Interior volume of a black hole

[PeterDonis]

This paper by Christodolou and Rovelli discusses how to define the interior volume of a black hole:

https://arxiv.org/abs/1411.2854

I'll try to condense their basic argument into a (heuristic) form that makes it easier to raise the issues I want to raise. The basic idea is to start with the following way of defining the interior volume of a 2-sphere in Minkowski spacetime: it is the maximum 3-volume of the interior of the 2-sphere obtained by considering all of the possible spacelike 3-surfaces that contain the 2-sphere. It is straightforward to show that this 3-volume is just the Euclidean 3-volume of the 2-sphere in the 3-surface of simultaneity defined by the 2-sphere, i.e., it's the volume we would intuitively arrive at anyway.

The paper then tries to generalize this idea to the case of a classical black hole formed by the collapse of a spherically symmetric object, so that the spacetime geometry of the vacuum region exterior to the object is Schwarzschild. (It also considers the "eternal" Schwarzschild case, but I'm not going to go into that here.) The basic reasoning proceeds as follows:

(1) The horizon is an outgoing null surface consisting of a continuous series of 2-spheres labeled by increasing values of an ingoing null coordinate $v$ (we are using Eddington-Finkelstein coordinates on the vacuum region).

(2) For each 2-sphere, we define its interior volume the same way as for Minkowski spacetime above: it's the maximum 3-volume of the interior of the 2-sphere obtained by considering all of the possible spacelike 3-surfaces that contain the 2-sphere. (Note that since we are not considering extended Schwarzschild spacetime, these 3-volumes will be finite.)

(3) However, because of the way the spacetime inside the horizon is curved, finding this maximum 3-volume is much more difficult than it is in Minkowski spacetime. Thus, the paper somewhat simplifies the problem by considering only the asymptotic volume, i.e., the volume for 2-spheres at very large values of $v$. They then derive the result that this volume increases linearly with $v$, which can be heuristically described as the interior of the hole growing larger linearly with time, even though the area of its horizon remains constant.

First, a pedantic point about how the coordinate $v$ is defined. In Fig. 1 of the paper, which is a Penrose diagram of the spacetime, the ingoing null curve labeled $v = 0$ is the start of the horizon--i.e., the point at $r = 0$, the center of the collapsing matter, at which a horizon forms and starts expanding outward. Or, to put it another way, this is the event on the worldline at the center of the collapsing matter at which a radially outgoing light ray emitted just then will reach the surface of the collapsing matter at just the point where that matter is at $r = 2M$ (where $M$ is the mass of the collapsing matter). After that, the same outgoing light ray will remain at $r = 2M$ forever.

In the paper, for simplicity, the "collapsing matter" is taken to be an ingoing null shell of zero thickness. In Fig. 1, and also in the text at the start of section III of the paper, the $v$ coordinate of the shell is taken to be $v = 0$. However, Fig. 1 is not consistent with this. As shown in Fig. 1, the $v = 0$ ingoing null curve reaches $r = 0$ at the start of the horizon (as described above): i.e., on the line at the left at the point where the 45 degree line going up and to the right, which is the horizon, starts. But the collapsing matter cannot reach $r = 0$ at that point, because if it did, the singularity would start there, and it doesn't. It starts at the top left corner of the diagram. So the worldline of the collapsing null shell must be an ingoing null worldline that hits the top left corner of the diagram.

In Fig. 5, however, which shows the "maximal volume" 3-surfaces in Eddington-Finkelstein coordinates, the worldline of the collapsing null shell (the red line) is shown consistently with its $v$ coordinate being $v = 0$. But note that in this figure, the horizon (the green line) intersects $r = 0$ at a negative $v$ coordinate. Since this figure is consistent with everything said in the text, I take it that Fig. 1 is in error, and that the line labeled $v = 0$ in Fig. 1 should really be labeled with the negative $v$ coordinate at which the green line intersects $r = 0$ in Fig. 5 (something like $v = - 4$). There should also be a line added to Fig. 1, corresponding to the red line in Fig. 5, which is a 45 degree line up and to the left that hits the top left corner, and which should be labeled $v = 0$.

With that out of the way, let me get to the main issues I want to raise. Consider what the 3-surfaces shown in Fig. 5 would look like if transferred to Fig. 1. All of the surfaces hit the singularity at $v = 0$, so they all hit the top left corner of Fig. 1. They simply "fan out" from that corner to all the different points on the horizon that are after (up and to the right of) the intersection of the $v = 0$ line (the worldline of the ingoing null shell that collapses to form the hole) with the horizon. The paper is simply deriving the result that, for large values of $v$, the 3-volumes of these surfaces from the singularity to the horizon increase linearly with $v$.

