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

[Tom Mcfarland]

LIGO is reputed to have detected gravitational waves from the merger of 2 black holes (BHs).

For an external observer of such an event, each BH would appear to approach the event horizon of the other, but never cross it in a finite time...is this correct so far? But the event horizons are roughly spheres, so this approach would appear to force at least a small intrusion of one or both BHs into a 4th spatial dimension, the merged object appearing as a topological 3-sphere to the external observer.

By analogy, in a 2-D universe, the merger of 2 disk-shaped BHs at their circular event horizons would
force the merged object into a 3rd spatial dimension, forming a 2-sphere (a balloon).

Am I imagining this scenario correctly, or if not, why not?

 

[Orodruin]

is this correct so far?

No. You need to be much more careful with what you mean by "finite time". Your language seems to suggest some assumption of an absolute time, which does not exist. Furthermore, a black hole merger is very far from the static solution to the EFEs that describes a Schwarzschild black hole.

 

[Tom Mcfarland]

Hmmm...admittedly I am not a trained cosmologist, but I thought I had used the same care which I have seen Susskind use.
That is, I am sitting at some point (without assuming any absolute coordinates) and at some other location I observe what
appears to be 2 black holes in the process of merging. At each stage of the merger, time is measured on my timepiece.
Also, I do not know what EFEs are. Am I resigned to abandon this question?

Tom McFarland

 

[kimbyd]

LIGO is reputed to have detected gravitational waves from the merger of 2 black holes (BHs).

For an external observer of such an event, each BH would appear to approach the event horizon of the other, but never cross it in a finite time...is this correct so far? But the event horizons are roughly spheres, so this approach would appear to force at least a small intrusion of one or both BHs into a 4th spatial dimension, the merged object appearing as a topological 3-sphere to the external observer.

What happens is the two black holes merge and form a single, larger black hole. The entire process is over in a few seconds once the black holes touch.

 

[Tom Mcfarland]

If the merger involves a BH and a chunk of normal matter S, like a star, then the external observer will see S approach the BH, moving ever more slowly as S nears the event horizon, but the external observer will never see S cross (or touch) the event horizon. If we now replace S with another BH, I would have (naïvely?) expected similar behavior, but now with more symmetry. Some kind of merged object results, with more mass, but if an external observer never sees the 2 event horizons cross, then what is the nature of the merged object? Will the math answer the question?...I do not have the needed math skills. However, an external observer would see two BHs with disjoint interiors but with event horizons which were very close together. Such a merged object is a 3-sphere and can only reside in 4-space, which string theory says must be there, but is degenerate.

Cheers, Tom McFarland

 

[kimbyd]

There is one major difference between a black hole and any other object: the black hole has no proper surface. The event horizon isn't anything physical. It's just a location past which nothing can return. You can't actually observe the event horizon in any meaningful sense: you can only observe matter fall into it (or rather, the effects of that happening).

So when two black holes merge, the horizons touch, quickly turning into a sort of barbell shape that then very rapidly vibrates until it settles into a new event horizon that's larger than either of the two original black holes' horizons.

 

[rootone]

Put a couple of drops of liquid Mercury on a surface that is slightly curved, so they are bound to shortly collide.
It's like that.

 

[Tom Mcfarland]

Kimbyd, you make a good point that I did not notice...the event horizon is not the equivalent of a physical object.

However, I would like to argue that an external observer still will not see two event horizons "touch" because the observed timing of movement near the two event horizons will be increasingly dilated (stretched) the closer the horizons get. For the external observer, the two event horizons ought to appear to hover near to each other, but this can only happen in 4-space. I am having trouble imagining what such a 4-dimensional configuration would look like to a 3-D external observer. Would you like to try?

Cheers, Tom McFarland

 

[Nugatory]

If the merger involves a BH and a chunk of normal matter S, like a star, then the external observer will see S approach the BH, moving ever more slowly as S nears the event horizon, but the external observer will never see S cross (or touch) the event horizon.

