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Black holes and evaporation

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13 hours ago, Markus Hanke said:

No, we have to take this locally. It’s Schwarzschild coordinate time, so this is what a far-away stationary clock measures locally in its own frame of reference. It is not what physically happens anywhere else. In GR, time is always a purely local concept.

Sure, but rjbeery is talking about multiple clocks (infalling A, far away B, etc). If the different observers remain able to communicate (A before reaching the event horizon), they must be able to relate their times to each other. Does "time is purely local" mean that GR just doesn't say anything about how the clocks relate, and doesn't depend on them relating? In reality, even using GR to model spacetime, observers still can relate their clocks in ways that GR doesn't care about.

13 hours ago, Markus Hanke said:

Again, this is not possible. Time is a purely local concept in GR; there is, in general, no notion of simultaneity across extended regions of curved spacetime, and you can’t map notions of space and time local to some far-away observer into anything that happens anywhere else. In particular not to test particles in free fall, which aren’t stationary. You can define static hypersurfaces of simultaneity based on the coordinate system you have chosen (in Schwarzschild, these will be nested spheres), but that is not the same thing.

But you can do that, even if not in all cases. If "Pick a method of determining simultaneity" is understood to mean that you're making a choice of what you mean by simultaneity and how you define it, then for example a clock hovering above a black hole, at rest relative to a distant clock, can use "radar time" to define simultaneity of events at the two clocks' locations. In this example, they can agree on simultaneity.

Eg. if the hovering clock is gravitationally time dilated so that its clock is ticking at half the rate of distant clock, the clocks can be set so that every tick of the hovering clock happens "at the same time" (by their choice of simultaneity definition) as every second tick of the distant clock, and both observers can agree, and the choice of simultaneity can remain consistent and useful indefinitely.

In the case described (A is infalling), each observer can have their own notion of simultaneity, but they won't agree with each other. I don't see how this is a problem in this thread, it's not like any claimed physical effects are based on simultaneity??

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Posted (edited)
18 hours ago, Halc said:

It seems that there is no coordinate system that foliates all of spacetime. This seems to be an interesting argument against any philosophy of time that posits an absolute coordinate system (a preferred frame of one sort or another).  Presentism is only a subset of these philosophies.

The inability to identify any coordinate system that can consistently map any pair of events as to which occurs first seems to me to be a fatal flaw in such a philosophy.

Well, I am not an expert in philosophy, so I can’t really comment on that.
So far as GR is concerned, depending on what kind of spacetime you are dealing with, it is often possible to foliate the entirety of the manifold. However, there will always be infinitely many possible foliation schemes, so there is never any preferred notion of time. This is in keeping with the principle of relativity, of course.

15 hours ago, md65536 said:

Sure, but rjbeery is talking about multiple clocks (infalling A, far away B, etc). If the different observers remain able to communicate (A before reaching the event horizon), they must be able to relate their times to each other. Does "time is purely local" mean that GR just doesn't say anything about how the clocks relate, and doesn't depend on them relating? In reality, even using GR to model spacetime, observers still can relate their clocks in ways that GR doesn't care about.

The relationship between these clocks is simply the coordinate transformation that relates the metrics. For example, the stationary far-away observer can use the Schwarzschild metric, whereas the observer in free fall uses the Gullstrand-Painleve metric. These are simply related by a coordinate transformation, since both frames are of course in the same physical spacetime. But they use different notions of what ‘time’ means, so defining a notion of simultaneity is not in general possible if the observers are separated in time and/or space, unless there are certain very specific symmetries present. At best, it might be possible to foliate spacetime in a manner that both observers can agree upon, using a suitable coordinate system and foliation parameter; but this works only in certain highly symmetric cases, and the foliation parameter is not something that any physical clock would actually show in either of the two frames, so I don’t see how it is helpful here.

15 hours ago, md65536 said:

But you can do that, even if not in all cases. If "Pick a method of determining simultaneity" is understood to mean that you're making a choice of what you mean by simultaneity and how you define it

You may be able to do this in certain special cases that are highly symmetric, which is why I put the qualifier “in general” as part of my original comment. Flat Minkowski spacetime is a trivial example. I don’t think it is possible in Vaidya spacetime though, which is what we would be talking about when it comes to evaporating black holes. Crucially, I don’t think it is helpful to even consider the concept of simultaneity in curved spacetimes, since it is not a generally applicable concept - in my experience, it is bound to lead to more confusion than clarity.

