A Spacetime model of evaporating black hole

[PeterDonis]

that would make the OV dust ordinary radiation from the collapsing matter

It starts out that way, but as I said, that won't account for all of the mass being radiated away. To accomplish that you need something like the ingoing negative energy dust.

For it to be Hawking radiation it needs to come from the horizon, in some sense.

Kinda sorta. One of the main issues with trying to model this is the fact that the event horizon is not locally detectable; it's a global property of the spacetime. So it's hard to see how it could locally generate radiation.

One of the lines of research that's been pursued is to try to connect the radiation to an apparent horizon (a marginally trapped surface), since those are locally detectable. In the model under discussion, the collapsing matter would form an apparent horizon just outside the event horizon, and that might somehow cause it to emit Hawking radiation or something like it. But AFAIK this line of research is still open and there is no general agreement on whether it works.

I am also skeptical of Hawking’s math itself.

You're not the only one. Many, if not most, researchers in the field, from what I can tell, seem to think his original math doesn't really work--it was a valuable heuristic guide to research, but it doesn't really work as an actual model. The problem is that nobody has come up with a better model that everyone can accept.

are any of the usual energy conditions violated in the actual math?

Yes. They have to be, because the model violates the area theorem: the area of the event horizon decreases. But my understanding is that the way they are violated in the actual math doesn't lend itself to any neat, simple picture like ingoing negative energy.

 

[PeterDonis]

How could the stress-energy tensor have a non-zero divergence?!

I'm not sure it can; see my post #20 for an alternate view of how the issue of "radiation has to have a source somewhere" would show up at the boundary (short version: in the junction conditions, not the divergence).

 

[PeterDonis]

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.

As I said in post #20, I think that issue shows up in the junction conditions, not in a non-vanishing divergence.

 

[Dale]

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

Unless the gluing causes a problem. To be clear @PeterDonis ‘s model does not have the problem I thought it did, because his model was different from what I thought.

The model I wrongly thought he was proposing has a boundary between a Schwarzschild vacuum and an outgoing Vaidya metric, with the boundary being located outside the horizon.

The OV metric has 0 divergence because all of the null flux going out one side comes in the other side. But at the “glue” there is null flux going out one side with no flux coming in the other side. So even though the EFE is solved on each side of the junction, right there it fails

 

[Dale]

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

I don’t know that it is more realistic by any means. But I can do a diagram later

 

[PeterDonis]

even though the EFE is solved on each side of the junction, right there it fails

This is not saying that the covariant divergence condition fails. It is saying that the junction conditions fail.

See, for example, the discussion in MTW, Section 21.13. The junction conditions treated there are derived from the EFE itself, not from its covariant divergence. Failure to meet the junction conditions means failure of the EFE itself to be satisfied at the boundary. Note that this can be true even if the covariant divergence of both sides of the EFE is still zero.

 

[Dale]

This is not saying that the covariant divergence condition fails. It is saying that the junction conditions fail.

Sure, I am fine with that. It is just another way of saying the same thing. Thinking about the divergence of the SET is the way that I recognize that the junction conditions must fail in this case

Note that this can be true even if the covariant divergence of both sides of the EFE is still zero

Yes. The junction conditions are more general and can fail in other ways too

 

[PeterDonis]

It is just another way of saying the same thing.

Not really. Saying that the junction conditions fail is another way of saying that the EFE is not satisfied at the boundary. But the EFE not being satisfied does not necessarily imply that the covariant divergence condition is not satisfied. The covariant divergence of the SET could still be zero even if the SET does not equal the Einstein tensor (times whatever constant factor your choice of units imposes).

However, if the EFE is satisfied, then the covariant divergence condition must also be satisfied, because the Bianchi identities ensure that the Einstein tensor always satisfies it. So, conversely, if the covariant divergence condition is not satisfied by the SET, then the EFE cannot be satisfied either.

So the EFE not being satisfied is the correct criterion, and is not saying the same thing as the covariant divergence condition not being satisfied; cases of the latter are a subset of cases of the former.

