Stellar-mass black hole formation sequence

  • #26
That's the point I'm questioning.
What point? That whatever quantum gravity effects become important at the singularity won't affect the formation of the horizon?
That the horizon will invariably form (and by "form" I mean "have enough matter fall into it to consist of a black hole even if all the matter still-outside of it suddenly disappeared") at a size much larger than quantum gravitational effects could matter.

No. You are still thinking about things from a static viewpoint--that's the viewpoint that says ingoing radiation gets blueshifted and outgoing radiation gets redshifted. But a collapsing configuration is not static, and your intuitions about a static viewpoint don't work.

Just as one example: a static observer very close to a static black hole's horizon sees incoming radiation highly blueshifted; but an infalling observer falling past that static observer sees incoming radiation redshifted, not blueshifted. This is still a static situation overall, so it doesn't fully capture what is going on in a collapsing star; but at least it illustrates that static intuitions can't be applied to infalling objects.
Ah, I see. I think I'm getting a better picture of things.

Would it perhaps be accurate to say something like "the solutions to the Oppenheimer-Snyder model of collapse prevent a discontinuous event horizon; all the components of a gravitationally collapsing object will perceive the event horizon swallowing them up simultaneously"? Or something along those lines?

Of course, I'm not interested in the event horizon for its own sake; I'm really wanting to know at what point the conditions for the creation of Hawking radiation are met. Because Hawking radiation can't be teleological. I guess this is a restatement of the firewall problem....
 
  • #27
PeterDonis
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That the horizon will invariably form (and by "form" I mean "have enough matter fall into it to consist of a black hole even if all the matter still-outside of it suddenly disappeared") at a size much larger than quantum gravitational effects could matter.
You mean, at a density much smaller than that at which quantum gravity effects could matter. According to our best current understanding of quantum gravity effects, yes, this is true; the density at ##r = 0## when the horizon forms there is much, much smaller than the Planck density. To put it another way, if we imagine an observer sitting at ##r = 0## and watching the density increase with time, the event on his worldline at which, if he emits a light signal, it will be just at the event horizon, has to be significantly earlier, according to his own clock, than the event at which the density reaches the Planck density (which will be just before the entire collapsing object reaches zero volume and the singularity forms).

Would it perhaps be accurate to say something like "the solutions to the Oppenheimer-Snyder model of collapse prevent a discontinuous event horizon
I'm not sure what you mean by a "discontinuous event horizon". The event horizon, considered as a boundary between two regions of spacetime, has to be a continuous 3-surface between two 4-volumes; it's not possible for it to be disconnected. That isn't a feature just of the O-S model; it's a requirement for any event horizon, no matter how it is formed. (More precisely, it's a requirement for any event horizon formed by the collapse of a single isolated system; there can be multiple such systems in the universe that never come together. But we're just talking about a single isolated system here.)

all the components of a gravitationally collapsing object will perceive the event horizon swallowing them up simultaneously"?
No. This can't be true, because the event horizon is a null surface, and different parts of the object cross it at different events, so those events must be null separated. Null separated events cannot be simultaneous in any coordinate chart; only spacelike separated events can be simultaneous (and only if you choose the appropriate coordinate chart).

(Btw, if you think about it, you will see that this is part of the reason why the statement I made about the density at ##r = 0## must be true. The other part is that, once collapsing matter crosses the horizon, it's on a timelike worldline, so the event of it crossing the horizon and the event of it reaching the singularity are timelike separated.)

I'm really wanting to know at what point the conditions for the creation of Hawking radiation are met.
I think the current best guess for that is that Hawking radiation is generated at outgoing trapped surfaces--these are 2-spheres at which radially outgoing light stays at the same radius instead of moving outward. An event horizon does not have to be always associated with an outgoing trapped surface; in the O-S model, an outgoing trapped surface only forms when the surface of the collapsing object falls below the event horizon. From then on, assuming nothing else falls in, the event horizon and the outgoing trapped surface coincide. But before that, there is no outgoing trapped surface anywhere, at least as I understand the model.
 
  • #28
You mean, at a density much smaller than that at which quantum gravity effects could matter. According to our best current understanding of quantum gravity effects, yes, this is true; the density at ##r = 0## when the horizon forms there is much, much smaller than the Planck density. To put it another way, if we imagine an observer sitting at ##r = 0## and watching the density increase with time, the event on his worldline at which, if he emits a light signal, it will be just at the event horizon, has to be significantly earlier, according to his own clock, than the event at which the density reaches the Planck density (which will be just before the entire collapsing object reaches zero volume and the singularity forms).
So there cannot be a case in which core collapse takes place with a high enough central density that the 9/8 condition would be met at a very very low (subatomic) radius?

