A Spacetime model of evaporating black hole

[Dale]

[Moderator's note: Thread spin-off due to more advanced sub-topic.]

there is the region occupied by outgoing Hawking radiation, where the metric, again in the idealized case of perfect spherical symmetry, is the outgoing Vaidya metric

I thought that the horizon for the outgoing Vaidya metric was a white hole horizon. Am I mistaken about that?

 

[PeterDonis]

I thought that the horizon for the outgoing Vaidya metric was a white hole horizon. Am I mistaken about that?

That's true of the maximal analytic extension of the outgoing Vaidya metric, yes. But only a portion of that spacetime is used in the model we are discussing here, and that portion is all outside the horizon. Roughly speaking, the horizon in the model we are discussing here is a Schwarzschild horizon.

 

[Dale]

That's true of the maximal analytic extension of the outgoing Vaidya metric, yes. But only a portion of that spacetime is used in the model we are discussing here, and that portion is all outside the horizon. Roughly speaking, the horizon in the model we are discussing here is a Schwarzschild horizon.

I don't think that works.

 

[PeterDonis]

I don't think that works

Can you be more specific about what you think doesn't work? (Note that I have marked this spin-off thread as "A" level as I think we'll need to get to that level to properly discuss the issue.)

 

[Dale]

Can you be more specific about what you think doesn't work?

If you have an outgoing Vaidya metric which is not a vacuum spacetime that borders a Schwarzschild metric which is a vacuum spacetime, then it seems like you cannot avoid having a non-vanishing divergence of the stress energy tensor at the border.

 

[PeterDonis]

If you have an outgoing Vaidya metric which is not a vacuum spacetime that borders a Schwarzschild metric which is a vacuum spacetime, then it seems like you cannot avoid having a non-vanishing divergence of the stress energy tensor at the border.

If this were a valid criticism, it would apply equally well to the 1939 Oppenheimer-Snyder model, which has a non-vacuum FRW region bordering a Schwarzschild vacuum region.

Or, for that matter, it would apply to a simple model of a spherically symmetric (non-rotating) planet or star, which has a spherical non-vacuum region bordering a Schwarzschild vacuum region.

This is one of those cases where intuition can lead one astray, and we need to actually look at the math. Fortunately, I did a good piece of it for the simplest case, a static, spherically symmetric spacetime, in this Insights article:

https://www.physicsforums.com/insig...-in-a-static-spherically-symmetric-spacetime/

As you can see from that article, the only significant nonzero component of the covariant divergence equation for this case becomes the Tolman-Oppenheimer-Volkoff equation. And that equation works just fine across the boundary. It's true that, in the idealized case where the boundary has zero thickness, there is a discontinuity in the derivatives of the mass (or density) and pressure as a function of radius. But that's just an artifact of the idealization, and doesn't affect the covariant divergence vanishing: it just means that the LHS and the RHS of the TOV equation both discontinuously change at the boundary from some finite value to zero. A more realistic model will have a boundary layer of finite thickness in which conditions change smoothly from the non-vacuum to the vacuum case, and the discontinuity is removed.

For now I'll leave the more complicated cases where things are not static as an exercise. But as my comments at the top of this post should indicate, there is no covariant divergence issue with those cases either. The math is just more complicated to work through.

 

[Dale]

If this were a valid criticism, it would apply equally well to the 1939 Oppenheimer-Snyder model, which has a non-vacuum FRW region bordering a Schwarzschild vacuum region.

I don’t think so. In the OS metric the dust has an inward flux. This inward flux of the dust is precisely what causes the sphere of dust to collapse, leaving vacuum behind.

That isn’t what is happening with the Vaidya metric. In that you have null dust that is being emitted which is coming from the star (or whatever other spherical source you want to consider). The emission of the null dust reduces the mass of the star. If you remove the star and replace it with vacuum then you don’t have anything that can produce the null dust.

I will read your article, but since you went to the OS metric as a counter example, I think you missed my point.