15 hours ago, md65536 said:

In the case described (A is infalling), each observer can have their own notion of simultaneity, but they won't agree with each other. I don't see how this is a problem in this thread, it's not like any claimed physical effects are based on simultaneity??

I agree, it has little to do with topic of the thread, so I’m not sure why it was brought up at all.

Edited by Markus Hanke

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6 hours ago, Markus Hanke said:

At best, it might be possible to foliate spacetime in a manner that both observers can agree upon, using a suitable coordinate system and foliation parameter; but this works only in certain highly symmetric cases, and the foliation parameter is not something that any physical clock would actually show in either of the two frames, so I don’t see how it is helpful here.

rjbeery's example didn't require the two observers (infalling A and distant B) to agree on simultaneity. Though, B and C (observer at location of black hole in B's coordinates, after it has evaporated) agree. It was brought up to illustrate the idea of A existing "forever" in B's coordinates, never passing the Schwarzschild BH event horizon.

I agree the topic really has nothing to do with simultaneity, and that's why I'm commenting on it. You've said Schwarzschild BHs don't evaporate, and it's not possible to determine simultaneity across extended regions of spacetime. If someone's not following the details of the thread, they might think those are equally problematic and that "GR says the example's not possible." But, not being able to unambiguously define simultaneity resolves nothing of rjbeery's paradox, while Schwarzschild BHs not evaporating completely destroys it. If anyone else is struggling to see and understand the resolution of the "paradox", simultaneity's not a problem, but evaporation is. (Because, rjbeery has both the event horizon disappearing, and existing forever for the infalling object to be caught above it, both in the distant observer's coordinates.)

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16 hours ago, Markus Hanke said:

So far as GR is concerned, depending on what kind of spacetime you are dealing with, it is often possible to foliate the entirety of the manifold. However, there will always be infinitely many possible foliation schemes, so there is never any preferred notion of time. This is in keeping with the principle of relativity, of course.

I'm talking about interpretations that deny the principle of relativity. There seem not to be infinitely many possible foliation schemes. As a matter of fact there doesn't seem to be any that foliate all of spacetime. The typical one suggested is the curved (non inertial) comoving frame corresponding locally to the inertial frame in which the CMB appears isotropic, but any frame like that does not properly foliate local deviations from flat space like black holes.  If they did, then rjbeery would have grounds to stand on when trying to objectively determine if event X inside a black hole occurs before or after event Y somewhere outside it, particularly after the BH has evaporated.

An objective foliation scheme should not in any way depend on an observer. Any two observers, no matter how separated and unable to communicate, should be able to sync their clocks simply by setting said clock to the current objective time, and then I suppose having the clock running at some rate which depends on the speed of the clock and its current gravitational potential.  The latter requires a standard 'zero', which also seems undefined. For example, what is the gravitational potential at the surface of Earth? Nobody publishes that. They only publish the potential if Earth was in an otherwise empty universe, which it obviously isn't.

Anyway, point is, there is no viable objective foliation scheme that includes all spacetime events.  The lack of a viable scheme means that time and motion cannot be objective. The principle of relativity cannot be denied.

Correct me if I'm wrong.

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17 hours ago, Markus Hanke said:

So far as GR is concerned, depending on what kind of spacetime you are dealing with, it is often possible to foliate the entirety of the manifold.

Can you please explain your thoughts on this matter.
I tend to agree with Halc, that a foliation ( Cauchy surface  ) is a surface in space-time which is like an 'instant in time'.
As such, a 'global'  ( entire manifold ) foliation is non-sensical, as it implies a universal now.

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4 hours ago, MigL said:

Can you please explain your thoughts on this matter.
I tend to agree with Halc, that a foliation ( Cauchy surface  ) is a surface in space-time which is like an 'instant in time'.
As such, a 'global'  ( entire manifold ) foliation is non-sensical, as it implies a universal now.