That is one of the most common and pervasive misunderstandings about falling into a black hole. It is true that the external observer will never see the infalling object reach and pass the event horizon; it will be redshifted to invisibility so will simply disappear instead.

However, the external observer will eventually (and fairly quickly, at that) observe that the mass of the black hole has increased by the mass of the infalling object, with commensurate increase in the Schwarzschild radius of the black hole. In most presentations of black holes, this effect is completely ignored because the infalling mass is assumed to be negligible compared with the mass of the black hole (exercise: how much does the Schwarzschild radius of a ten-solar-mass black hole change when you drop a 100-kilogram astrophysicist into it?)... But that's not the case when we're dropping one black hole into another.

 

[Tom Mcfarland]

Nugatory:

You write "the external observer will eventually (and fairly quickly, at that) observe that the mass of the black hole has increased by the mass of the infalling object"

Are you claiming that information about the change in mass can travel faster than the (redshifted) light which an external observer uses to observe the change?

 

[Vanadium 50]

travel faster than the (redshifted) light

Redshifted light still travels at c.

 

[Tom Mcfarland]

My point was that time dilation near the event horizons will give an external observer (using light) the vision of two event horizons drawing ever closer, ever more slowly, and never touching in a finite time.

Yes,the external observer sees the red-shifted light traveling with speed c, but time dilation near the horizon only allows light from that location to arrive at the external observer much more slowly. Nugatory seems to claim that information on BH mass can somehow arrive faster than it would if that same information had been carried by light ??

 

[PeterDonis]

Moderator's note: Moved to relativity forum.

 

[PeterDonis]

Are you claiming that information about the change in mass can travel faster than the (redshifted) light which an external observer uses to observe the change?

No. The information about the change in mass is not coming from the black hole event horizons. It is coming from the past, from the objects that originally formed the hole.

This is actually the case for ordinary massive objects, but we normally don't notice it because our normal experience is with situations that are so close to static that the light speed travel time doesn't make a dfiference. For example, the effect of the Sun's mass on the Earth's orbit is not due to the Sun "now"; it's due to the intersection of the Sun with the Earth's past light cone. But that's only 500 seconds ago (by Earth clocks), and the changes in that time are very small anyway (though there are observable effects due to this, such as the perihelion shift of the orbits of the planets, which I believe has now been measured for all the planets out to Mars). And the gravitational redshift due to the Sun's mass has negligible effect on all of this.

In the case of a black hole, things are very different. Because of the extreme redshift of light coming from just above the hole's horizon, and the accompanying time delay, 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. A little bit in the future from now, you will be feeling the gravity from that collapsing matter a little bit closer to the horizon--and so on indefinitely into the future. But you will never feel gravity from the collapsing matter inside the horizon, for the very reason you give: gravity can't travel faster than light. (Note that this is still a heuristic picture, and has a number of simplifications in it, but it's the best I can do in a "B" level thread.)

Similar remarks apply to the case of two holes merging.

 

[kimbyd]

Kimbyd, you make a good point that I did not notice...the event horizon is not the equivalent of a physical object.

However, I would like to argue that an external observer still will not see two event horizons "touch" because the observed timing of movement near the two event horizons will be increasingly dilated (stretched) the closer the horizons get. For the external observer, the two event horizons ought to appear to hover near to each other, but this can only happen in 4-space. I am having trouble imagining what such a 4-dimensional configuration would look like to a 3-D external observer. Would you like to try?

Cheers, Tom McFarland

I don't think this is accurate at all. I'm not really sure precisely what the merger would look like, but I do know the beginning and end states of the merger.

First, it may be useful to listen to this, which is the waveform that was detected from the merger seen two years ago:

(Note: the description below the video is pretty comprehensive, and should clear up any questions about it you might have)

That "blip" at the end of the wave is what happens when the event horizons touch (I don't know the precise moment that the touch happens, it might be right at the peak or a little before, but probably not after). After that touch, the gravity waves become undetectable within a fraction of a second. After that time, the resulting black hole has settled into a nearly-spherical shape. If the touch itself cannot be observed, how would it be possible to observe the nearly-spherical resulting black hole? There has to be some sort of transition between those two states.