 

[Dale]

@PeterDonis talking with you can be very annoying. You insist on making a big deal correcting things that are not wrong. As you said

cases of the latter are a subset of cases of the former

I never said that in general every failure of every junction condition is due to non-vanishing divergence of the SET. I merely said that in this specific case the SET has divergence at the boundary, which is indeed one way that the junction conditions can fail at the junction.

Talking with you is almost not worth it. Yes, you know a lot. But trying to actually learn anything involves a relentless slog through a huge mass of unnecessary correction.

Just look at this thread. You first spent several posts harping on irrelevant metrics where there was a matter-vacuum boundary. As I told you, that was not the issue though you continued to pursue it anyway.

We found the crux of the matter, which was my misunderstanding of the model, and began to have an interesting conversation.

Now we are back to another morass of pointless criticism and correction. What I am saying here is not wrong. Restrain your knee-jerk criticism for places where I am actually wrong, like my understanding of the model you were using.

When discussing a particular case that fails in a particular way it is not wrong to identify the specific cause of failure. I tried to be gracious to you and instead of just letting it be you insist on wasting further effort here.

 

[PeterDonis]

SET has divergence at the boundary, which is indeed one way that the junction conditions can fail at the junction.

No, it isn't. That was my point. The junction conditions do not contain the covariant divergence at all. I gave a specific reference to support that point.

I'm sorry if you find my correcting you on that point annoying.

 

[PeterDonis]

You first spent several posts harping on irrelevant metrics where there was a matter-vacuum boundary. As I told you, that was not the issue though you continued to pursue it anyway.

Yes, I agree that I misunderstood the actual issue you were raising.

However, the fact that you kept focusing on the covariant divergence, when in fact, as I have said, that is not the right thing to be looking at, contributed to my confusion.

I'm glad I was able to make the structure of the model clear once I understood the issue you were raising, which, as I have said in previous posts, is a genuine one.

 

[Dale]

The junction conditions do not contain the covariant divergence at all.

The covariant divergence not vanishing at the junction implies that there is no solution to the EFE, which in turn implies that there are no junction conditions which satisfy the EFE. The one implies the other. The reverse is not true, but I made no claim to the reverse and need no correction on the correct claim I did make.

And the slog continues

 

[PeterDonis]

The covariant divergence not vanishing at the junction

I'm still not convinced it actually does, which is why I looked for a condition that does not depend on demonstrating this.

I know you think the Oppenheimer-Snyder case is irrelevant, but I disagree. If the SET can have a vanishing divergence on the vacuum to non-vacuum boundary between the FRW matter region and the Schwarzschild vacuum region in that case, how is it a slam dunk that it must have a non-vanishing divergence on the vacuum to non-vacuum boundary in the model you were imagining, where the OV region is bounded at one end by a Schwarzschild vacuum? What is the crucial difference between those two cases that you see and I do not?

 

[Dale]

I'm still not convinced it actually does

Perhaps we should discuss that then instead of wasting time correcting something that doesn’t need correction.

Remember, the model in question is Schwarzschild vacuum outside of the event horizon surrounded by an OV spacetime.

Imagine a small “box” of spacetime, in all four dimensions, which straddles the junction. There is no flux of four-momentum across the $\theta$ or $\phi$ surfaces. If the mass is decreasing at a constant rate then the flux out of the future surface is the same as the flux into the past surface. Finally, the flux into the inner surface is zero, but the flux out of the outer surface is non-zero. So there is a net flux of four-momentum out of the box, so the SET has a non-vanishing divergence.

If the SET can have a vanishing divergence on the vacuum to non-vacuum boundary between the FRW matter region and the Schwarzschild vacuum region in that case, how is it a slam dunk that it must have a non-vanishing divergence on the vacuum to non-vacuum boundary in the model you were imagining

Consider the same approach for the OS case, with a box of spacetime straddling the edge of the dust. Here again there is no flux through the angular surfaces. There is no four momentum flux across the outer surface, but there is inward four momentum flux across the inner surface as the dust falls through that surface. Now examine the past and future surfaces, there is a larger amount of four-momentum flowing in through the past surface than flows out through the future surface.

