Why wouldn't black hole singularity evaporate before it can form?

[lukesfn]

Let me add another voice to the discussion. The free falling observers never see any Hawking radiation. Not before they cross the Horizon, not while they cross the Horizon, not after they cross the horizon. That's a simple consequence of the equivalence principle valid for any B-hole no matter its mass.

Thanks, but this appears to be a common miss-perception but as far as I can tell is only true in one specific situation rather then being the general case. (After all, why would Hawking radiation disappear if you started free falling) This has already been discussed in this thread in detain referencing several papers with the math, but see below for the most relevant bit.

(2) Hawking radiation as seen by an in-falling observer

According to [2], the only a free falling observer with zero instantaneous radial velocity at the horizon will see no hawking radiation.

Do you have any references where you've seen this discussed? I haven't seen any calculations of the behavior of quantum fields close to the singularity, so I'm not sure what the basis is for the temperature behavior you're referring to.

I'm referring to the discussion we had earlier in this thread some time ago now. See below where I extrapolated a formula for Hawking radiation inside the BH. I guess the accuracy of the formula would come under question at some point near the singularity, but it is still a curious question.

I will return to the formula I gave earlier. Under certain assumptions, this gives observed hawking radiation for a free faller where m is mass of the BH, r is the observers radius, and /r₀ is the radius the object was dropped from. (Before being dropped, the object was hovering at a constant radius)
1/(4m) (1 + 2m/r) (1 - 2m/r)^1/2
×{(1 - 2m/r₀)^1/2 + (2m/r×(1-r/r₀))^1/2 }^1/2
×{(1 - 2m/r₀)^1/2 - (2m/r×(1-r/r₀))^1/2 }^-1/2

Now if we simplify things by taking the limit /r₀→∞
1/(4m) (1 + 2m/r) (1 - 2m/r)^1/2
×{1 + (2m/r)^1/2 }^1/2
×{1 - (2m/r)^1/2 }^-1/2

In the approximation in the paper it came from, this formula now gives a temperature of hawking radiation for an observer free falling from ∞. This formula can be continued all the way to the singularity. (Note that I don't think that the paper ever tried to extend this formula inside the horizon, I am doing that on my on initiative) Anyway, towards the singularity, the observed temperature tends towards ∞. This implies that at some finite radius, the observer will see a radiation with energy greater then the mass of the BH. Therefore, I've shown that the speculation in my original post has some mathematical basis. I still find it quite an interesting question.

This isn't really a meaningful question; to an infaller that's falling through the horizon, the BH isn't an "object" that the concept of "relativistic mass" or energy can be applied to; that sort of concept can only be meaningfully applied to the BH by an observer very far away, to whom the BH just looks like an ordinary gravitating object. There is an $M$ parameter in the metric, but that parameter doesn't depend on the observer's state of motion.[/QUOTE]
I suppose that would make sense why my head hurts when I try to think about it.

I assume M is the rest mass, which obviously should not depend on motion.

So you are saying the concept of "relativistic mass" breaks down strongly curved space? I must say that I am struggling with the concept of velocity differences in strongly curved space.

Even so, I am not convinced question is completely without meaning. Perhaps it is time for some more reading...

 

[PeterDonis]

I'm referring to the discussion we had earlier in this thread some time ago now. See below where I extrapolated a formula for Hawking radiation inside the BH.

Hm, I must have missed that, I'll have to take another look.

I assume M is the rest mass

To a faraway observer, yes, $M$ will look like the rest mass of an isolated object--i.e., it will be the invariant length of the object's 4-momentum vector in the asymptotically flat coordinates of someone moving inertially very far away from the hole.

So you are saying the concept of "relativistic mass" breaks down strongly curved space?

I don't really find the term "relativistic mass" useful anyway, since we have another perfectly good word for the same thing: energy. But either way, the concept as you're using it only applies to an isolated object, when you're far enough away from it (compared to its size) for it to behave like an isolated object relative to you. That can still apply in a curved spacetime, but it obviously doesn't if the object is a black hole and you are inside its horizon, since you are no longer far enough away from it for it to behave like an isolated object relative to you.