 

[Tom Mcfarland]

Kimbyd and Peter Donis:

I would like to thank you gentlemen for your patience in expressing your thoughts in language I can understand. I am a bit disappointed in losing the case for the merger of 2 BHs being a 3 sphere in 4-space, but honestly, I think you folks have a stronger argument. I see no way to further argue in favor of a 3-sphere as the topology for 2 merged BHs, in spite of it being such a beautiful concept.

Goodbye for now, Tom McFarland

 

[pervect]

LIGO is reputed to have detected gravitational waves from the merger of 2 black holes (BHs).

There are two different of gravitational waves reported in the literature, that I'm aware of. There are numerous papers on both, some of the peer-reviewed papers would be https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102 and https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.241103. So yes, they've been detected - twice. And the gravitational waves have been attributed to black holes, and not other compact massive objects, based on the characteristics of the recived signal and using General Relativity as a model.

For an external observer of such an event, each BH would appear to approach the event horizon of the other, but never cross it in a finite time...is this correct so far?

"Never cross it" is rather a subjective concept, and is probably not entirely accurate depending on one's interpretation of "never." For instance, Schwarzschild coordinates never assign a time coordinate to the event of an infalling test particle reaching the event horizon of a black hole, putting the time coordinate at infinity. Other coordinates would assign finite time coordinates to this event, and hypothetical clocks on test particles falling into the event horizon would have finite readings.

The binary inspiral case is more complicated in detail than a test particle falling into the Schwarzschild black hole, requiring numerical simultaions to analyze, but the same basic idea is there.

The issues associated with "never" can be avoided by looking at the General relativistic predictions for a signal emitted from a binary inspiral. Regardless of coordinates use, everyone predicts that the received signal will initially increase in amplitude, as the two black holes approach each other, then reach a peak, then start to die off.

The logical necessity of the die-off follows from the fact that no signal can escape the event horizon of a black hole. So if we use a viewpoint where we assign finite coordinates to the event (which is helpful in analyzing it!), using this viewpoint we can say that the event "happens", but the signals from the actual crossing of the event horizon can't reach us. But signals are emitted before the event horizon crossing takes place, and we can analyze and measure those.

 

[timmdeeg]

The logical necessity of the die-off follows from the fact that no signal can escape the event horizon of a black hole. So if we use a viewpoint where we assign finite coordinates to the event (which is helpful in analyzing it!), using this viewpoint we can say that the event "happens", but the signals from the actual crossing of the event horizon can't reach us. But signals are emitted before the event horizon crossing takes place, and we can analyze and measure those.

Lets consider the very short time to reach spherical symmetry of the new black hole after the event horizons of the two black holes have just "touched" each other.
I'm reasoning whether it makes a difference regarding the redshift of gravitational waves if they are emitted locally close to the event horizon or by the extended system consisting of two black holes. The radiation stops after spherical symmetry is reached. Is the radiation emitted in said short time period highly redshifted?

 

[PeterDonis]

I see no way to further argue in favor of a 3-sphere as the topology for 2 merged BHs

I haven't actually addressed this point explicitly. Even leaving aside what has already been said, your argument is based on a misconception. The event horizon of a single black hole, as a surface in spacetime, is not a 2-sphere. It's a 3-surface composed of an infinite series of 2-spheres, and the third "dimension" of the surface is null--heuristically, a curve that links the series of 2-spheres together is a null curve. You can think of this surface as a "hypercylinder" (and visualize it as a cylinder, a stack of circles representing the 2-spheres), as long as you keep in mind that the "axial" dimension of the cylinder is null (lightlike). The description of the horizon as a 2-sphere depends on slicing up the cylinder into a stack of 2-spheres and picking out only one of them. But the horizon, as a surface in spacetime, is not just one of those slices; it's the whole cylinder.