 

[PeterDonis]

I don’t think so. In the OS metric the dust has an inward flux. This inward flux of the dust is precisely what causes the sphere of dust to collapse, leaving vacuum behind.

None of this changes the fact that the model has two spacetime regions, an FRW dust region and a vacuum region, with a boundary between them that has a discontinuity in density. And if your argument were correct, that would mean the covariant divergence of the SET would not vanish at the boundary.

That isn’t what is happening with the Vaidya metric. In that you have null dust that is being emitted which is coming from the star (or whatever other spherical source you want to consider). The emission of the null dust reduces the mass of the star. If you remove the star and replace it with vacuum then you don’t have anything that can produce the null dust.

Again, you are describing the maximal analytic extension of the outgoing Vaidya metric. But that is not what is being used in the model under discussion.

In the model under discussion, the source of the outgoing null dust in the Vaidya region is the matter in the collapsing FRW region. But none of that changes the fact that there is a boundary between the outgoing Vaidya region and the Schwarzschild vacuum region in the model. Note that the Schwarzschild region is to the past of the outgoing Vaidya region. It is not "inside" the outgoing Vaidya region, i.e., it is not in any sense supposed to contain a "source" of the outgoing null dust.

(Note that in the maximal analytic extension of the outgoing Vaidya metric, there is no Schwarzschild region at all. The source of the outgoing null dust is not vacuum.)

I will read your article, but since you went to the OS metric as a counter example, I think you missed my point.

I don't think I did. I think you are not understanding how the covariant divergence of the SET works, and how general the wrong argument you are making would be if it were correct. If it were correct, it would apply to any spacetime model that has a boundary between a non-vacuum region and a vacuum region.

Note also that the Insights article I referenced is not about the OS model. It is about a generic static, spherically symmetric spacetime.

 

[PeterDonis]

That isn’t what is happening with the Vaidya metric. In that you have null dust that is being emitted which is coming from the star (or whatever other spherical source you want to consider).

Note that in the maximal analytic extension of the outgoing Vaidya metric, there is no "star" or other matter region. There is a white hole horizon, as you said, but "inside" that horizon there is just more outgoing null dust. There is no other "source" anywhere.

In a model of a spherically symmetric star emitting radiation that is idealized as outgoing null dust, the outgoing Vaidya region is only a portion of the maximal analytic extension of the outgoing Vaidya metric, which is joined to a spherically symmetric matter region whose radius, surface area, and mass decreases with time. There is no horizon anywhere in such a model.

 

[Dale]

None of this changes the fact that the model has two spacetime regions, an FRW dust region and a vacuum region, with a boundary between them that has a discontinuity in density.

It isn’t the discontinuity in the density that concerns me. It is the divergence of the SET. Divergence and discontinuity aren’t the same thing. But patching the two spacetimes together leads to a non-vanishing divergence at the discontinuity.

Please stop for a minute and actually think about my concern. I am not an idiot nor a novice here.

And if your argument were correct, that would mean the covariant divergence of the SET would not vanish at the boundary.

I have already explained that is not my argument.

In the model under discussion, the source of the outgoing null dust in the Vaidya region is the matter in the collapsing FRW region.

OK, I think that is more likely the issue. I think I misunderstand the model you were describing.

Is there a Penrose diagram like the one in the other thread where the various regions are labeled?

No, I didn't. I think you are not understanding how the covariant divergence of the SET works, and how general the wrong argument you are making would be if it were correct. If it were correct, it would apply to any spacetime model that has a boundary between a non-vacuum region and a vacuum region

Yes, you are completely misunderstanding my argument. You are focusing on the discontinuity. The discontinuity isn’t the issue, it is only the location where the issue happens.

Divergence of the SET is locally the conservation of energy and momentum. In the Vaidya metric the energy and momentum locally flow outward from the star to the surrounding region, locally decreasing the mass of the star in the process.

That is not an issue with OS, nor with other familiar metrics with a boundary between vacuum and matter.

My (possibly incorrect) understanding of the model is that we have a Vaidya metric surrounding a Schwarzschild black hole. So energy is flowing out without any source. How do you have the twinkle twinkle without the little star? Or how is the model constructed if I have that part wrong?

