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

[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.

 

[PeterDonis]

Those solutions don't describe any mass distribution at all.

But they are idealized solutions. 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.

Very likely GR breaks down some distance outside the singularity.

Yes, this is the current belief of most physicists. But we won't know until we have a better theory and a way to test it experimentally.

How far outside is at this point unknown.

This is a more contentious area, since there are different conflicting opinions on how strong quantum gravity effects are at or near the horizon. Again, we won't know until we have a better theory and a way to test it experimentally.

 

[kimbyd]

Yes, this is the current belief of most physicists. But we won't know until we have a better theory and a way to test it experimentally.

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.

This is a more contentious area, since there are different conflicting opinions on how strong quantum gravity effects are at or near the horizon. Again, we won't know until we have a better theory and a way to test it experimentally.

This is why I don't try to make any statements about what the mass distribution anywhere inside the horizon is. It's very conceivable that quantum effects differ strongly from GR right up to the horizon itself. We don't yet know.

Yes, General Relativity has almost all of the interior of the black hole in vacuum, with an infinitesimal point that GR can't describe near the center being the only possible place mass can exist for any appreciable length of time. But given the potential possibility that quantum gravity may impact the horizon itself, we can't take any of that very seriously.

 

[PeterDonis]

It's very conceivable that quantum effects differ strongly from GR right up to the horizon itself.

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, because the light would not stay trapped at that surface forever; quantum gravity effects would eventually allow that light to escape to infinity.

If the above is the case, then much of what we have been saying in this thread is not applicable, because it only applies to a model in which true black holes and event horizons exist. A key feature of such a model is that event horizons are null surfaces, which have certain useful properties. Apparent horizons are not necessarily null surfaces, so those properties cannot be relied on in a model in which there are no true event horizons but only apparent horizons. So, for example, descriptions like the one I gave earlier, of the event horizon being shaped like a pair of trousers, might not be applicable to an apparent horizon in a model where quantum gravity effects were significant enough to prevent a true event horizon from forming.

If we want to discuss possible quantum gravity models like the one I described above, that probably belongs in the Beyond the Standard Model forum. This forum is supposed to be restricted to discussion of SR and GR as they currently exist, since they are our best current theories that have experimental confirmation. It's good to be aware of possible limitations, but for a given discussion we need to pick a particular model, and in this thread that model should be based on GR as it currently stands.

 

[kimbyd]

Honestly, it'd be really exciting to me if the event horizon of a black hole was only an apparent horizon, and the quantum gravity effects were observable at or near the event horizon itself. This would mean that we could potentially obtain evidence of quantum gravity by observing black holes near their horizon, e.g. from observing matter falling into a black hole, observing BH-BH mergers, or observing the silhouette of a black hole.

As a semantic matter, I'm pretty certain that the astrophysics community would continue to call them black holes even if they don't quite match the idealized picture we have of them from General Relativity, as long as they adhere pretty closely to the GR black hole's main properties: apparent horizon size/mass relationship, horizon temperature, and apparent loss of information into the horizon (even though the information obviously wouldn't actually be lost in this case: it would be thermalized enough to be unmeasurable in practice for us to continue to call it a black hole).

 

[PeterDonis]

it'd be really exciting to me if the event horizon of a black hole was only an apparent horizon, and the quantum gravity effects were observable at or near the event horizon itself

Yes, it would be, but so far we don't have that. The LIGO observations were consistent with a standard GR model of a black hole merger, which is at least some evidence against quantum gravity effects being observable near the horizon.

As a semantic matter, I'm pretty certain that the astrophysics community would continue to call them black holes even if they don't quite match the idealized picture we have of them from General Relativity

As a matter of terminology, I agree.

 

[kimbyd]

Yes, it would be, but so far we don't have that. The LIGO observations were consistent with a standard GR model of a black hole merger, which is at least some evidence against quantum gravity effects being observable near the horizon.

I don't think that conclusion can be drawn. It's only one signal, and not all that far above the noise. Not far enough to detect any subtle effects anyway. I'd expect any quantum effects to be very subtle.

It's evidence against gross deviation, but not more than that.

 

[PeterDonis]

I'd expect any quantum effects to be very subtle.

If they're subtle, they can't prevent an event horizon from forming. Quantum effects that were significant enough to do that would have to lead to significant differences from the standard GR model near the horizon (with one caveat--see below). Whether they would have to be significant enough to show up in LIGO data is another question; of course we can't know that if we don't have an actual quantum gravity model. I don't think the LIGO data is very strong evidence on the question, but I don't think it's zero evidence either.

The one caveat is that I have seen some hypothetical proposals that essentially say the deviation from standard GR near the horizon would only have to be for a short time, in which case we would not see them unless we happened to be observing during that short time. But they're only hypothetical proposals.