I must say that I am struggling with the concept of velocity differences in strongly curved space.

That's understandable, since there is no longer a unique definition of "relative velocity" for spatially separated objects in curved spacetime, because there's no unique way to compare 4-vectors at different events.

 

[lukesfn]

That's understandable, since there is no longer a unique definition of "relative velocity" for spatially separated objects in curved spacetime, because there's no unique way to compare 4-vectors at different events.

It is starting to become clearer what kind of reasons might lead to energy conservation in GR not well defined in the global case.

Perhaps there is a way to define the energy BH from from the point of view of a BH that would at least work in this specific case.

Ether way, if the energy of the BH is not well defined for a moving observer inside the horizon, then, it certainly gives a loop hole where an in-falling observer seeing radiation with more energy then the mass of the BH may not be inconsistent with the singularity being there when the observer reaches it.

 

[lukesfn]

I was trying to think about this from other angles.

There will always be a point where an in falling observer will observe it's self passing radiation greater then it's own mass, so how can it ever add to the mass of the singularity? Unfortunately, this line of thought suffers from the same issue of the observers confusion over the energy of the singularity. Or maybe any mass of the observer is destroyed and converted to radiation, which makes things less clear.

Considering the Bekenstein bound and BH entropy, the math showing that the BH should look like it evaporates before the in-falling observer reaches the singularity might make some kind of sense. If the sphere of a BH Event Horizon is right at the Bekenstein Bound, and all of the mass of the energy, and entropy of a BH is at the singularity, the Bekenstein Bound would be violated for any sphere inside the Event Horizon. But if an in-falling observer itself passing all the engery of the BH by the time it reaches the center, then the Bound might not be violated.

Could be as if space being ripped apart causes a pressure that would prevent a singularity?

Still... I can't say that this really makes a lot of sense, since any observed hawking like radiation inside the Event Horizon must be in-falling, so it isn't really intuitive that a singularity might be prevented by in-falling radiation.

 

[dauto]

Thanks, but this appears to be a common miss-perception but as far as I can tell is only true in one specific situation rather then being the general case. (After all, why would Hawking radiation disappear if you started free falling) This has already been discussed in this thread in detain referencing several papers with the math, but see below for the most relevant bit.

How is that not violating the equivalence principle?

 

[stevendaryl]

Considering the Bekenstein bound and BH entropy, the math showing that the BH should look like it evaporates before the in-falling observer reaches the singularity might make some kind of sense.

From the perspective of a distant observer watching someone fall into a black hole, it looks like this, described using Schwarzschild coordinates:

The infalling observer starts slowing down as he gets closer to the horizon. Meanwhile, the horizon gradually shrinks due to Hawking radiation. At some finite time, the radius of the horizon shrinks to zero, and both the infalling observer and the black hole vanish in a burst of radiation. So the three events--(1) the infalling observer crosses the event horizon, (2) the infalling observer hits the singularity, and (3) the black hole disappears in a burst of radiation--all happen at the same time coordinate t. But that's only because the Schwarzschild time coordinate is notoriously unreliable near the event horizon. From the point of view of the infalling observer, (1) happens before (2), which happens before (3).

I have seen an explanation for what's going on in an evaporating black hole that uses Penrose diagrams, but I can't seem to find it on the internet.

 

[lukesfn]

How is that not violating the equivalence principle?

The equivalence principal is only meant to hold locally, however the origin of the observed hawking radiation is not local so there is no violation.

Doesn't a stationary observer at infinity see hawking radiation? Couldn't this observer also be considered in free fall? But this isn't a problem for the equivalence principal is it?

The equivalence principal states approximately that "The outcome of any local experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime."

Here a local experiment does not include measuring hawking radiation that comes from outside the laboratory. If the horizon was in the middle of the lab, and you observed hawking radiation appearing at the horizon, it would violate the principal, however, this is not what was being described.

Inside the EH, the radiation would probably appear to come from the location of the observed visual horizon which would be further inside the BH then the observer, so not local, and although it would also be in-falling, the observer would catch up to it.