Note that the Schwarzschild region is to the past of the outgoing Vaidya region. It is not "inside" the outgoing Vaidya region, i.e., it is not in any sense supposed to contain a "source" of the outgoing null dust.

I thought it was inside. So I do not understand how the different patches go together.

 

[Dale]

Roughly speaking, there is a region where, as I said before, the metric is Schwarzschild; there is the region occupied by the collapsing matter, where the metric in the idealized case of perfectly spherically symmetric collapse is a portion of a closed collapsing FRW metric (as in the 1939 paper by Oppenheimer and Snyder); there is the region occupied by outgoing Hawking radiation, where the metric, again in the idealized case of perfect spherical symmetry, is the outgoing Vaidya metric; and there is the region after the final evaporation of the hole, in which the metric is Minkowski. You then have to impose appropriate junction conditions at the boundaries between these regions.

There are a lot of regions. What parts are where and which pieces are connected to each other?

The piece that I am most focused on is the spacetime immediately near the evaporating horizon. In which region is the horizon itself and which other regions are important during the evaporation process?

 

[PeterDonis]

In the Vaidya metric the energy and momentum locally flow outward from the star to the surrounding region, locally decreasing the mass of the star in the process.

Ok, this helps me to understand what issue you are raising.

In the model under discussion, energy and momentum flow from a region containing stress-energy, to another region containing stress-energy. There is no place where energy and momentum flow from a vacuum region to a region containing stress-energy, or vice versa. I agree such a model would not make sense. However, that doesn't happen in the model under discussion. See further comments below.

My (possibly incorrect) understanding of the model is that we have a Vaidya metric surrounding a Schwarzschild black hole.

Your understanding is incorrect, at least as far as inferring that there is an issue with conservation of energy and momentum.

I don't have ready access to digital tools for drawing spacetime diagrams, so I drew one by hand and scanned it:

The shaded region on the left is the collapsing matter that forms the hole.

The "SV" region is the Schwarzschild vacuum region outside the collapsing matter, before the evaporation process starts. This is the region I referred to before as being to the past of the outgoing Vaidya region.

The "OV" region is the outgoing Vaidya region. Note that all null lines through this region have endpoints at the collapsing matter region.

The "BH" region is the black hole. The geometry in this region, outside the collapsing matter, is Schwarzschild vacuum interior to the event horizon. However, as noted below, the radiation from the hole's evaporation is not coming from this region! (If you think about it, you'll realize that it can't, because the radiation from the hole's evaporation goes out to future null infinity, and the black hole, by definition, is not in the causal past of future null infinity.)

The "M" region is the flat Minkowski region that is left once the hole has completely evaporated and the last bit of radiation has gone outward to infinity.

The boundary between "BH" and "OV" is the black hole event horizon. The boundary between "OV" and "M" is the last bit of radiation from the hole's final evaporation, going out to infinity.

Note that, while it's true that the "r" coordinate everywhere in the "OV" region is greater than the "r" coordinate everywhere in the "BH" region, it still doesn't really make sense to say that the "BH" region is "inside" the "OV" region. As noted above, the outgoing null dust in the "OV" region is coming from the collapsing matter region, not the "BH" region.

 

[PeterDonis]

the evaporating horizon

The horizon itself doesn't evaporate. It can't, by definition: it can't send radiation out to infinity, which is what it would have to do to evaporate.

As the diagram I posted shows, all of the radiation comes from outside the horizon.

 

[PeterDonis]

Another note on the diagram I posted: it illustrates why you can't think of a black hole as an "object" that can be "inside" something else. You have to resist all such intuitions, and look at the actual causal structure, and realize that the black hole itself is best viewed as in the future, not as "inside" anything. It's true that the "future" inside the hole is avoidable (just be sure not to fall in), but it's still to the future, not "inside".

 

[Dale]

I don't have ready access to digital tools for drawing spacetime diagrams, so I drew one by hand

Thanks, yes, this does resolve my confusion. I had misunderstood the arrangement of the various regions.

