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

[kimbyd]

During the visit at LIGO Lousiana quite recently (organized by the german Journal "Bild der Wissenschaft" for our group) there was the opportunity to ask the scientist who spoke to us what happens with regard to the singularities after the two black holes have "just touched". He confirmed that this is an open question yet and that the answer should be somehow encoded in the ring down signature. I failed however to ask whether it is possible to at least in principle calculate the expected signatures of what happens here or if one attempts to search for a not yet existing theoretical foundation of this high-energy regime by analizing the measured ring down signature.
Any comment?

That seems really, really unlikely to me. At least in classical General Relativity, precisely zero information about the behavior of matter inside the event horizon can be gleaned from measuring anything outside of the event horizon.

I suppose if there were some measurable difference from the prediction of General Relativity that could be measured in the ringdown, that might provide further insight. But that's really a shot in the dark. There's a fair chance that no deviation from GR will be measured during BH-BH mergers.

 

[timmdeeg]

That seems really, really unlikely to me. At least in classical General Relativity, precisely zero information about the behavior of matter inside the event horizon can be gleaned from measuring anything outside of the event horizon.

I suppose if there were some measurable difference from the prediction of General Relativity that could be measured in the ringdown, that might provide further insight. But that's really a shot in the dark. There's a fair chance that no deviation from GR will be measured during BH-BH mergers.

The question was what happens to the mass distribution, taking about the merger of "real black holes" containing stress energy. I think the vacuum solution based on General Relativity can't answer that. But it provides information about the tilted lighcones inside the horizons, so what do you think happens to those after the BHs have just merged? If I see it correctly such information if existing should at least provide hints regarding the behaviour in time of the masses in their centers.

From the short discussion we had at LIGO it seemed to me however that these questions are hardly solvable theoretically, whereby the speaker mentioned inter alia the problem of "simultaneity convention" as discussed by @pervect in #21. So hopefully the ring down date will provide insights on what goes on with regard to the mass distribution..

 

[PeterDonis]

"real black holes" containing stress energy

"Real" black holes are still vacuum; they don't contain stress-energy.

I think the vacuum solution based on General Relativity can't answer that.

All of the calculations being compared with the LIGO observations are using the vacuum solution based on GR. So this belief of yours is simply false.

it provides information about the tilted lighcones inside the horizons, so what do you think happens to those after the BHs have just merged?

They are still tilted, inside the merged horizon.

the behaviour in time of the masses in their centers

Your picture of black holes is incorrect. They are not vacuums with horizons and "masses in their centers". They are vacuum solutions all the way down. The singularities at the centers are moments of time that are to the future of all events inside the horizon; they are not places in space containing mass.

 

[timmdeeg]

"Real" black holes are still vacuum; they don't contain stress-energy.

I meant "real black hole" in the sense you used

A real black hole that forms by gravitational collapse will have some nonzero stress-energy inside the horizon, if we consider the entire 4-dimensional spacetime region inside the horizon:

here.

All of the calculations being compared with the LIGO observations are using the vacuum solution based on GR. So this belief of yours is simply false.

This belief considers "real black holes", see above.

They are still tilted, inside the merged horizon.

I have the notion that the newly formed black hole is largely deformed and therefor the tilting is location-dependent in contrast to the tilting in a spherical symmetric black hole (*).

Your picture of black holes is incorrect. They are not vacuums with horizons and "masses in their centers". They are vacuum solutions all the way down. The singularities at the centers are moments of time that are to the future of all events inside the horizon; they are not places in space containing mass.

Do you agree that if we talk about the merger of "real black holes" (see above) we then implicitly assume a mass distribution which changes over time? The signature LIGO receives originates from the merger of "real black holes" and said discussion regarding the ring down was based on that and wouldn't have made sense otherwise.

(*) I'm aware of that in this case the tilting of the light cones varies with $r$. But is the same for $r=const.$

 

[Tom Mcfarland]

Kimbyd (#38):
It's very conceivable that quantum effects differ strongly from GR right up to the horizon itself.

reply by PeterDonis (#38):
If this is the case, it is almost certainly also the case that quantum gravity effects prevent an event horizon from forming at all.
If this is the case, there is no such thing as a "black hole" in the sense of a region inside an event horizon. There could still
be apparent horizons--surfaces where outgoing light locally stays in the same place--but they would not be absolute (event) horizons

Earlier, PeterDonis (#19) drew an analogy between BH mergers in space-time and slicing up trousers, with a (viewer's) time axis parallel to the trouser leg axes.
In the case of merging black holes, the slices orthogonal to the time axis were 3-manifolds containing two orbiting 3-dimensional BHs in the process of merging.