Also, the equivalence principal is only an approximation, and once you account for the fact space is curved, then other effects are possible, it is just that if the BH is big enough, you won't be able to notice at the EH, which makes sense because there would not be much Hawking radiation to notice in that case.

As far as I have been able to understand, the only observer to not see hawking radiation coming from a BH, is one at the horizon, in free fall, but with a radial velocity of zero, which means they could be moving at the speed of light, which I am not sure is really a valid observer anyway.

 

[lukesfn]

But that's only because the Schwarzschild time coordinate is notoriously unreliable near the event horizon. From the point of view of the infalling observer, (1) happens before (2), which happens before (3)./QUOTE]

I did previously starting a thread asking more about why the out side observer seems to disagree with the in-falling observer, however after that, it makes perfect sense to me that the in-falling observer has a path beyond the EV even if this is not seen by the distant observer.

However, this thread isn't concerned with the distant observer. Following the maths from earlier in the thread, it appears that the temperature of hawking radiation as observed by an in-faller, should increase inside the EH all the way to the singularity, and at some point the observer might see radiation with higher energy then the BH, which doesn't seem obviously consistent with a singularity being there.

If you have any explanation, I would love to see it.

 

[Dale]

There will always be a point where an in falling observer will observe it's self passing radiation greater then it's own mass so how can it ever add to the mass of the singularity

I don't think this is correct in all cases, however even where is correct it does not support the inference you are trying to make.

Consider a massive target and a relativistic projectile in flat spacetime in the target's frame. Suppose the target emits a pulse of light. For any mass of projectile and any energy of light pulse there always exists some velocity of projectile at which the light pulse is sufficiently blueshifted in the projectile's frame to have more energy than the projectile has mass, but in all frames the target will gain mass from the collision.

 

[lukesfn]

Consider a massive target and a relativistic projectile in flat spacetime in the target's frame. Suppose the target emits a pulse of light. For any mass of projectile and any energy of light pulse there always exists some velocity of projectile at which the light pulse is sufficiently blueshifted in the projectile's frame to have more energy than the projectile has mass, but in all frames the target will gain mass from the collision.

That kind of reasoning is why I included the following.

Unfortunately, this line of thought suffers from the same issue of the observers confusion over the energy of the singularity.

In your example, the energy of the target, in the frame of the projectile could be easily calculated due to relative velocity differences in the flat space time, however, it doesn't seem to be so easy to do the same calculation for an observer free falling into a BH because it is a strongly curved space time.

Perhaps the observed energy of a BH should increase due to acceleration during free fall, and the increasing observed energy of hawking radiation is relative to this, but, is there anyway to do the maths to check this?

 

[Dale]

it doesn't seem to be so easy to do the same calculation for an observer free falling into a BH because it is a strongly curved space time.

Which makes the attempted inference even less justified.

 

[lukesfn]

Which makes the attempted inference even less justified.

Still, for me, explaining things away by simply saying, that the energy of the BH is not well defined for the in-falling observer isn't very satisfying. I often don't mind a bit of hand waving in an explanation, but in this case, for me, int this case, I'm left wanting a deeper understanding and a more convincing proof.

At least that want might motivate me to learn a bit more about GR.

I guess it must be true, that for any in-falling observer nearing the singularity who witnesses hawking radiation of a greater energy then the rest energy of the BH, it must be possible to define another observer, sharing the same location, but with a different velocity, who sees radiation energies that are much lower then the rest energy of the BH. Given the force needed to arrive at such a location with such a velocity, it may not be practical to arrive at such a frame in some cases, however, perhaps that is not relevant.

 

[PAllen]

Still, for me, explaining things away by simply saying, that the energy of the BH is not well defined for the in-falling observer isn't very satisfying. I often don't mind a bit of hand waving in an explanation, but in this case, for me, int this case, I'm left wanting a deeper understanding and a more convincing proof.

At least that want might motivate me to learn a bit more about GR.