The boundary between "BH" and "OV" is the black hole event horizon.

This boundary does concern me. The OV metric has a white hole horizon. Is it even possible to find a consistent junction condition with a Schwarzschild BH event horizon?

 

[PeterDonis]

The OV metric has a white hole horizon.

Not the OV region in the model, no. There is no white hole horizon anywhere.

The maximal analytic extension of the OV metric has a white hole horizon, but that portion of the maximal extension is not used in this model.

In a Penrose diagram of the type shown, a white hole horizon would go up and to the left. It would be the boundary of a region that an observer starting from past timelike infinity $i^-$ or a null line starting from past null infinity could never get into. Or, to put it another way, it would be the boundary of a region that is not in the causal future of past null infinity (the time reverse of a black hole horizon, which is a region that is not in the causal past of future null infinity).

 

[PeterDonis]

Is it even possible to find a consistent junction condition with a Schwarzschild BH event horizon?

In the "OV" region, the "mass" decreases to zero as you go from the SV boundary to the BH boundary. (If you imagine the region covered by null lines in between the two boundaries, with each null line having its own label $u$, the mass is a decreasing function of $u$, and goes to zero in the limit as $u$ approaches the value it has on the BH horizon.) So there isn't even a discontinuity in the density at the BH boundary; it's zero.

 

[Tomas Vencl]

In the "OV" region, the "mass" decreases to zero as you go from the SV boundary to the BH boundary. (If you imagine the region covered by null lines in between the two boundaries, with each null line having its own label $u$, the mass is a decreasing function of $u$, and goes to zero in the limit as $u$ approaches the value it has on the BH horizon.) So there isn't even a discontinuity in the density at the BH boundary; it's zero.

Maybe I am mising something important. Imagine green observer with its proper time. You say that mass of bh decreases to zero between points (and greens times) 1 and 2 ? Then, how he can see some photons at point 3 eventually emmited from collapsing surface ? Or he cant ?

 

[PeterDonis]

how he can see some photons at point 3 eventually emmited from collapsing surface ? Or he cant ?

In the model as I described it, he can't, because there is no radiation in the BH region. As I said in post #15, the geometry of that region outside the collapsing matter is the vacuum Schwarzschild geometry interior to the horizon. That means there can't be any radiation present.

One could construct a model in which the collapsing matter does emit radiation inside the BH horizon, and that radiation goes into the singularity and gets destroyed. The geometry in the BH region in such a model would be more complicated, and it wouldn't change what's visible from outside the horizon, so I didn't bother considering that complication.

 

[PeterDonis]

patching the two spacetimes together leads to a non-vanishing divergence at the discontinuity.

On thinking this over, I'm not sure the genuine issue that you describe (which, as you now agree, is not present in the model under discussion, but is still a genuine issue, in the sense of a constraint that any valid model must satisfy) would be visible as a non-vanishing covariant divergence.

Consider a simpler example: a spherically symmetric, static planet or star that just appears out of nowhere. We usually say that's not possible because stress-energy can't be created or destroyed. But let's look at how this would be modeled.

The object would be a spacetime region occupied by matter, surrounded by Schwarzschild vacuum. If it appears out of nowhere, that means the spacetime region occupied by the matter has a past boundary that's spacelike, and to the past of that boundary there's vacuum. That vacuum would be Minkowski vacuum. (There would also be Minkowski vacuum to the past of the future light cone of the events on the past boundary of the matter region that are on the surface of the object that appears out of nowhere--heuristically, the fact that the object is now present has to propagate outward at the speed of light, so the transition from Minkowski to Schwarzschild vacuum will be along a null surface that's the future light cone of the transition events on the object's surface. But we can ignore that part here.)