PeterDonis ((#14)
the gravity you would feel "now" from the hole is coming from very, very far in the past--from the collapsing matter that formed the hole,
just before it reached the event horizon

From our perspective as a distant viewer of merging BHs ("seeing" electromagnetic or gravitational wave evidence) , the view
would likewise be from the past states of the system (lower down Peter's trousers).. What might appear to the distant observer as a
culminated merger would rather be the progressively red-shifted and attentuated evidence arriving at the viewer from before the BHs formed or merged

That leaves the topology of the time slices unresolved for the upper part of Peter's trousers. I would prefer to accept an existential
approach, rather than manipulate metrics,, that is, if we are forever unable to view evidence of a consummated merger, then the merger does not occur,
which adds a 2-way symmetry to behavior across any hypothetical event horizon, and God likes symmetry, right?

What we can observe, then, is gravity quickly "sewing together" the boundaries or horizons of the BHs without merging the interiors, creating a 3-sphere
as the topology of the upper part of PeterDonis' trousers, rather than merely a bigger 3-ball. The new space-time is now 5-dimensional (4 space and one time).

 

[PeterDonis]

Do you agree that if we talk about the merger of "real black holes" (see above) we then implicitly assume a mass distribution which changes over time?

Not in the sense you mean. The "mass" of a black hole is not a local property; you can't look at particular locations and say that some of the mass is here, some is there, etc. The "mass" of a black hole is a global property of the spacetime geometry.

In the case of two black holes merging, seen from far away, there is just a single system with a single mass. The merger is an internal detail of the system. It is true that mass (energy) can be carried away from the system by radiation--for example, the gravitational waves produced by the merger--and that this can be detected from far away (basically, as you see the radiation pass you, you also see the mass of the system decrease, as shown for example by the change in orbital parameters of a test object orbiting the system). So in that sense, the mass of the system does change with time. But making that precise is more complicated than you appear to recognize.

 

[PeterDonis]

That leaves the topology of the time slices unresolved for the upper part of Peter's trousers

How so?

I would prefer to accept an existential
approach, rather than manipulate metrics,, that is, if we are forever unable to view evidence of a consummated merger, then the merger does not occur,

Sorry, you don't get to pick and choose here. You have to solve the Einstein Field Equation with appropriate initial conditions. The scientists who run LIGO have done this (numerically, using supercomputers, since the equations in question can't be solved analytically), and we now have three observed events which match up nicely with the theoretical solutions, with appropriate choices of parameters.

What we can observe, then, is gravity quickly "sewing together" the boundaries or horizons of the BHs without merging the interiors, creating a 3-sphere
as the topology of the upper part of PeterDonis' trousers, rather than merely a bigger 3-ball.

I have no idea how you are coming up with this. The "trousers" are just a way of describing the shape of the horizon in the case of a black hole merger; that shape is a shape in 4-d spacetime (in the "trousers" visualization I suppressed one space dimension, but that is the same for the "legs" part and the "upper" part of the trousers, nothing changes when the legs merge).

 

[kimbyd]

Kimbyd (#38):
It's very conceivable that quantum effects differ strongly from GR right up to the horizon itself.

Right, but as long as they don't differ anywhere outside the horizon, no deviation can be detected.

It's possible for quantum effects to change the behavior of the horizon (especially if quantum effects make it so that the horizon isn't real). But we'd have to get lucky.

 

[timmdeeg]

Not in the sense you mean. The "mass" of a black hole is not a local property; you can't look at particular locations and say that some of the mass is here, some is there, etc. The "mass" of a black hole is a global property of the spacetime geometry.

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

I don't think that the point that GR is incorrect some distance outside the singularity is very controversial: there's likely no way to avoid the singularity without having a mass distribution spread out over some finite region of space.

From my perspective this conclusion made by kimbyd in #37 seems convincing. But then it seems to me that the "mass distribution" is "spread out" in the center of the black hole in order to avoid the singularity. If correct so far, isn't the center a "particular location"?

 

[Nugatory]

But then it seems to me that the "mass distribution" is "spread out" in the center of the black hole in order to avoid the singularity. If correct so far, isn't the center a "particular location"?

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.

 

[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.