I guess it must be true, that for any in-falling observer nearing the singularity who witnesses hawking radiation of a greater energy then the rest energy of the BH, it must be possible to define another observer, sharing the same location, but with a different velocity, who sees radiation energies that are much lower then the rest energy of the BH. Given the force needed to arrive at such a location with such a velocity, it may not be practical to arrive at such a frame in some cases, however, perhaps that is not relevant.

Hawking radiation (so far as I know) doesn't come from the singularity. It originates at the horizon for observers remaining outside the BH. I have never seen anyone argue that it exists inside the horizon for any observer. Do have any reference for this, or and derivation for such a claim?

 

[PAllen]

Also, the question of a race between BH formation and BH evaporation is complicated, unsettled question well outside classical GR (for which there is no such thing as Hawking radiation or BH evaporation). With quantum effects considered, there is the fundamental problem that no complete theory exists to analyze this. Quantum effects change both horizon formation and radiation (the majority of experts analyze that horizons form 'faster' than classical GR, beating evaporation). But pending a real resolution of quantum gravity there is no final answer.

Here is one major paper in the debate (arguing for collapse beats evaporation). It references papers on the other side of the debate (whose authors, to the best of my knowledge, have not conceded). There may be a majority view that collapse wins, but this is not currently a fully settled question:

http://arxiv.org/abs/0906.1768

 

[lukesfn]

Hawking radiation (so far as I know) doesn't come from the singularity. It originates at the horizon for observers remaining outside the BH. I have never seen anyone argue that it exists inside the horizon for any observer. Do have any reference for this, or and derivation for such a claim?

I was extrapolating a formula from one of the following papers. The derivation is earlier in this thread.

(2) Hawking radiation as seen by an in-falling observer

(3) Hawking radiation as perceived by didifferent
observers

Whether or not this derivation is valid is a good question, but it all seems perfectly consistent to my naive point of view. It makes sense to me that hawking radiation would be seen by an in-falling observer inside the EH, however, any such radiation would also be in-falling, and would have originated away from the singularity at a location at a greater radial distance to where it was observed.

Also, the question of a race between BH formation and BH evaporation is complicated, unsettled question well outside classical GR (for which there is no such thing as Hawking radiation or BH evaporation). With quantum effects considered, there is the fundamental problem that no complete theory exists to analyze this. Quantum effects change both horizon formation and radiation (the majority of experts analyze that horizons form 'faster' than classical GR, beating evaporation). But pending a real resolution of quantum gravity there is no final answer.

Here is one major paper in the debate (arguing for collapse beats evaporation). It references papers on the other side of the debate (whose authors, to the best of my knowledge, have not conceded). There may be a majority view that collapse wins, but this is not currently a fully settled question:

http://arxiv.org/abs/0906.1768

Thanks for the links, that all sounds interesting in it's self, but it isn't 100% relevant, since my questions are about a BH with a horizon. edit: To be fair, I suppose it is somewhat relevant.

I guess the question is now, can the energy of a BH be defined from the frame of a radially in-falling observer, so that the energy levels of observed radiation do not exceed the energy of the BH, on the path to the singularity.

 

[PeterDonis]

can the energy of a BH be defined from the frame of a radially in-falling observer

I still don't think this is a meaningful question (and I don't think the papers that PAllen linked to approach the general question of which wins out, collapse or evaporation, in this way). The only quantity related to the black hole that the infalling observer can observe is $M$, the parameter that appears in the metric, and that parameter is the same for all observers, since they're all measuring the same metric (i.e., the same spacetime geometry). Asking what "the energy of the BH is" in the infalling observer's frame treats the BH as if it were an external object that the infalling observer could "measure", when in fact it's not an "object" to him at all; it's the spacetime all around him.

The presence of Hawking radiation, if it's present inside the horizon, wouldn't change the above. (If it turns out that the quantum question gets resolved as "evaporation beats collapse", then there would never be a horizon or a region inside it to begin with.) The emission of Hawking radiation by the BH as a whole would (I think) make $M$ time-dependent, so the infalling observer might be able to measure that; but measuring a local radiation energy that was, in the infalling observer's frame, larger than $M$ would not, so far as I can see, be any indication of the overall state of the BH--at best, it would be an indication of how close to the singularity the infalling observer was.