When you have two black holes merging, there aren't two event horizons becoming one. There is one event horizon, but its "shape" in spacetime (using the heuristic visualization I gave above) is now like a pair of trousers instead of a cylinder. The description "two horizons merge into one" depends on slicing up the pair of trousers into a stack of individual "layers"; then some layers (the ones "before" the merger) consist of two 2-spheres, and other layers (the ones "after" the merger) consist of only one. But the horizon, as a surface in spacetime, is not any single slice; it's the whole pair of trousers.

 

[kimbyd]

Kimbyd and Peter Donis:

I would like to thank you gentlemen

Not a man.

 

[pervect]

I would think so, but I'm not sure this is a rigorous way of looking at it. The more rigorous approach can be described in words by using the block universe approach, where one assumes that the entire history of the black merger can be described in the block universe, via a metric. The metric, then contains the entire history, past and future, of the merger.

From this metric, containing the entire history of the merger, one computes the signal in "space" as a function of "time", in some small region of space-time far away from the black hole. The problem of splitting space-time into space plus time in a small local region has a generally accepted solution, which can be described concisely by saying that the process involves using projection operators to projecting the 4 dimensional space-time into the appropriate subspaces. (This isn't very detailed, but I hope it's sufficient).

This approach doesn't have any simultaneity conventions for it to depend on, at least not until the very end of the process. At the end of the process, one usually imposes the condition that time is orthogonal to space, and that the different spatial directons are also orthogonal.

When one talks about a scenario where "the black holes just touch", my interpretation of this is that one is imposing some specific simultaneity convention, some specific coordinates, to describe the state of the black hole at some "instant in time", so one can single out a specific instant in time "where the event horizons touch".

One might ask - why be so careful about waiting until the end to spit up space-time into space plus time? What's wrong with doing the split earlier? The short answer here is that while the splitting of space-time into space+time in a small local region is well understood, there are some well-known difficulties in doing so over a larger region. Much depends on exactly what characteristics one demands of the split. The particular assumption that's problematical is that there exists a split that is hypersurface orthogonal. This problem arises in, for example, the rotating disk - where no hypersurface orthogonal split can exist. Similar issues and confusions will happen when one tries to describe what one means by "an instant of time" in the binary inspiral. We don't expect a hypersurface-orthogonal split to exist there, either.

Recall that in my description of how we made the local split for the observer a long way from the black hole who was observing the inspiral,, we did impose an orthogonality condition between time and space.

Without a specific answer to this question, of how to single out some specific hypersurface of the inspiral that corresponds to "the instant when the horizons touch", I feel that it is best to be cautious about proceeding further. To really be sure, one would actually have to rigorously define how one was making this split to be able to describe the inspiral in this manner, and do the necessary calculations using the resulting coordinates. The point is that one doesn't have to do this to figure out what the signal at infinity is - one can defer the issue of the splitting until later, and when one does so, the mechanics of how to do the split are generally well accepted.

This brings up the question of how the calculations for the expected signal out of an inspiral were actually done. There are certainly papers in the literature that describe the process, but I can't say I'm familiar with them. I believe developing the numerical codes and ensuring their accuracy and convergence was a very long and challenging process. I would expect, though, that conceptually they used the general framework I outlined earlier, a framework that can be described as a 4-dimensional "block universe" view of the problem, that doesn't single out specific instants of time for the inspiral process, but instead comes up with an abstract mathematical representation that does not depend on such a split.

 

[PeterDonis]

The more rigorous approach can be described in words by using the block universe approach, where one assumes that the entire history of the black merger can be described in the block universe, via a metric. The metric, then contains the entire history, past and future, of the merger.

This is the basis for my description in post #19.