I think the current best guess for that is that Hawking radiation is generated at outgoing trapped surfaces--these are 2-spheres at which radially outgoing light stays at the same radius instead of moving outward. An event horizon does not have to be always associated with an outgoing trapped surface; in the O-S model, an outgoing trapped surface only forms when the surface of the collapsing object falls below the event horizon. From then on, assuming nothing else falls in, the event horizon and the outgoing trapped surface coincide. But before that, there is no outgoing trapped surface anywhere, at least as I understand the model.
I highlighted the point I was particularly interested in, because that's really where the crux of my question (and this whole thread) lies.

If Hawking radiation is generated from an outgoing trapped surface (OTS) as soon as an OTS forms, and an OTS forms as soon as an object falls within its own event horizon (presumably in its own reference frame), then what is the earliest point in the collapse of a star that Hawking radiation can be generated?

In particular, if the surface of the core has fallen within the core's own event horizon but the rest of the star is only beginning to collapse, can Hawking radiation be generated from an OTS at the core event horizon before the rest of the star has collapsed very much?
 
  • #29
PeterDonis
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So there cannot be a case in which core collapse takes place with a high enough central density that the 9/8 condition would be met at a very very low (subatomic) radius?
The 9/8 condition applies to a static object. A collapsing object is not static. You need to stop trying to apply static reasoning to a non-static situation.

what is the earliest point in the collapse of a star that Hawking radiation can be generated?
When the outer surface of the star reaches the event horizon. In other words, when all of the matter that is going to collapse into the forming black hole has reached the event horizon. There is no OTS until then.

if the surface of the core has fallen within the core's own event horizon but the rest of the star is only beginning to collapse, can Hawking radiation be generated from an OTS at the core event horizon before the rest of the star has collapsed very much?
Basically, it looks like you're envisioning a scenario where we have a core that's collapsing, then a significant region of empty space, and then the rest of the star. In that case, yes, you could view the collapse process in two separate stages: first the core collapses to a black hole, with a region of empty space around it, and Hawking radiation begins when all of the matter of the core has reached the horizon of this initial hole.

Then, some time later, an additional shell of matter falls into the hole. This is a different kind of process and a different kind of spacetime model than we have been talking about, and we should defer discussion of it until we've got the original model clear.

The question you appear to be asking is, how small could the core be in this type of scenario? And the answer is, unless it is larger than the maximum mass limit for a neutron star (1.5 to 3 solar masses), it won't collapse to a black hole. And Hawking radiation for a hole of that size is completely negligible. So Hawking radiation can't have a significant effect on such a collapse. If the object is below the maximum mass limit for a neutron star, it will form a neutron star (it could even form a white dwarf if it is below the maximum mass limit for that), and no Hawking radiation will be generated at all.
 
  • #30
The 9/8 condition applies to a static object. A collapsing object is not static.
Ah, yes, my bad. Forget I put it in those terms.

Basically, it looks like you're envisioning a scenario where we have a core that's collapsing, then a significant region of empty space, and then the rest of the star. In that case, yes, you could view the collapse process in two separate stages: first the core collapses to a black hole, with a region of empty space around it, and Hawking radiation begins when all of the matter of the core has reached the horizon of this initial hole.

Then, some time later, an additional shell of matter falls into the hole. This is a different kind of process and a different kind of spacetime model than we have been talking about, and we should defer discussion of it until we've got the original model clear.
Well, that's the process I've been aiming for, more or less. You don't have to have a region of empty space per se; it can simply be a region of lower density. All you need to meet this condition is that once neutron degeneracy pressure breaks down and the collapse begins, the core collapses more rapidly than the outer layers.

Is there some aspect of the Oppenheimer-Snyder model which would prevent the core collapse from "outrunning" the collapse of the rest of the star?
 
  • #31
PeterDonis
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You don't have to have a region of empty space per se; it can simply be a region of lower density.
I suggest taking some time to work out the math. It isn't as simple as you are assuming it is.

(As I note below, there is no known analytical solution to the differential equations for the case of nonzero pressure; but you can still look at the equations themselves and work out some qualitative features.)

Is there some aspect of the Oppenheimer-Snyder model which would prevent the core collapse from "outrunning" the collapse of the rest of the star?
Yes; but that aspect is that the O-S model assumes zero pressure, so it eliminates the only possible thing that could slow down the collapse of any part of the star.

AFAIK there is no analytical solution for the case of nonzero pressure, so the only way to study that case would be to do so numerically. I know such numerical simulations have been done, but unfortunately I'm not familiar enough with the numerical relativity literature to be able to point to specific results.
 

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