 

[lukesfn]

I still don't think this is a meaningful question

To be fair, I rephrased my question to make it meaningful, however, I am arguing over semantics now, and your point is understood.

(and I don't think the papers that PAllen linked to approach the general question of which wins out, collapse or evaporation, in this way). The only quantity related to the black hole that the infalling observer can observe is $M$, the parameter that appears in the metric, and that parameter is the same for all observers, since they're all measuring the same metric (i.e., the same spacetime geometry). Asking what "the energy of the BH is" in the infalling observer's frame treats the BH as if it were an external object that the infalling observer could "measure", when in fact it's not an "object" to him at all; it's the spacetime all around him.

The presence of Hawking radiation, if it's present inside the horizon, wouldn't change the above. (If it turns out that the quantum question gets resolved as "evaporation beats collapse", then there would never be a horizon or a region inside it to begin with.) The emission of Hawking radiation by the BH as a whole would (I think) make $M$ time-dependent, so the infalling observer might be able to measure that; but measuring a local radiation energy that was, in the infalling observer's frame, larger than $M$ would not, so far as I can see, be any indication of the overall state of the BH--at best, it would be an indication of how close to the singularity the infalling observer was.

Throughout this thread, I have been taking the assumption that any hawking radiation perceived with in the EH, would have no effect on the BH to an outside observer, or any effect on the EH.

Right now, I'm only concerned with the amount of energy observed coming from a region of space with in the BH EH, not the energy of the entire BH. Perhaps I should ask, is it possible to define the energy with in a sphere defined a radius from the singularity, from the point of view of an in falling observer, inside the horizon.

 

[stevendaryl]

I was extrapolating a formula from one of the following papers. The derivation is earlier in this thread.

My intuition (and I don't have the mathematical skills to back this up) is that the situations outside the event horizon and inside the event horizon are sufficiently different that you can't extrapolate from one region to the other. From what I've read, Hawking radiation is associated with a horizon, and for an infalling observer, there is no horizon once you've passed the Schwarzschild radius. Inside the Schwarzschild radius, the coordinate r becomes time-like and so the horizon, which is at a fixed value of r, becomes something in the past, rather than something that persists.

 

[Dale]

I'm left wanting a deeper understanding and a more convincing proof.

And I am left wanting a more convincing proof that there is a problem. We have clear solutions showing collapse and aggregations in a finite amount of time as measured by the collapsing/aggregating matter, and Hawking radiation becomes arbitrarily small as M gets large. So to me what seems unconvincing is the implication that Hawking radiation could prevent collapse or aggregation.

At least that want might motivate me to learn a bit more about GR.

That certainly is a good outcome, regardless of the correctness of the argument.

 

[PAllen]

Whether or not this derivation is valid is a good question, but it all seems perfectly consistent to my naive point of view. It makes sense to me that hawking radiation would be seen by an in-falling observer inside the EH, however, any such radiation would also be in-falling, and would have originated away from the singularity at a location at a greater radial distance to where it was observed.

Neither of the papers you linked makes any claim that there is any Hawking radiation observed inside the horizon. You can't apply formulas developed outside the horizon, for what is accepted to be a horizon based phenomenon, inside the horizon just because you want to.

 

[PAllen]

And I am left wanting a more convincing proof that there is a problem. We have clear solutions showing collapse and aggregations in a finite amount of time as measured by the collapsing/aggregating matter, and Hawking radiation becomes arbitrarily small as M gets large. So to me what seems unconvincing is the implication that Hawking radiation could prevent collapse or aggregation..

What makes it non-obvious is that all parties agree on:

- radiation begins before the horizon forms (sometimes called pre-Hawking radiation)
- all BH theoretically disappear due to evaporation

So, "was there ever really a horizon - or an ever shrinking almost horizon" is a non-trivial question of competing rates. As the paper I linked concludes:

"The above description does not take into account the effect of backreaction on the formation of event horizon. If
radiation of energy can decrease the mass (and hence the effective radius of the event horizon) at a sufficiently rapid
pace, then the collapsing shell might never catch up with the event horizon and a black hole may never form. This,
however, does not happen essentially because the amount of radiation emitted during the collapse is not enough to
decrease the effective radius of the event horizon sufficiently fast. This result is nontrivial and requires the careful
computations which we have performed in this paper."