I would expect, though, that conceptually they used the general framework I outlined earlier, a framework that can be described as a 4-dimensional "block universe" view of the problem, that doesn't single out specific instants of time for the inspiral process, but instead comes up with an abstract mathematical representation that does not depend on such a split.

My understanding is somewhat different. My understanding is that initial conditions are specified on a spacelike hypersurface and then evolved forward in "time" (where this "time" is just a coordinate and is not assumed to have any physical meaning--see below). In other words, there is an implicit "split" of spacetime into space and time that is adopted at the start of the process (this is just a choice of coordinates).Since the values of all actual observables are independent of the choice of coordinates, this is not a problem. However, there is no attempt made, AFAIK, to associate any physical meaning with this split (i.e., with the coordinates); they are chosen for ease of calculation and nothing else. So I think the possible issue with the OP is not so much picking coordinates as trying to give them physical meaning.

 

[kimbyd]

One might ask - why be so careful about waiting until the end to spit up space-time into space plus time?

The selected time slicing for the merger should ultimately not impact the results. The underlying picture of the space-time is, as PeterDonis has mentioned, rather like a pair of pants (though with the legs twisted in a helix, because the black holes will be orbiting). I don't know precisely how the calculations of how this occurs are done. Certainly they could be done without any reference to any specific time-slicing at all, just taking a specific boundary condition and computing the rest of the shape of the space-time such that it is consistent with Einstein's equations and that boundary condition.

The time-slicing of the system is only really critical as a visualization tool, as humans generally conceive of time very differently from space, so it's often easiest for us to visualize these things as a sequence of individual time slices. I'm assuming that when that time slicing is selected, it's done so that it makes the resulting visualization easy to understand. If a time slicing is used in the calculations, it's done so that the overall coordinate system chosen avoids any singularities that would mess with the calculation (and there may potentially also be computational efficiency considerations on the coordinate choice).

 

[Tom Mcfarland]

To Peter Donis

I understand and enjoy your Cartesian product portrayal of an event horizon, and I have accepted the cosmologist's preference for thinking of space-time as a 4-manifold product space. However, as a mathematician with little training in physics, I have long felt that space-time is merely a convenient framework on which to hang the math of cosmology. Space-time seems designed to obscure intuition, because "time" and "space" are so different. I am much more comfortable with the multiple dimensions of string theory than with the treatment of "time" as a space-like 4th dimension. Please do not take offense...I still love you guys.

 

[kimbyd]

To Peter Donis

I understand and enjoy your Cartesian product portrayal of an event horizon, and I have accepted the cosmologist's preference for thinking of space-time as a 4-manifold product space. However, as a mathematician with little training in physics, I have long felt that space-time is merely a convenient framework on which to hang the math of cosmology. Space-time seems designed to obscure intuition, because "time" and "space" are so different. I am much more comfortable with the multiple dimensions of string theory than with the treatment of "time" as a space-like 4th dimension. Please do not take offense...I still love you guys.

It's very clear that space-time obeys symmetries such that time is a dimension very similar to space. The main difference, in terms of relativity, is that space-time distances show an opposite sign for time distances as with spatial distances (which is positive and which negative depends upon your convention). Whatever the underlying physics, there's no question that time behaves like a dimension.

The statement that time is something completely different from space is not really supported by any evidence (though I am aware of some physicists that are investigating this possibility on a theoretical level).

 

[Orodruin]

I am much more comfortable with the multiple dimensions of string theory than with the treatment of "time" as a space-like 4th dimension.

You do realize that string theory also involves space- and time-like dimensions? Nature does not care what you are comfortable with. To argue against time and space being similar is essentially to argue against relativity.

 

[PeroK]

However, as a mathematician with little training in physics ...

Says it all!

 

[timmdeeg]

Thanks for your detailed answer; sorry for being late.

When one talks about a scenario where "the black holes just touch", my interpretation of this is that one is imposing some specific simultaneity convention, some specific coordinates, to describe the state of the black hole at some "instant in time", so one can single out a specific instant in time "where the event horizons touch".