 

[kimbyd]

If they're subtle, they can't prevent an event horizon from forming. Quantum effects that were significant enough to do that would have to lead to significant differences from the standard GR model near the horizon (with one caveat--see below). Whether they would have to be significant enough to show up in LIGO data is another question; of course we can't know that if we don't have an actual quantum gravity model. I don't think the LIGO data is very strong evidence on the question, but I don't think it's zero evidence either.

The one caveat is that I have seen some hypothetical proposals that essentially say the deviation from standard GR near the horizon would only have to be for a short time, in which case we would not see them unless we happened to be observing during that short time. But they're only hypothetical proposals.

I think we'd need to have a concrete model for the deviations from GR for this discussion to continue in a productive manner.

 

[Denis]

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).

If you think this is the main difference, I recommend to try to travel back and forth in any of the three spatial dimension, and then in the time dimension. The result will be sufficiently different.
The Lorentz symmetry is simply the symmetry group of the wave equation $\square u = \frac{1}{c^2} \partial_t^2 - \sum_i \partial_i^2 u = 0$. This wave equation plays an important role in particle physics, because all the fields of the SM follow variants of this equation. Why people think that this particular wave equation is somehow fundamental, given that QFT, at least if one includes gravity, is at best an effective, large distance approximation, is beyond me. Moreover given that classical condensed matter theory already contains examples how such wave equations can appear as large distance approximations.

 

[kimbyd]

If you think this is the main difference, I recommend to try to travel back and forth in any of the three spatial dimension, and then in the time dimension. The result will be sufficiently different.

Your statement doesn't have any bearing on the symmetries that exist between time and space, which are only apparent at very high velocities.

The Lorentz symmetry is simply the symmetry group of the wave equation $\square u = \frac{1}{c^2} \partial_t^2 - \sum_i \partial_i^2 u = 0$. This wave equation plays an important role in particle physics, because all the fields of the SM follow variants of this equation. Why people think that this particular wave equation is somehow fundamental, given that QFT, at least if one includes gravity, is at best an effective, large distance approximation, is beyond me. Moreover given that classical condensed matter theory already contains examples how such wave equations can appear as large distance approximations.

Examining possible violations of Lorentz invariance is a pretty popular avenue for experimental physics. Despite covering a huge range of energy scales (up to a few times the Planck scale) and with tremendous experimental accuracy, there hasn't been any such violation found yet. See here for a summary of the results of the many searches:
https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation

 

[Denis]

Your statement doesn't have any bearing on the symmetries that exist between time and space, which are only apparent at very high velocities.

Feel free to ignore the obvious asymmetries I have described. As long as you do not think that such ignorance is an argument, I have no problem with this.

The people who have developed atomic theory have not waited until the continuous field theories failed, but developed atomic theory during the time the continuous large distance approximations were completely satisfactory from point of view of observation. Today, we have to wait until the experiment shows a Lorentz violation before we start to develop theories with broken Lorentz covariance?

Not really a good idea, simply because developing such theories (and looking at their weak places) is the classical scientific way to guide experimenters where to start their experiments.

 

[kimbyd]

Feel free to ignore the obvious asymmetries I have described.

The future/past asymmetry with regards to time is a different issue altogether, one which doesn't really have any bearing on the rest of your question. I didn't want to go too much into it, because it's a rather complicated topic.

The super short version is that all fundamental laws that we know of are symmetric in time. The asymmetry you describe only appears to arise out of complex systems in an environment with a past low-entropy state. It simply is not a component of fundamental law, but appears to arise out of an environment with a low-entropy boundary condition in the past.

But that discussion simply has no bearing whatsoever on whether or not the Lorentz symmetry is obeyed, which is more about effects like the speed of light being the same for all observers, momentum/velocity relationship holding at very high momenta, and other such effects.

 

[Denis]

There is no disagreement about the point that Lorentz symmetry is obeyed for some parts of reality, in particular for everything we are able to observe up to now.

The question is if it is obeyed for really everything. It cannot be obeyed for everything which exists, or everything which exists now, because this phrase presupposes a notion of existence, thus some sort of realism, but even the extremely weak notion of realism used by the EPR criterion of reality is, in combination with Lorentz symmetry of everything real, sufficient to prove Bell's inequality, and is empirically falsified.

But even if one ignores the problems of fundamental relativity with realism and causality because of Bell's theorem, there are not even nice candidates for relativistic theories. GR is full of singularities and incompatible with quantum theory, relativistic QFT has a problem with infinities too, there is Haag's theorem. All this is not even a problem if the aim is an effective theory, which fails below a critical length - but this requires to give up fundamental relativity.

And without relativity being fundamental, it is quite probable that one will return to space and time being very different.

 

[PeterDonis]

The question is if it is obeyed for really everything.

And since nobody knows the answer to this question, it's speculation either way at this point, and therefore off topic here at PF.

All, please keep the discussion on the specific topic of the thread.

 

[timmdeeg]

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".

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?

 

[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.