I Stress-Energy of Star Collapsing into Black Hole

[timmdeeg]

As a black hole is described by a vacuum solution of Einstein's field equations the stress-energy tensor vanishes identically zero. If one assumes the mass of a black hole to be "within" the singularity then it seems there is no sensible way to apply the stress-energy tensor to a point.

So what happens to the initial stress-energy after the horizon is formed? Can it just be assigned to the spacetime curvature of the black hole?

 

[Ibix]

So what happens to the initial stress-energy after the horizon is formed? Can it just be assigned to the spacetime curvature of the black hole?

All the matter ends up in the singularity - which is to say, we don't know what actually happens to it because the singularity is GR's maths giving up. So the stress-energy isn't assigned to anything in spacetime - you just have a singularity whose mass can be deduced from the spacetime curvature.

It's analogous to a point mass in Newtonian gravity - an object of zero size and infinite density with a finite mass doesn't really make physical sense, but the gravitational field is just fine. Black hole structures aren't point masses, but similarly have a thing with implausible physical characteristics that nevertheless generates a plausible field.

 

[timmdeeg]

All the matter ends up in the singularity - which is to say, we don't know what actually happens to it because the singularity is GR's maths giving up. So the stress-energy isn't assigned to anything in spacetime - you just have a singularity whose mass can be deduced from the spacetime curvature.

The singularity is not part of the spacetime, its rather a point in time. Should the stress-energy be assigned to the singularity from a purely formal point of view (having in mind that it doesn't make sense physically)?

What is the best way to express the "fate" of the stress-energy ?

 

[Ibix]

What is the best way to express the "fate" of the stress-energy ?

You can't express it, not until we get a better theory of gravity. The model says that all infalling matter ends up in the singularity, which most physicists take to mean "it does something we don't know how to describe yet".

 

[phyzguy]

Although I would add that since the singularity is within the event horizon, and there is no way we can observe inside the event horizon, in some sense it doesn't matter. The interior of the event horizon is no longer part of our observable universe.

 

[timmdeeg]

@phyzguy Hm, does "within the event horizon" fit with the notion that the singularity is not a part of the manifold?

You can't express it, not until we get a better theory of gravity. The model says that all infalling matter ends up in the singularity, which most physicists take to mean "it does something we don't know how to describe yet".

Yes I understand that.

Is it at least not false saying that according to present theory (until we know better) the stress-energy is not a part of the manifold, resp. not a part of the spacetime?

 

[Ibix]

Is it at least not false saying that according to present theory (until we know better) the stress-energy is not a part of the manifold, resp. not a part of the spacetime?

Yes. As I understand it, singularities are not part of spacetime and what happens there is beyond the theory.

It's worth noting that the usual models of eternal black holes (Schwarzschild, Kerr, etc) don't have "initial stress energy". They always existed and the question doesn't make sense. There are semi-realistic models of black hole formation, such as Oppenheimer-Snyder, in which a spherically symmetric mass distribution collapses into a black hole. If you want to know why the curvature is what it is at a given event then it can only depend on stress-energy in the past lightcone of that event, since nothing is superluminal. But the past lightcone of an event outside the horizon cannot include anything inside the horizon - so the curvature now can always be traced back to the matter distribution somewhere outside the horizon in the past. And hence what happens inside the horizon is irrelevant.

This becomes slightly doubtful when one considers evaporating holes, some of which don't have true horizons. In the very far future, exactly what happens in a black hole may be very relevant to observers outside - but not today.

 

[Ibix]

The interior of the event horizon is no longer part of our observable universe.

This isn't quite right, I think. Or at least there's a distinction you need to be careful with. Observing inside the horizon is easy: fall in. This is rather different to the unobservable parts of FLRW spacetime, which we can never see, not even for a short time before our deaths.

 

[PeterDonis]

This is rather different to the unobservable parts of FLRW spacetime, which we can never see

Note that we can never see them even if we get into a rocket and fly as fast as we possibly can in their direction; there is no analogue to "just fall into the black hole" that allows us to see these regions in FLRW spacetime.

 

[timmdeeg]

It's worth noting that the usual models of eternal black holes (Schwarzschild, Kerr, etc) don't have "initial stress energy". They always existed and the question doesn't make sense.

Yes but the black hole originating from a gravitational collapse and the eternal black are both described by the vacuum solution and thus shouldn't be any different. So the question regarding the stress-energy (related to $M$) shouldn't be different too? Or am I missing something?

 

[PeterDonis]

the black hole originating from a gravitational collapse and the eternal black are both described by the vacuum solution

No, this is not correct. The solution for the eternal black hole is vacuum everywhere. The solution for the black hole originating from gravitational collapse is not: the spacetime region occupied by the object that collapses to form the hole is not vacuum.

So the question regarding the stress-energy (related to ) shouldn't be different too?

Certainly it is: a spacetime that is vacuum everywhere (zero stress energy everywhere) is not the same as a spacetime that is only vacuum in a portion, and has nonzero stress-energy in another portion.

 

[timmdeeg]

No, this is not correct. The solution for the eternal black hole is vacuum everywhere. The solution for the black hole originating from gravitational collapse is not: the spacetime region occupied by the object that collapses to form the hole is not vacuum.

Ok, I see the difference, thanks. Can we assign stress-energy to an eternal black hole somehow corresponding to its mass?

 

[PeterDonis]

Can we assign stress-energy to an eternal black hole somehow corresponding to its mass?

No. Vacuum means vacuum: zero stress-energy. The mass of the eternal black hole is a global property of the spacetime geometry; there is no stress-energy associated with it.

 

[timmdeeg]

Thanks for this enlightening answer.