A foliation is a slicing-up of all of 4D spacetime into such 3D hypersurfaces. Since you can slice it up in infinitely different ways, that implies there's no 'universal now'; such a thing would be arbitrarily chosen. It's a mathematical thing. A surface of a typical foliation corresponds to an instant in time on a local scale, but just because it's mathematically possible to slice up spacetime doesn't mean that an entire surface meaningfully represents a moment in time. As long as spacetime obeys some reasonable rules, it's possible to foliate it... eg. the surfaces can't intersect. But if spacetime *needed* intersecting surfaces, I think that would imply some really weird physical consequences?

I don't know the other mathematical rules, just adding 2 cents. My understanding is that if you have causally disconnected regions of spacetime, you can foliate it however you want because you'll never get things out of order. Like, if you took two different books and pushed them together so their pages interleaved randomly, and then glued them together, you're not going to have any pages out of order no matter how you put them together. But by analogy, the relative order of pages in different books is generally meaningless, as with foliations of all of spacetime.

I'm not sure, but a foliation might require that a spacetime is connected. In the case of a black hole, would that require that spacetime is multiply connected? Which is not prohibited by GR. Or can you just take partial foliations using the world lines of multiple observers (like a distant observer and an infalling one) and combine them into one like gluing books?

However, this isn't an issue in this thread. I think OP's example can be completely described using a distant observer's coordinate time, and only events outside of the black hole's event horizon. I may have earlier misunderstood that the example was relating interior and exterior events.

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Posted (edited)
20 hours ago, md65536 said:

I agree the topic really has nothing to do with simultaneity, and that's why I'm commenting on it. You've said Schwarzschild BHs don't evaporate, and it's not possible to determine simultaneity across extended regions of spacetime. If someone's not following the details of the thread, they might think those are equally problematic and that "GR says the example's not possible." But, not being able to unambiguously define simultaneity resolves nothing of rjbeery's paradox, while Schwarzschild BHs not evaporating completely destroys it. If anyone else is struggling to see and understand the resolution of the "paradox", simultaneity's not a problem, but evaporation is. (Because, rjbeery has both the event horizon disappearing, and existing forever for the infalling object to be caught above it, both in the distant observer's coordinates.)

Yes, well put.

20 hours ago, md65536 said:

It was brought up to illustrate the idea of A existing "forever" in B's coordinates, never passing the Schwarzschild BH event horizon.

Ok, I see.
In Vaidya spacetime this issue never arises, since (unlike with Schwarzschild) this coordinate time for a far-away observer remains finite.

11 hours ago, Halc said:

Correct me if I'm wrong.

No, you are correct. What I meant is that it is sometimes possible to foliate all of spacetime given a particular coordinate choice, i.e. from the point of view of a particular observer. There are infinity many possible observers, and each one of them will use a different foliation scheme; hence the foliation is never objective and shared by everyone, it is always observer-dependent, even if it spans the entire spacetime. There is no such thing as universal time, of course.
BTW, slicing up 4D spacetime into an ordered sequence of 3D hypersurfaces is called the ADM formalism of GR.

Edited by Markus Hanke

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2 hours ago, Markus Hanke said:

No, you are correct. What I meant is that it is sometimes possible to foliate all of spacetime given a particular coordinate choice, i.e. from the point of view of a particular observer. There are infinity many possible observers, and each one of them will use a different foliation scheme; hence the foliation is never objective and shared by everyone, it is always observer-dependent, even if it spans the entire spacetime. There is no such thing as universal time, of course.

Just to add to that, it's not like in SR where each observer also has a different notion of simultaneity, but each of those is physically meaningful. Eg. in flat spacetime, any two events that can be considered simultaneous by someone will have intersecting future light cones, where different future observers can agree or disagree on whether the events were simultaneous. In GR you must make a choice of how to define the surfaces of a foliation, that's not just based on a physically meaningful connection between its events. You'd choose it to make a useful tool, not a 'real' representation of simultaneity throughout the universe for a given observer.

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2 hours ago, Markus Hanke said:

What I meant is that it is sometimes possible to foliate all of spacetime given a particular coordinate choice, i.e. from the point of view of a particular observer.

...

There is no such thing as universal time, of course.