Perhaps Eddington-Finkelstein coordinates should be used to have a better imagination. PeterDonis has mentioned the "pair of trousers" in #19.

Similar issues and confusions will happen when one tries to describe what one means by "an instant of time" in the binary inspiral. We don't expect a hypersurface-orthogonal split to exist there, either.

Here I suspect that the trousers picture isn't helpful, unless one is able to imagine the winding of the two cylinders in 3 dimensions. But I think irrespective of that an instant of Eddinton-Finkelstein time is something else than an "instant of time" based on "a hypersurface-orthogonal split".

This brings up the question of how the calculations for the expected signal out of an inspiral were actually done. There are certainly papers in the literature that describe the process, but I can't say I'm familiar with them. I believe developing the numerical codes and ensuring their accuracy and convergence was a very long and challenging process. I would expect, though, that conceptually they used the general framework I outlined earlier, a framework that can be described as a 4-dimensional "block universe" view of the problem, that doesn't single out specific instants of time for the inspiral process, but instead comes up with an abstract mathematical representation that does not depend on such a split.

This is an interesting question. I will be visiting LIGO together with a group of "interested layman" in August, perhaps I can pose this question on this occasion.

 

[pervect]

This is the basis for my description in post #19.
My understanding is somewhat different. My understanding is that initial conditions are specified on a spacelike hypersurface and then evolved forward in "time" (where this "time" is just a coordinate and is not assumed to have any physical meaning--see below). In other words, there is an implicit "split" of spacetime into space and time that is adopted at the start of the process (this is just a choice of coordinates).Since the values of all actual observables are independent of the choice of coordinates, this is not a problem. However, there is no attempt made, AFAIK, to associate any physical meaning with this split (i.e., with the coordinates); they are chosen for ease of calculation and nothing else. So I think the possible issue with the OP is not so much picking coordinates as trying to give them physical meaning.

You make some good points. Specifying a metric will imply an implicit split. But I agree that the resulting split doesn't have any obvious physical significance, especially since it won't be unique. Whether or not some splits (metric choices) are "better" for the stability and accuracy of the simulation is an interesting question that I don't know the answer to. Probably there is some interaction, though.

I did think of one more wrinkle. If we had a stationary system, we could look at the redshift factor in coordinate independent terms as being derived from the length of the timelike Killing vector. The legnth of the Killing vector goes to zero at the horizon in the Schwarzschild case (I think it does in the stationary case, too, though I'm not sure), which implies the redshift goes to infinity.

[add]. The Killing vector will vanish at the Killing horizon. I don't think it will necessary vanish at the absolute or apparent horizons - it'd be best to assume it didn't until proven otherwise.

Exactly how to apply this idea to the non-static case (some sort of quasi-stationary analysis, perhaps) isn't clear. There couldn't be a true Killing vector, but there might be some approximation of one. If this works it, it could be an approach that might be able to answer the original question and provide a justification for saying that the redshift goes to infinity.

 

[PeterDonis]

Perhaps Eddington-Finkelstein coordinates should be used to have a better imagination. PeterDonis has mentioned the "pair of trousers" in #19.

Yes, this description is based on Eddington-Finkelstein coordinates (Painleve coordinates also lead to a similar description). However, strictly speaking, those coordinates describe a single static black hole, not a merger. I am assuming that coordinates with similar properties can be defined on the spacetime geometry describing a black hole merger. That assumption seems quite plausible to me, but no exact solution for the black hole merger case is known, so I can't exhibit the coordinates explicitly.

Here I suspect that the trousers picture isn't helpful

It gets more complicated, yes, because you have to imagine the legs of the trousers twisting around each other.

an instant of Eddinton-Finkelstein time is something else than an "instant of time" based on "a hypersurface-orthogonal split"

That's correct in the sense that the fundamental property being used to define Eddington-Finkelstein coordinates--using ingoing null geodesics--does not require the spacetime to have a timelike vector field that is hypersurface orthogonal. (Similar remarks apply to Painleve coordinates--there the property being used is ingoing timelike geodesics).