 

[Naty1]

PAllen:

Hawking radiation (so far as I know) doesn't come from the singularity. It originates at the horizon for observers remaining outside the BH. I have never seen anyone argue that it exists inside the horizon for any observer. Do have any reference for this, or and derivation for such a claim?

I checked half a dozen books and again conclude as does PAllen.

Here is an example: Leonard Susskind, THE BLACK HOLE WAR, PG 377

... The picture pf the black hole horizon that was emerging was a tangle of string flattened out onto the horizon by gravity...quantum fluctuations...would cause some parts of the string to stick out a it, and these bits would be the mysterious horizon-atoms. Roughly speaking, someone outside the black hole would detect bits of string, each with two ends firmly attached to the horizon...these are open strings...and could break loose from the horizon and that would explain how a black hole radiates and evaporates. It seems John Wheeler was wrong; black holes are covered with hair.

And here are a couple of additional descriptions relative to discussions here:

Black Hole Complementarity proposed by Susskind:

{slightly edited}

“..A stationary observer external to the horizon observes a stretched horizon, a hot layer which absorbs, scrambles and eventually emits the information the falls into the BH. To a freely falling observer, the horizon ‘appears’ as empty space…falling observers detect nothing at the horizon. They only encounter a destructive environment much later when the eventually approach the singularity.

Unruh showed that thermal and quantum jitters get mixed up in an odd way. Jitters that appear innocent quantum fluctuations to someone falling through the horizon become exceedingly dangerous thermal fluctuations to someone that remains suspended outside the BH {outside the horizon}

 

[Naty1]

Lo and behold, see what I found in my notes and had completely forgotten:

from Steve Carlip...
http://www.physics.ucdavis.edu/Text/Carlip.html#Hawkrad

...here's a way to understand Hawking radiation. Picture a virtual pair created outside a black hole event horizon. One of the particles will have a positive energy E, the other a negative energy -E, with energy defined in terms of a time coordinate outside the horizon. As long as both particles stay outside the horizon, they have to recombine in a time less than h/E. Suppose, though, that in this time the negative-energy particle crosses the horizon. The criterion for it to continue to exist as a real particle is now that it must have positive energy relative to the timelike coordinate inside the horizon, i.e., that it must be moving radially inward. This can occur regardless of its energy relative to an external time coordinate.

So the black hole can absorb the negative-energy particle from a vacuum fluctuation without violating the uncertainty principle, leaving its positive-energy partner free to escape to infinity. The effect on the energy of the black hole, as seen from the outside (that is, relative to an external timelike coordinate) is that it decreases by an amount equal to the energy carried off to infinity by the positive-energy particle. Total energy is conserved, because it always was, throughout the process -- the net energy of the particle-antiparticle pair was zero.

Note that this doesn't work in the other direction -- you can't have the positive-energy particle cross the horizon and leaves the negative- energy particle stranded outside, since a negative-energy particle can't continue to exist outside the horizon for a time longer than h/E. So the black hole can lose energy to vacuum fluctuations, but it can't gain energy...

thoughts?? My amateur conclusion is that Carlip doesn't think any Hawking radiation exists inside a BH horizon.

 

[Dale]

So, "was there ever really a horizon - or an ever shrinking almost horizon" is a non-trivial question of competing rates.

Since you can make the Hawking radiation rate arbitrarily small by having a sufficiently large M it seems clear to me that "pre-evaporation" cannot be a general case.