I am talking about universal time, and while I agree that there's no such thing, there seems to be no contradiction arising by postulating it. I'm proposing such a contradiction here.

If the choice is relative to a particular observer, it's hardly objective.  Another clock cannot be set to the universal time without agreeing on this privileged observer or privileged location in space.  Points in space 50 billion light years away do not exist at all relative to a given observer, so his personal choice of coordinates do no in fact foliate all of spacetime. There is a choice that does foliate all points in an arbitrarily large scale, but as pointed out in this topic, it doesn't work for excessive local curvature such as black holes. There seems in fact to be no possible coordinate system that does, and that's the contradiction that arises from postulating universal (objective/absolute) time/space.

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11 hours ago, md65536 said:

Just to add to that, it's not like in SR where each observer also has a different notion of simultaneity, but each of those is physically meaningful. Eg. in flat spacetime, any two events that can be considered simultaneous by someone will have intersecting future light cones, where different future observers can agree or disagree on whether the events were simultaneous. In GR you must make a choice of how to define the surfaces of a foliation, that's not just based on a physically meaningful connection between its events. You'd choose it to make a useful tool, not a 'real' representation of simultaneity throughout the universe for a given observer.

Very well put +1

12 hours ago, Halc said:

There seems in fact to be no possible coordinate system that does, and that's the contradiction that arises from postulating universal (objective/absolute) time/space.

True, if you have large local variations in curvature (as would be the case in the actual universe), such a global foliation will not in general be possible.

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The paper I referred to has addressed my issues and acknowledged the contradictions I believed to exist in the conventional model. It was a relief to find, actually, being published in a prominent journal, Nuclear Physics B, in 2016.

Quote

More recently, through an analysis of the dynamics of the interior of the collapsing star including the back-reaction of Hawking radiation, the same conclusion of no horizon was achieved [10].

Of course, even in the absence of both event and apparent horizon, the appearance of a collapsing star to a distant observer can approximate a real black hole to arbitrary accuracy as the surface of the collapsing sphere is arbitrarily close to the Schwarzschild radius. These objects are sometimes called incipient black holes, black stars, or just black holes. The observational evidence of black holes does not immediately invalidate theories without horizons.

The seminal work of Kawai, Matsuo and Yokokura [1] gives the cleanest and clearest evidence that the conventional Penrose diagram Fig. 1 should be discarded. They found that neither apparent horizon (a trapped region) nor event horizon form through the collapse of matter. This was shown elegantly through the consideration of a spherical distribution of matter collapsing at the speed of light. The crucial point is that the formation and evaporation of a black hole are not two separate processes distinct from each other. Hawking radiation should be taken into consideration before the horizon appears, as long as the surface of the collapsing matter is very close to the Schwarzschild radius.1

My frustrations were doubled up in this thread because some posters' responses implied that 1) the issues I raised were trivially explained, 2) the physics community was well-aware of them, 3) the responders personally understood the explanations but, 4) the math was too complex to explain it to laymen. My BS-meter wouldn't stop blinking.

Anyway, thanks for the back-and-forth on this.

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I would love to have posted for four lol.

Let's try a starter

Schwartzchild metric

Vacuum solution $T_{ab}=0$ which corresponds to an unaccelerated freefall frame $G_{ab}=dx^adx^b$ if

$ds^2> 0$ =spacelike propertime= $\sqrt{ds^2}$

$ds^2<0$ timelike =$\sqrt{-ds^2}$

$ds^2=0$ null=lightcone

spherical polar coordinates $(x^0,x^1,x^2,x^3)=(\tau,r,\theta,\phi)$

$G_{a,b} =\begin{pmatrix}-1+\frac{2M}{r}& 0 & 0& 0 \\ 0 &1+\frac{2M}{r}^{-1}& 0 & 0 \\0 & 0& r^2 & 0 \\0 & 0 &0& r^2sin^2\theta\end{pmatrix}$

line element

$ds^2=-(1-\frac{2M}{r}dt)^2+(1-\frac{2M}{r})^{-1}+dr^2+r^2(d \phi^2 sin^2\phi d\theta^2)$

Stress tensor Dust solution no force acting upon particle (not converted to polar coordinates)