Now, consider again the TOV equation, which is what the covariant divergence equation amounts to for a static, spherically symmetric object. There will be a discontinuity in this equation as the spacelike past boundary of the object is crossed (i.e., when the object appears out of nowhere). But the discontinuity will appear equally in the LHS and the RHS of the equation; both sides will change discontinuously from zero to the same finite value, so they will still balance and the covariant divergence (which is basically the LHS minus the RHS of the TOV equation) will still vanish. Note that this is the same reason that the covariant divergence still vanishes across a timelike or null boundary between matter and vacuum, as in the boundary between a static object and the Schwarzschild vacuum outside it. So the fact that this model is not valid will not show up as a nonzero covariant divergence.

The problem will be in the junction conditions: it is not possible to match a Minkowski vacuum to a spherically symmetric, static region containing matter across a spacelike boundary and satisfy those conditions. (Heuristically, the Minkowski vacuum wants the spacelike boundary to be spatially flat, whereas the matter region wants it to be spatially curved; the junction conditions require those two curvatures to match, but they don't.) The junction conditions across such a boundary are what require that the matter (or radiation) has to come from a source. Or, to put it another way, they require that the worldlines of the matter must either go to infinity to both the past and future, or must end on a singularity somewhere; they can't just stop in the middle of an otherwise nonsingular region.

 

[Dale]

So the fact that this model is not valid will not show up as a nonzero covariant divergence.

My misunderstanding of the model was worse than that. I was imagining that the boundary of the Vaidya metric was timelike. Specifically, I had thought that the boundary of the OV metric was outside the event horizon some finite (quantum mechanical scale) distance. That is why I was speaking of a star. For that model it would literally be a shine without the star.

 

[PeterDonis]

I had thought that the boundary of the OV metric was outside the event horizon some finite (quantum mechanical scale) distance.

I didn't really draw the OV region the way the production of the Hawking radiation seems to be imagined by most physicists in the field. To match what they seem to imagine (at least for this model taken as an abstract model, apart from whether it's actually physically reasonable--see further comments below), I should have drawn it as a very thin region, so that, for example, an observer falling in with the collapsing matter would see all of the radiation that will ever be emitted go out over a very short interval of their proper time. The far distant observer sees it take a huge amount of time (about $10^{67}$ years for a one solar mass hole), heuristically, because of the huge time dilation between them and, say, a Planck length above the horizon.

Even that doesn't necessarily make the model as I drew it (which is basically the way things would have to work if Hawking's original model in the paper he published in the 1970s were correct) physically reasonable. AFAIK most workers in the field don't think it actually is, but there doesn't seem to be any consensus about what model is physically reasonable.

 

[Dale]

So I was imagining some finite (quantum) distance outside of the horizon you would have an outgoing Vaidya metric bordering an ingoing Vaidya metric. With the mass parameter for both being equal and decreasing by the usual Hawking radiation formula.

That at least is the closest implementation I could think of for the half-quantum half-classical picture that is usually presented. The outgoing dust is pretty ordinary, but the ingoing dust is weird negative-energy dust. But that is in keeping with the usual half-and-half picture.

 

[Dale]

So in that scenario the Schwarzschild vacuum is only outside the initial collapsing dust and the outgoing null dust. The boundary between the OV and IV metrics begins with the beginning of the horizon. So the interior of the horizon is never Schwarzschild, but always IV with decreasing mass. And then expanding Minkowski inside the expanding OV dust shell after evaporation.

In my mind the OV-IV boundary had to be outside the horizon. Both in keeping with the usual half-and-half description and because only the IV horizon is a black hole horizon. So we couldn’t have the boundary on the horizon or in the OV region where the horizon would be a white hole.

 

[PeterDonis]

The outgoing dust is pretty ordinary, but the ingoing dust is weird negative-energy dust.

That would indeed match the usual heuristic picture of Hawking radiation, but I'm not sure that heuristic picture is actually a very good one. The "ingoing negative-energy dust" part, AFAIK, doesn't actually correspond to anything in the actual math that Hawking presented. It does, however, address another heuristic issue; see further comments below.

the interior of the horizon is never Schwarzschild, but always IV with decreasing mass.

Yes, that would be the case in the model you propose. But, as above, I'm not sure how well that heuristic picture actually matches the math.