 

[atyy]

Singularities are a prediction of classical general relativity. Hawking radiation is a prediction of general relativity as a quantum effective field theory, which breaks down at high curvatures near the singularity. So the singularity might be thought of as simply where the theory, or at least our understanding of it, breaks down. Thus we don't understand the end stages of black hole evaporation when black holes are very small and curvatures are very high. Then the question is why do we still think Hawking radiation and black hole evaporation are reliable predictions for big black holes? That's because they are features of the horizon where curvature is low, in a big black hole, and where the effective field theory is believed to be predictive. (By "predictive", I mean the theory is sensible and can be falsified, whereas at high curvatures the theory seems to be "not even wrong".) This is analogous to quantum electrodynamics which works at low energies, but fails to be a sensible theory at high energies. The difference is we have experiemental verification of many low energy predictions of quantum electrodynamics, but Hawking radiation has not (yet?) been observed.

Fig. 2 of http://arxiv.org/abs/0704.1814 gives an estimate of how effective field theory fails at late stages of black hole evaporation.

 

[PAllen]

Since you can make the Hawking radiation rate arbitrarily small by having a sufficiently large M it seems clear to me that "pre-evaporation" cannot be a general case.

I lean toward the majority view on this, but I don't think this argument is enough to settle the question. For a big black hole, suppose you are seeing an incredibly low rated of pre-hawking radiation for 10^^(10^^(10^^...)) years before the horizon has formed. Then, without comparing rates and a model of horizon formation subject to decreasing mass, you still don't know which wins. There exists an actual computation claimed to show evaporation always wins (reference 17, I believe) in the paper I cite. The paper I referenced aims to refute that, but neither side claims the question is subject to trivial argument.

 

[Dale]

For a big black hole, suppose you are seeing an incredibly low rated of pre-hawking radiation for 10^^(10^^(10^^...)) years before the horizon has formed.

As long as the BH is sufficiently massive that it is colder than the CMB I don't think that matters.

 

[PAllen]

As long as the BH is sufficiently massive that it is colder than the CMB I don't think that matters.

The CMB gets colder over time, eventually below the Hawking temperature of any size BH. Also, note that the question of whether a horizon forms is a global, invariant question. Thus, analysis in any coordinate system is valid. In particular, analysis using and SC type time foliation is valid. This gives infinite time before horizon formation for evaporation to succeed. So you have to show how horizon formation wins the race even in these coordinates.

 

[Dale]

The CMB gets colder over time, eventually below the Hawking temperature of any size BH.

I don't see how future cooling is relevant to the collapse. As long as the BH is colder today than the CMB today then it has been colder than the CMB for the past "10^^(10^^(10^^...)) years" too. So what if that changes in the future, it doesn't impact the collapse.

 

[PeterDonis]

is it possible to define the energy with in a sphere defined a radius from the singularity, from the point of view of an in falling observer, inside the horizon.

I don't think so, because inside the horizon there are no static objects, and that includes spheres of constant radius. So there's no way to add up the "energy inside a sphere of constant radius" at an instant of time inside the horizon.

 

[stevendaryl]

Lo and behold, see what I found in my notes and had completely forgotten:

from Steve Carlip...
http://www.physics.ucdavis.edu/Text/Carlip.html#Hawkrad



thoughts?? My amateur conclusion is that Carlip doesn't think any Hawking radiation exists inside a BH horizon.

I don't think that intuitive description of black hole radiation is original to Carlip; I think that Hawking himself may have come up with it.

However, as I understand it, physicists have not convincingly proved that the intuitive description in terms of virtual pairs actually connects with the more careful analysis, having to do with the different definitions of "vacuum" and "particle number" used by stationary and infalling observers. In other words, I don't think that the intuitive description has been shown to be correct.

 

[PAllen]

I don't see how future cooling is relevant to the collapse. As long as the BH is colder today than the CMB today then it has been colder than the CMB for the past "10^^(10^^(10^^...)) years" too. So what if that changes in the future, it doesn't impact the collapse.

I thought we were discussing collapse, not eternal black holes. The debate on this concerns collapse models. Thus, eternal past is irrelevant. Thus you have a collapse, with infalling matter and also a tiny bit of CMB. The question remains: for a distant observer, the collapse process 'never' completes, the evaporation does (even accounting for CMB). At least that is the argument of Krauss et. al. I think this is most likely wrong, but reading over the respective papers, I don't think the debate can be trivialized. I think you have to model both horizon formation and evaporation as quantum processes, and there won't be a final answer until there is an accepted theory of quantum gravity.