$T^{\mu\nu}=\rho_0\mu^\mu\nu^\mu$

$T^{\mu\nu}x=\rho_0(x)\mu^\mu(x)\mu^\nu(x)$

Rho is proper matter density

Four velocity

$\mu^\mu=\frac{1}{c}\frac{dx^\mu}{d\tau}$

Leads to

$ds^2=-c^2d\tau^2=-c^2dt^2+dx^2+dy^2+dz^2=-c^2dt^2(1-\frac{v^2}{c^2})^\frac{1}{2}=\frac{1}{\gamma}$

$T^{00}=\rho_0(\frac{dt}{d\tau})^2=\gamma^2\rho_0=\rho$ $\rho$ is mass density in moving frame.

$T^{0i}=\rho_0\mu^o\mu^i=\rho^o\frac{1}{c^2}\frac{dx^o}{d\tau}\frac{dx^2}{d\tau}=\gamma^2\rho_0\frac{\nu^i}{c}=\rho\frac{\nu^i}{c}$

$\nu^i=\frac{dx^i}{dt}$

$T^{ik}=\rho_0\frac{1}{c^2}\frac{dx^i}{d\tau}\frac{dx^k}{d\tau}=\gamma^2\rho\frac{\nu^i\nu^k}{c^2}=\rho\frac{\nu^i\nu^k}{c^2}$

Thus

$T^{\mu\nu}=\begin{pmatrix}1 & \frac{\nu_x}{c}&\frac{\nu_y}{c} &\frac{\nu_z}{c} \\\frac{\nu_x}{c}& \frac{\nu_x^2}{c} & \frac{\nu_x\nu_y}{c^2}& \frac{\nu_x\nu_z}{c^2}\\ \frac{\nu_y}{c}& \frac{\nu_y\nu_z}{c^2} & \frac{\nu_y^2}{c^2}& \frac{\nu_y\nu_z}{c^2}\\ \frac{\nu_z}{c} &\frac{\nu_z\nu_x}{c^2}&\frac{\nu_z\nu_y}{c^2}&\frac{\nu_z}{c^2}\end{pmatrix}$

Now how much of the above did you understand ? I still haven't included the apparent horizon which is not necessarily the same as the event horizon nor did I include Hawking radiation at this time.

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21 minutes ago, Mordred said:

Now how much of the above did you understand ?

I understand the theory of studying "how {variable} changes with respect to {dimension 1} in the {dimension 2} direction". I can read Einstein tensor notation. I know the Schwarzschild metric pretty well. I understand Kruskal coordinates.

This is neither here nor there, though, because I tend to analyze Physics on a more philosophical level. When the math "says something" I try very hard to understand what it's saying physically, and when I don't I try to find someone who does and ask them to explain it.

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Posted (edited)

I never look at the math philosophically. I look at what the math predicts will occur.

However I also treat Hawking radiation in the QFT regime. Apparent and event horizons do cause some differences.

That paper has a key caveat in that it is only valid for an infalling observer. A BH will not evaporate in a finite time for an observer at infinity.

Here is the Arxiv.

The other important detail is that Hawking radiation only occurs if the Blackbody temperature of the horizon exceeds the blackbody temperature of the surrounding universe.

So how would you get a back reaction ? ( Temperature varies according to the observer)

Edited by Mordred

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2 hours ago, Mordred said:

That paper has a key caveat in that it is only valid for an infalling observer. A BH will not evaporate in a finite time for an observer at infinity.

I also work under the presumption that there is a single reality. I have a real problem (again, philosophical, but I believe that it has merit) with Kruskal coordinates and their supposed resolution to the so-called mathematical singularity problem at the event horizon.

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Posted (edited)

The reality is that you will have observer effects with time, energy etc.

There is no philosophy behind that but well tested applications of GR. The paper you linked discusses some of those observer effects...

Edited by Mordred

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15 hours ago, rjbeery said:

The paper I referred to has addressed my issues and acknowledged the contradictions I believed to exist in the conventional model. It was a relief to find, actually, being published in a prominent journal, Nuclear Physics B, in 2016.

