more directly reply. But this comes to mind - surely, time dilation in a gravitational field is real. Doesn't that make the outside observer's view a consistent way of seeing things?

The way of seeing things adopted by the outside observer is not unique - there is more than one that they can pick. If they pick one that fails at the event horizon they will not be able to describe a trip that crosses the event horizon in a coherent way.

Gravitational time dilation is real in the sense that of I watch a clock lower in a gravity well than I am it will tick slower. However, I cannot watch a clock that crosses the horizon, so it's really the wrong tool to use to think about this kind of trip.

It sounds like you are saying the outside observer does not see what really happens which is that Bob freely falls through.

The outside observer can't see Bob passing through the event horizon, no. And if he sets up a coordinate system that relies on observing hovering clocks he can't describe the trip either. But that's because those are bad tools for studying a horizon crossing trip.

 

[brooknorton1]

Would it be incorrect to say... Bob starts out far from the black hole at t=0. I'll use "t" to mean as Bob measures it. As Bob approaches the horizon that he will see outside events accelerating. At some point in approaching the horizon he will see events that are a million years later than t=0. Is this description already wrong?

 

[Ibix]

As Bob approaches the horizon that he will see outside events accelerating.

No. Assuming he starts far from the horizon, he will see distant events slower and slower due to distant objects being redshifted from his speed. Falling clocks do not behave the same as hovering ones, I'm afraid. Bob sees very little history after crossing the horizon.

Is this description already wrong?

Yes, I'm afraid so.

 

[brooknorton1]

Interesting about falling vs hovering clocks. Thanks.

 

[PeterDonis]

In an evaporating black hole there is no event horizon for geodesics to cross

If there is no event horizon, there is no black hole; the definition of a black hole is a region of spacetime that cannot send light signals to future null infinity, and the event horizon is the boundary of that region. In the original evaporating black hole model that Hawking proposed, there certainly is an actual black hole region bounded by an event horizon.

In other models, such as the "Bardeen black hole", there is no actual event horizon and any event anywhere in the spacetime can send light signals to future null infinity, so "black hole" is a misnomer for these models. In these models the horizon is only an apparent horizon, and it eventually goes away when the "hole" evaporates. The causal structure of these models is the same as Minkowski spacetime, so there are no issues with "information loss" or quantum unitarity, which makes them attractive alternatives to models like Hawking's original one.

 

[PeterDonis]

https://arxiv.org/abs/1102.2609

I'm not too sure about this paper. I seem to remember it coming up in previous PF threads, but I can't find them right now. The title claim, in particular, appears to me to contradict much literature from Hawking on, since in models like Hawking's original one, a BH does form and does evaporate. Many physicists believe models like Hawking's original one will not turn out to be physically realized in our actual universe, but mathematically AFAIK they are perfectly self-consistent, and they falsify the title claim of the paper.

 

[Dale]

In the original evaporating black hole model that Hawking proposed, there certainly is an actual black hole region bounded by an event horizon.

Do you have a reference for the metric for that? I have asked for that before but never found it. AFAIK in Hawking’s original model he never actually derived a full spacetime metric but only used local perturbation arguments.

 

[PeterDonis]

surely, time dilation in a gravitational field is real

You have to be very careful when making statements like this. Yes, there is a thing that is real which can be described as "time dilation in a gravitational field", but it doesn't mean what you appear to think it means.

Here is what is real: suppose I have two observers who are hovering at different constant altitudes above a non-rotating gravitating mass, which could be a planet or star or black hole. These observers exchange round-trip light signals and time the round trips with their clocks. The observer who is at a lower altitude will find that their clock shows less elapsed time for a round trip than the observer who is at a higher altitude. Informally, we say that the lower observer's clock is "running slower" and the upper observer's clock is "running faster".

But note the key qualification: both observers have to be hovering. They can't be free-falling towards the mass--in fact they can't be moving relative to the mass at all. Otherwise you don't have a "pure" case of gravitational time dilation.

Doesn't that make the outside observer's view a consistent way of seeing things?

In general, the outside observer's view only works outside the horizon, and it gets more and more distorted the closer you get to the horizon.

In terms of "gravitational time dilation", as above, the outside observer's view, as you are stating it, only works for themselves (assuming they are "hovering") and other observers who are "hovering". It doesn't work for observers who are free-falling into the hole.

 

[PeterDonis]

It sounds like you are saying the outside observer does not see what really happens which is that Bob freely falls through.

The outside observer cannot see Bob at or below the horizon, because the light signals Bob emits at or below the horizon can never reach the outside observer. Just as light signals from someone below your horizon on Earth cannot reach you. But the fact that you don't see things that happen below your horizon on Earth doesn't mean those things don't happen; and similarly, the fact that the outside observer can't see Bob at or below the horizon does not mean Bob does not fall there.

 

[PeterDonis]

Schwarzschild coordinates apply to an eternal black hole, not one where the black hole evaporates

Btw, the paper you reference appears to make this error when concluding that nothing can fall into an evaporating black hole.

 

[Ibix]

Do you have a reference for the metric for that? I have asked for that before but never found it. AFAIK in Hawking’s original model he never actually derived a full spacetime metric but only used local perturbation arguments.

There are certainly Penrose diagrams of evaporating black holes with singularities. I don't know their degree of reliability. Figure 1 here, for example: https://arxiv.org/abs/1506.07133.