 

[lukesfn]

Neither of the papers you linked makes any claim that there is any Hawking radiation observed inside the horizon. You can't apply formulas developed outside the horizon, for what is accepted to be a horizon based phenomenon, inside the horizon just because you want to.

I can't think of a reason why the formulas wouldn't extend beyond the event horizon. I don't know what the difference would be there. They predict a non zero hawking radiation observed by a free faller at the horizon. If it changes in a non-continuous way at the horizon, it would seem to me to break the equivalence principal.

My intuition (and I don't have the mathematical skills to back this up) is that the situations outside the event horizon and inside the event horizon are sufficiently different that you can't extrapolate from one region to the other. From what I've read, Hawking radiation is associated with a horizon, and for an infalling observer, there is no horizon once you've passed the Schwarzschild radius. Inside the Schwarzschild radius, the coordinate r becomes time-like and so the horizon, which is at a fixed value of r, becomes something in the past, rather than something that persists.

Obviously I am not an expert, but there is no reason obvious to me why the math should suddenly change at the EH, however, the hawking radiation observed inside the BH comes from a non-static falling "Horizon" inside the static EH that the in-faller overtook in it's past. Non of the observed radiation can escape the BH, everything is in-falling. It all seems to fit together perfectly and make perfect sense to me, however, the in-faller will beat the radiation in the race to the singularity. Which implies it overtakes some of the energy in that space, which implies that not all of the energy of a BH is concentrated at the singularity, however, it seems that GR has trouble with locating energy in this situation anyway, so perhaps this is all ok.

And I am left wanting a more convincing proof that there is a problem.

Fair enough. Even if my calculations of hawking radiation are correct, I can't be sure they are a problem if I don't have a way to calculate the energy of the region they come from.

We have clear solutions showing collapse and aggregations in a finite amount of time as measured by the collapsing/aggregating matter, and Hawking radiation becomes arbitrarily small as M gets large. So to me what seems unconvincing is the implication that Hawking radiation could prevent collapse or aggregation.

But I assume the hawking radiation you are talking about here is as observed at infinity, not as observed by the collapsing matter at it's location inside the EH, which increases as the radius gets smaller, there for the second half of that argument is not relevant to preventing a singularity, even if there is a EH.

I don't think so, because inside the horizon there are no static objects, and that includes spheres of constant radius. So there's no way to add up the "energy inside a sphere of constant radius" at an instant of time inside the horizon.

I just starting to realize this could be a problem my self. I'm doing some reading. It is surprising how many different definitions of energy there are in GR.

Ironically, perhaps, the observed Hawking radiation gives an estimate of the energy of the region it came from. Interestingly, even if it doesn't make sense to talk about the energy of such a region the radiation wouldn't cause that region to loose any energy, because it is radiating into its self.

I was just trying to imagine a region of space, moving out at the speed of light, starting from just inside the EH, and falling to the singularity, and calculating how much energy it looses, however, I think I would just run into the same problem because the energy loss needs to be calculated by an observer at infinity, but the radiation won't get there.

 

[PAllen]

I can't think of a reason why the formulas wouldn't extend beyond the event horizon. I don't know what the difference would be there. They predict a non zero hawking radiation observed by a free faller at the horizon. If it changes in a non-continuous way at the horizon, it would seem to me to break the equivalence principal.

The point is that the derivation of the Hawking radiation assumes it is observed outside the horizon, and is caused by the horizon. In the realm of quantum theory (unlike classical) the horizon is not necessarily locally undetectable. Look up black hole firewalls for the current hot (pun intended) debate on the quantum nature of horizons.

More importantly, I asked you for references to anyone proposing Hawking radiation is detectable inside the horizon. You gave links to two papers that say no such thing. That leaves you with nothing but personal opinion in disagreement with all derivations and discussions of Hawking radiation.

 

[Dale]

I thought we were discussing collapse, not eternal black holes.

Yes, me too.