One other note: as I mentioned above, having the region inside the horizon be IV instead of vacuum addresses another issue with the diagram as I drew it, namely: how does the mass become zero? In the diagram as I drew it, the collapsing matter, when it crosses the horizon, still has nonzero mass--and there's no way for that mass to radiate anything more away to infinity, because it's now at (or inside) the horizon. So in the diagram as I drew it, it doesn't seem like the hole could completely evaporate away.

But if the interior outside the collapsing matter is IV, then the "ingoing negative-energy dust" basically cancels out the positive energy of the collapsing matter region inside the horizon, so the mass ends up being zero. This also has to increase the energy density in the OV region from what it is at the boundary of the collapsing matter. The ingoing negative energy dust effectively transfers the rest of the mass from the collapsing matter inside the horizon to the OV region outside the horizon.

Again, I don't know how well this matches the actual math that Hawking presented in his original paper. But as far as I can tell, it's mathematically consistent. Whether "ingoing negative energy dust" is physically reasonable is a separate question.

 

[Tomas Vencl]

Would it be possible to draw diagrams of the more realistic models considered by both of you? Thank you.

 

[martinbn]

I might be repeating something that was already discussed but I don't understand what the potential issue is! How could the stress-energy tensor have a non-zero divergence?! You glue to portions of two spacetimes to obtain a new one, which is a manifold with a specific metric. The SET of the new space time is obtained from the Einstein's equations, so it will definitely have zero divergence.

 

[Dale]

In the diagram as I drew it, the collapsing matter, when it crosses the horizon, still has nonzero mass--and there's no way for that mass to radiate anything more away to infinity, because it's now at (or inside) the horizon

Hmm, yes, that is a problem. I was focused more on my understanding of the region near the horizon, but that is highly problematic.

Also, that would make the OV dust ordinary radiation from the collapsing matter, rather than Hawking radiation. For it to be Hawking radiation it needs to come from the horizon, in some sense.

I don't know how well this matches the actual math that Hawking presented in his original paper. But as far as I can tell, it's mathematically consistent. Whether "ingoing negative energy dust" is physically reasonable is a separate question

Indeed, on both points. Although, of the two I personally am more concerned about the negative energy dust.

I am also skeptical of Hawking’s math itself. It is not a fault of his, I just don’t like the half-and-half hodgepodge of classical and quantum. I am not fundamentally opposed to such things when justified, but the way to justify it is to do the full calculation and show that it approximates the half-and-half calculation closely. That currently cannot be done.

One question for you: are any of the usual energy conditions violated in the actual math? If so, at least the negative energy null dust makes that violation clear.

 

[DrGreg]

I might be repeating something that was already discussed but I don't understand what the potential issue is! How could the stress-energy tensor have a non-zero divergence?! You glue to portions of two spacetimes to obtain a new one, which is a manifold with a specific metric. The SET of the new space time is obtained from the Einstein's equations, so it will definitely have zero divergence.

As I understand it, the question is whether two different solutions from different spacetimes are sufficiently mutually compatible to allow them to be glued together to form a valid spacetime.

The issue arises at the boundary between the two regions. The divergence at an event is an average calculated on a small sphere surrounding the event, in the limit as the sphere shrinks to zero. For an event on the boundary, the divergence is the sum of two hemispheres, one in each region. The question is, do these two "half-divergences" add up to zero or not?

 

[martinbn]

As I understand it, the question is whether two different solutions from different spacetimes are sufficiently mutually compatible to allow them to be glued together to form a valid spacetime.

The issue arises at the boundary between the two regions. The divergence at an event is an average calculated on a small sphere surrounding the event, in the limit as the sphere shrinks to zero. For an event on the boundary, the divergence is the sum of two hemispheres, one in each region. The question is, do these two "half-divergences" add up to zero or not?

Yes, but one matches the metrics of the two spacetimes, and enough derivatives. And if they match you obtain a smooth matric. The SET is calculated from the Einstein's equations so it will definitely have zero divergence.