It is plausible to believe that the reduced amount of four momentum flux across the future surface is exactly equal to the flux across the inner surface, and in fact since we know OS is a solution to the EFE it is guaranteed.

 

[PeterDonis]

the model in question is Schwarzschild vacuum outside of the event horizon surrounded by an OV spacetime.

To be sure I'm clear: basically you are envisioning a model in which we take the diagram I drew, but replace the shaded matter region with a Schwarzschild vacuum region? In other words, the "source" of the OV radiation is now supposed to be Schwarzschild vacuum instead of the matter in my diagram?

If the mass is decreasing at a constant rate

But the rate is not constant within the box you are using, since that box straddles the boundary. The rate is zero in the Schwarzschild region, and nonzero in the OV region.

Also, the $r$ coordinate of the boundary will be decreasing with time since the OV radiation is carrying away mass.

I'm going to need to work the math for this when I get a chance.

 

[Dale]

In other words, the "source" of the OV radiation is now supposed to be Schwarzschild vacuum instead of the matter in my diagram?

Yes

But the rate is not constant within the box you are using, since that box straddles the boundary. The rate is zero in the Schwarzschild region, and nonzero in the OV region.

This mass is the mass parameter of the OV spacetime which is a function of time (or more easily a function of the null parameter). It isn’t something that is in the box.

There does exist a mass function which leads to the flux through the future surface being equal to the flux from the past surface.

 

[PeterDonis]

Yes

Ok, good.

There does exist a mass function which leads to the flux through the future surface being equal to the flux from the past surface.

Again, I need to work through the math when I get a chance. I'm always leery of intuitive arguments in curved spacetimes, because the spacetime geometry changes, and that has to be factored in, since the divergence we're computing is a covariant divergence.

I'll post again when I've had a chance to work the math.

 

[Dale]

I need to work through the math when I get a chance. I'm always leery of intuitive arguments in curved spacetimes

Sounds good. If you do it rigorously, you may find it easier to align the box with the Vaidya $u$ and $v$ (inward and outward null coordinates) rather than aligned with the spherical $r$ and $t$ coordinates that I described verbally. Just make sure that the past outward null surface is entirely in the Schwarzschild region and the future outward null surface is entirely in the Vaidya region, and the calculation should be tractable.

 

[PeterDonis]

Just make sure that the past outward null surface is entirely in the Schwarzschild region and the future outward null surface is entirely in the Vaidya region

I'm not sure I understand, but let me describe the box I'm visualizing--maybe we're thinking of the same thing, just in different words.

The box I'm visualizing has four null sides. Its left corner and bottom corner are in the Schwarzschild region, and its right corner and top corner are in the Vaidya region. So the boundary between the regions passes through the bottom right and top left sides of the box.

 

[PeterDonis]

you may find it easier to align the box with the Vaidya and (inward and outward null coordinates)

Hm--actually that might make your argument much simpler and not even require any math, because all of the flux would be through one side of the box--the top right side of the one I described in my previous post. The flow lines of the null dust are parallel to the bottom right and top left sides of the box, so there is no flux through those sides, and the bottom left side is entirely in the Schwarzschild region, so there's no flux through it. So with that box it's obvious that the divergence can't be zero, even when we factor in the spacetime geometry change, since that won't change which sides of the box flux goes through.

If I've got that right, then you just saved me a bunch of tedious calculation.

 

[PeterDonis]

The flow lines of the null dust are parallel to the bottom right and top left sides of the box

That also makes it simpler for me to see the key difference between this and the OS case: in the OS case, the boundary is itself a fluid flow line! Whereas in this case, of course, it isn't.

That might be a valid general condition for spotting cases where a vacuum to non-vacuum boundary won't work, at least for models where the nonzero stress-energy can be modeled as a fluid.

 

[Dale]

The box I'm visualizing has four null sides. Its left corner and bottom corner are in the Schwarzschild region, and its right corner and top corner are in the Vaidya region. So the boundary between the regions passes through the bottom right and top left sides of the box.

Yes. That will make the calculations easier.

If I've got that right, then you just saved me a bunch of tedious calculation.

You are welcome!