Your initial claim in this thread was that evaporating black holes cannot exist (see very first sentence of OP). This paper does not support such a claim - in fact it is actually about the information loss paradox. Furthermore, it makes it explicitly clear that evaporating black holes are not Schwarzschild, which is what we have been attempting to explain all along; it does not attempt to dispute that they exist, as you seem to do.

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11 hours ago, rjbeery said:

I have a real problem (again, philosophical, but I believe that it has merit) with Kruskal coordinates and their supposed resolution to the so-called mathematical singularity problem at the event horizon.

You don't need to reference any particular coordinate system for this, you just look at the curvature invariants of the Riemann tensor. They are all finite at the horizon, unlike is the case for the curvature singularity at r=0. You can also look directly at whether the region is geodesically complete or not, which, again, is independent of any particular coordinate choice.

11 hours ago, rjbeery said:

I also work under the presumption that there is a single reality.

GR is purely classical, so it does not say anything different.

14 hours ago, Mordred said:

That paper has a key caveat in that it is only valid for an infalling observer.

I'm at a loss on a different point - the paper talks about a geometry that resembles a 'decaying white hole', but to my understanding there is no white hole counterpart in Vaidya spacetime, unlike in the Schwarzschild case. Or am I getting this wrong?

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Posted (edited)

To be honest I'm not sure myself how they are involving a white hole myself. There is quite a few assumptions expressed in the paper. As stated by myself earlier I wouldn't place too much faith in its accuracy. There is also details with regards to how the Penrose diagrams apply in different regions missing in that paper.

Edited by Mordred

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1 hour ago, Mordred said:

To be honest I'm not sure myself how they are involving a white hole myself. There is quite a few assumptions expressed in the paper. As stated by myself earlier I wouldn't place too much faith in its accuracy. There is also details with regards to how the Penrose diagrams apply in different regions missing in that paper.

Yes. I have been thinking about this some more, and there is something else that has been omitted - the fact that, if a horizon forms during the collapse process, it will initially do so in the interior of the collapsing mass. This divides the overall spacetime into three distinct regions:

1. The interior of the collapsing mass below the horizon - containing only mass-energy
2. The interior of the collapsing mass between horizon and surface of the mass - this contains both the mass distribution and Hawking radiation
3. The exterior spacetime - containing only Hawking radiation

Each of these regions has a distinct energy-momentum tensor, and thus its own metric as a solution to the field equations in that region. The overall solution for the entire spacetime is a metric that is has to ensure that spacetime remains smooth and differentiable at the boundaries between these regions, which introduces additional constraints on the overall geometry. None of this has been accounted for by the aforementioned paper.

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I agree with those details being missing. I was thinking along the same lines.

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9 hours ago, Markus Hanke said:

Your initial claim in this thread was that evaporating black holes cannot exist (see very first sentence of OP). This paper does not support such a claim - in fact it is actually about the information loss paradox. Furthermore, it makes it explicitly clear that evaporating black holes are not Schwarzschild, which is what we have been attempting to explain all along; it does not attempt to dispute that they exist, as you seem to do.

The information paradox is resolved in this (and other) papers by claiming that the event horizon does not, ever, form, which is exactly what I've been saying. My objection to the original diagram used in the Hawking paper remains valid, and the idea that the Vaidya analysis solves this is false -- Vaidya black holes do not allow the infinite observer to see the event horizon in finite time, and that's the crux of the problem.

If I were going to criticize anything about this and related papers, it's the fact that they are using Hawking "back reaction" to discuss the idea that the event horizon cannot form, but the rate of Hawking radiation is a function of the radius of the event horizon -- in other words, the black hole is a prerequisite for any back reaction of Hawking radiation to exist. This is why I'm careful to talk about "evaporating processes" without referring to Hawking radiation explicitly.

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Posted (edited)

You can choose to believe what you wish to believe. Both Marcus and I agree the paper isn't a good examination and glosses over essential details.

The paper is also very clear on the observer limitation.

Quite frankly I have studied far better examinations on BH event and apparent horizons. In particular numerous dissertations on the topic.

Edited by Mordred

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If you don't feel event horizons exist you might want to review this development.

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38 minutes ago, Mordred said:

If you don't feel event horizons exist you might want to review this development.

Accretion disks are still expected from a very compact, high mass area.

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