 

[PeterDonis]

Figure 1 here

That diagram is basically the one that Hawking originally proposed for an evaporating black hole.

Regarding @Dale's question about the metric, if you can draw a Penrose diagram of it, you can in principle write down some metric for it. Whether that metric will be easily understandable is a different question. But AFAIK nobody has disputed that that diagram represents some valid solution to the EFE. The question is whether it represents a solution that is ever physically realized in our actual universe.

 

[PeterDonis]

AFAIK in Hawking’s original model he never actually derived a full spacetime metric but only used local perturbation arguments.

His argument for why Hawking radiation should occur at all was based on an early version of quantum field theory in curved spacetime. It took a couple of decades for that to be developed to the point where Wald could write his 1993 monograph Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (an excellent if advanced read if you can find it--I was lucky enough to find a copy years ago) with the expectation that he would be describing a framework that was generally accepted (at least by Wald's fellow experts in the field), and that the derivation of Hawking radiation within that framework was generally accepted.

As far as relating the actual spacetime metric for such a solution to something well known, I believe there are papers that use the outgoing Vaidya metric for this purpose; I think we had a thread on that some years ago. I'll see if I can dig up a reference.

 

[PeterDonis]

Well, it looks like I asked whether anyone at PF knew of papers using an outgoing Vaidya metric to describe an evaporating black hole ten years ago :

https://www.physicsforums.com/threads/vaidya-metric-to-describe-evaporating-black-hole.731420/

Some other more recent posts/threads on the topic that reference relevant papers:

https://www.physicsforums.com/threa...e-for-a-vaidya-black-hole.992211/post-6379816

https://www.physicsforums.com/threads/black-hole-horizon-and-hawking-evaporation.995084/post-6407119

https://www.physicsforums.com/threads/a-proposed-black-hole-entropy-calculation.995519/

 

[Dale]

As far as relating the actual spacetime metric for such a solution to something well known, I believe there are papers that use the outgoing Vaidya metric for this purpose; I think we had a thread on that some years ago. I'll see if I can dig up a reference.

I don’t think that the Vaidya metric works. My understanding is that the outgoing Vaidya metric represents a white hole. Timelike objects cannot enter the horizon, only leave it.

 

[PeterDonis]

My understanding is that the outgoing Vaidya metric represents a white hole.

The outgoing Vaidya metric has a white hole in the past of its maximal extension. But, as I commented in one of the previous threads I linked to, we don't have to use the entire maximal extension of the outgoing Vaidya metric, any more than we have to use the entire maximal extension of the Schwarzschild metric when we model static gravitating masses.

In a spacetime modeling an evaporating spherically symmetric black hole, as I think I also described in one of those previous threads, we would expect to have the following regions:

A collapsing FRW region describing the matter that originally collapses to form the hole.

An exterior Schwarzschild region around the collapsing matter, as in the 1939 Oppenheimer Snyder model.

An outgoing Vaidya region describing the evaporation phase.

A final Minkowski region describing the flat spacetime left behind inside the last shell of radiation emitted at the hole's final evaporation.

If you pick an event inside the outgoing Vaidya region, and look into its past, instead of having a white hole, as you would if the spacetime were the maximal extension of outgoing Vaidya, you would find the Oppenheimer-Snyder region: collapsing matter surrounded by Schwarzschild vacuum. This region would take the place of the white hole region--which it already does if we compare the 1939 O-S model to maximally extended Schwarzschild spacetime.

The transition in the region outside the collapsing matter from Schwarzschild, to outgoing Vaidya, to final Minkowski, seems simple: if we foliate the spacetime by outgoing null surfaces labeled by $u$, then we have some value of $u$, call it $u_1$, at which the mass $M$ of the spacetime stops being constant and starts decreasing with $u$; and we have another greater value of $u$, call it $u_2$, at which the mass $M$ reaches zero. As long as both of those transitions are smooth, which is just a matter of finding a function $M(u)$ with the appropriate properties, the joining of the regions will work.

 

[Bandersnatch]

1993 monograph Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (an excellent if advanced read if you can find it--I was lucky enough to find a copy years ago)

Internet Archive's library has a copy.

 

[Dale]

The outgoing Vaidya metric has a white hole in the past of its maximal extension. But, as I commented in one of the previous threads I linked to, we don't have to use the entire maximal extension of the outgoing Vaidya metric

Right, but if we want to model Hawking radiation we do have to use the horizon with outgoing null geodesics and a decreasing mass parameter. I thought that part is a white hole.

An outgoing Vaidya region describing the evaporation phase.

I am fine limiting attention to this region. My understanding is that this region is a white hole horizon. Timelike paths exit, not enter.

 

[PeterDonis]

if we want to model Hawking radiation we do have to use the horizon with outgoing null geodesics and a decreasing mass parameter. I thought that part is a white hole.

No, it isn't. It is often said that Hawking radiation is "emitted from the horizon", but that's not actually the case. The only horizon present in Hawking-type models is a black hole horizon, i.e., an outgoing null surface. A white hole horizon is an ingoing null surface.

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. The "FEH" line going up and to the right is the "future event horizon", but there is nothing behind it; it's a boundary of the spacetime. What this maximal extension is describing is an object that emits outgoing null dust eternally, i.e., it starts with infinite mass at $u = - \infty$, emits null dust for an infinite time, and ends up with zero mass at $u = + \infty$.

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.

 

[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.)