B Exploring the Merger of 2 Black Holes: A 4th Dimensional Perspective

[timmdeeg]

If it were a particular location, then you could draw a timelike worldline for it in a spacetime diagram. Try this with a Kruskal diagram and you'll quickly see the problem.

Ah, that's a good hint. Thanks.

 

[PeterDonis]

Is it correct that this statement is true for both, "idealized" and "real" (containing stress energy) black holes?

You keep saying "real (containing stress-energy)". Can you clarify the distinction you are making?

it seems to me that the "mass distribution" is "spread out" in the center of the black hole in order to avoid the singularity

No. In the models @kimbyd is referring to, there are quantum field effects that come into play when spacetime curvature gets sufficiently large; those effects can be thought of, in the classical approximation, as creating an effective stress-energy that violates the energy conditions on which the classical GR singularity theorems are based. But this effective stress-energy isn't anything like a "mass distribution" (because it violates the energy conditions).

isn't the center a "particular location"?

No. A "location", in geometric GR terms, means "a timelike worldline" (or a set of closely spaced timelike worldlines). The singularity at $r = 0$ in the Schwarzschild geometry is spacelike, not timelike. The closest intuitive concept to such a thing is a moment of time, not a location in space.

 

[timmdeeg]

You keep saying "real (containing stress-energy)". Can you clarify the distinction you are making?

I referred to your post 36 " A realistic solution would contain a region of nonzero stress-energy, joined by a boundary to a vacuum region with Schwarzschild or Kerr geometry. An example of such a solution (still idealized, but less so than the pure vacuum solutions) is the Oppenheimer-Snyder model of a spherically symmetric collapsing object. Even in such a solution, an observer falling through the event horizon long after the collapse of the object will not pass through any region of nonzero stress-energy; that was the scenario I was describing in my previous post."
and am recognizing now (very late), that you implicitely said that there is no timelike worldline into the region of "nonzero stress-energy". This clarifies my misconception in this matter and I think one should avoid to talk about "mass distribution" in this context at all. Thanks for your efforts.

 

[PeterDonis]

I referred to your post 36

Ok. But just to clarify, in the model I was describing in that post, all of the stress-energy is far in the past. It is not "inside the black hole" except in a very limited sense. It certainly cannot be thought of as spread out through the interior of the black hole and providing its mass, the way it would in an ordinary object like a planet or star.

you implicitely said that there is no timelike worldline into the region of "nonzero stress-energy".

Not if you fall into the hole after it forms, no. If you happened to be sitting inside the star that originally collapsed to form the black hole, you could fall in and be inside the region of nonzero stress-energy. But not after the hole is formed.

 

[timmdeeg]

Ok. But just to clarify, in the model I was describing in that post, all of the stress-energy is far in the past. It is not "inside the black hole" except in a very limited sense. It certainly cannot be thought of as spread out through the interior of the black hole and providing its mass, the way it would in an ordinary object like a planet or star.

Yes, important point, thanks for clarifying. And after rereading your posts it seems that the Oppenheimer-Snyder model is more realistic that the Schwarzschild black hole.

 

[PeterDonis]

it seems that the Oppenheimer-Snyder model is more realistic that the Schwarzschild black hole.

A better way of putting this would be that the Oppenheimer-Snyder model is more realistic that the maximally extended Schwarzschild geometry (where "maximally extended" means the spacetime is vacuum everywhere, no stress-energy anywhere, and the black hole is "eternal", with both a future and a past singularity--as in the Kruskal diagram). The vacuum region of the Oppenheimer-Snyder model is a "piece" of the Schwarzschild geometry; it just doesn't contain all of it.

 

[timmdeeg]

The vacuum region of the Oppenheimer-Snyder model is a "piece" of the Schwarzschild geometry; it just doesn't contain all of it.

According to this source
http://www.aei.mpg.de/~rezzolla/lnotes/mondragone/collapse.pdf
the interior "piece" resembles a collapsing FRW-universe with constant positive curvature. I was perhaps erroneously thinking that the Oppenheimer-Snyder model avoids the singularity, but the radius of the dust shell goes to zero in finite time. So, GR should break down an $R=0$ also here. But I'm not sure, kindly correct.

 

[PeterDonis]

the interior "piece" resembles a collapsing FRW-universe with constant positive curvature

That's correct. More precisely, it's a portion of such a universe (because the matter region stops at the surface of the collapsing star and does not cover the entire "universe" of the FRW model).

 

[PeterDonis]

I was perhaps erroneously thinking that the Oppenheimer-Snyder model avoids the singularity, but the radius of the dust shell goes to zero in finite time.

That's correct, the O-S model does have a singularity, so it is subject to all the concerns about singularities in GR.

 

[timmdeeg]

That's correct, the O-S model does have a singularity, so it is subject to all the concerns about singularities in GR.

Which clarifies that there is stress energy only during the collapse, thanks for new insights.