The legnth of the Killing vector goes to zero at the horizon in the Schwarzschild case (I think it does in the stationary case, too, though I'm not sure)

It does--more precisely, in Kerr spacetime, the timelike Killing vector field vanishes on the outer horizon (and the inner one too, but we don't need to go into that here).

The Killing vector will vanish at the Killing horizon. I don't think it will necessary vanish at the absolute or apparent horizons

In the idealized cases of Schwarzschild and Kerr spacetime, the Killing, absolute, and apparent horizons are all the same. In more realistic models, strictly speaking, there is not a Killing horizon, because the spacetime is not stationary (for example, if objects fall into the hole or Hawking radiation comes out). But there can still be absolute and apparent horizons.

 

[Tom Mcfarland]

Earlier, Kimbyd made a good point that I had not noticed...the event horizon is not the equivalent of a physical object.

Approaching the "merger" issue from a different angle, where does the mass of a BH reside...inside, on, or outside the event horizon?
Or if this is not a well-formed question, can you re-phrase it?

 

[PeterDonis]

where does the mass of a BH reside

It's a property of the spacetime geometry as a whole; it doesn't "reside" at any particular place.

 

[kimbyd]

Approaching the "merger" issue from a different angle, where does the mass of a BH reside...inside, on, or outside the event horizon?

Nobody knows the mass distribution inside the event horizon of the black hole. General Relativity explicitly cannot describe the part of the black hole that contains mass (in technical terms, the "singularity" predicted by GR is not on the manifold). We'd need to know the correct theory of quantum gravity to state what the mass distribution inside the black hole is (if that statement even makes sense).

 

[PeterDonis]

Nobody knows the mass distribution inside the event horizon of the black hole.

If we are talking about idealized Schwarzschild or Kerr black holes, there is no mass distribution inside the horizon, or outside it for that matter: these are vacuum solutions.

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: but an observer falling into the hole well after it forms will never see any of that stress-energy, since it will have collapsed into the singularity long before. So the interior is still vacuum in this case. It's true that in general we won't know the exact manner in which the stress-energy collapsed, so we won't know the exact "mass distribution" in the region of spacetime occupied by the stress-energy, but that doesn't change what I just said.

General Relativity explicitly cannot describe the part of the black hole that contains mass (in technical terms, the "singularity" predicted by GR is not on the manifold)

The mass of the hole, the $M$ that appears in the metric, is not located at the singularity. It is, as I said before, a property of the spacetime as a whole. We can measure it by measuring the orbital parameters of objects well away from the horizon.

Plus, the singularity is not the entire interior of the hole inside the event horizon, so even if we don't have a good model for what happens close enough to the singularity (see below), that doesn't mean we don't have a good model of anywhere inside the horizon.

We'd need to know the correct theory of quantum gravity to state what the mass distribution inside the black hole is (if that statement even makes sense)

No, we'll need to know the correct theory of quantum gravity in order to know what replaces the singularity, assuming our current conjecture is correct that the presence of the singularity in the classical GR solution indicates a breakdown of GR in this regime. But that doesn't change what I said above either.

 

[kimbyd]

If we are talking about idealized Schwarzschild or Kerr black holes, there is no mass distribution inside the horizon, or outside it for that matter: these are vacuum solutions.

Right. Those solutions don't describe any mass distribution at all. Presumably there is one that is non-singular, but we'd need quantum gravity to determine what that distribution is.

No, we'll need to know the correct theory of quantum gravity in order to know what replaces the singularity, assuming our current conjecture is correct that the presence of the singularity in the classical GR solution indicates a breakdown of GR in this regime. But that doesn't change what I said above either.

We know for sure that GR cannot describe the singularity itself, but that doesn't mean that GR is accurate all the way to the singularity. Very likely GR breaks down some distance outside the singularity. How far outside is at this point unknown.