One possible issue with this is that none of these 3-surfaces intersect the collapsing null shell. But that's really a side effect of the fact that all of them intersect the same point at $r = 0$, namely, the upper left corner of the diagram, where the singularity begins. So the first main issue I have is, is that reasonable? (Note that this is not disputing that all of the surfaces are spacelike, or that they are the "maximum volume" ones intersecting each of the points on the horizon. In other words, I am not disputing that the paper's results are mathematically correct. I am questioning whether those results are physically reasonable as defining what "the interior volume of a black hole" means.) Note that the Minkowski spacetime scenario from which all this is drawn does not have this problem: if we take any series of 2-spheres along a timelike or null curve, the "maximum volume" 3-surfaces that contain them will be non-intersecting, even if we extend the spacetime to include its conformal boundaries as Fig. 1 does. (Note that this last point addresses a possible technicality with saying that the "maximum volume" 3-surfaces in the black hole case all "intersect" at the point $r = 0$, $v = 0$, since that point, being on the singularity, is not actually part of the original spacetime; it's only part of the conformally extended spacetime.) It would seem to me to be more reasonable physically to impose some kind of constraint on the 3-surfaces so that, heuristically, they "act like" the ones in the Minkowski case in a way that the ones given in the paper do not.

After deriving the above result, the paper then goes on to discuss potential implications for the generalized second law of thermodynamics (i.e., the second law applied to systems containing black holes). The key point of this section (section VIII) seems to be that the paper's result, that the interior volume of a black hole increases with time, at least for "late times", conflicts with the Bekenstein rule for assigning entropy to a black hole, which is that its entropy is the area of its horizon (more precisely, 1/4 of that area in Planck units). The Bekenstein rule makes the entropy of a hole with constant mass constant, whereas the paper's result seems to be saying that, since the interior volume of such a hole increases with time, the amount of information it can store should as well.

It seems to me, however, that this claim depends on shifting the meaning of the paper's result. The paper's result, as described above, is simply a result on a particular counterintuitive property of a set of spacelike 3-surfaces in the spacetime geometry of the vacuum region inside the horizon. But the claimed implication for black hole thermodynamics depends on an implicit claim that those surfaces are the right ones for assessing "how much information" a black hole can hold as a function of time. Not only is no argument for this given in the paper, but it seems to me that such a claim is physically unreasonable if the choice of 3-surfaces is itself unreasonable, on grounds such as those I gave above.

In fact, the paper itself seems somewhat ambivalent about this, because it seems to be arguing that the "growth" property shown by its result on the volumes of spacelike 3-surfaces is more suited to an apparent horizon than to an event horizon. In the text just before and after equation 28, the paper says that "an event horizon obeys, most likely, the generalized second law", and then says "but quantum effects both inside and at the horizon are likely to make event horizons unphysical". But the whole derivation of the mathematical result in the main part of the paper is for an event horizon! (Actually, strictly speaking, the apparent horizon and the event horizon coincide for this spacetime, but that means the paper's result applies to both kinds of horizons equally, which still makes it problematic, in my view, to argue that the two cases should be treated differently with regard to the paper's result.)

Questions and comments are welcome!

 

[Dale]

I also skimmed through that paper once. I didn’t analyze it at the same level of depth as you, but I had essentially the same “gut reaction”. Yes, they were able to construct a quantity V in curved spacetime and in flat spacetime V reduces to the volume. But I saw no justification that their V should be at all related to thermodynamics/entropy/information in curved spacetime.

 

[Demystifier]

I've just seen something very interesting in Susskind's lectures on complexity and black holes https://arxiv.org/abs/1810.11563. On page 31 he says:

"When a star collapses a horizon forms and its area grows until it reaches its final value. This is visible from outside the horizon and is known to be a manifestation of a statistical law, the second law of thermodynamics: Entropy increases until thermal equilibrium is reached.

There is another similar but less well-known phenomenon: the growth of the spatial
volume behind the horizon of the black hole. At first sight this could be another example of the increase of entropy, but more careful thought shows that this is not so. The growth of the interior continues long past the time when the black hole has come to thermal equilibrium. Something else—not entropy—increases. What that something else is should have been one of the deepest mysteries of black hole physics—if anyone had ever thought to ask about it.

Complexity is of course that something."

 

[PeterDonis]

Complexity is of course that something

Without a confirmed theory of quantum gravity I don't see how he can back up that statement. It might be a plausible speculation, but that is all it can be at our current state of knowledge.

 

[Dale]

I did realize several months ago that for a standard Schwarzschild black hole the volume inside the horizon should be infinite since $\partial_t$ is a spacelike vector and also a Killing vector. Clearly that doesn't translate directly to a collapse spacetime.

 

[Demystifier]

Without a confirmed theory of quantum gravity I don't see how he can back up that statement. It might be a plausible speculation, but that is all it can be at our current state of knowledge.

Susskind, of course, makes a conjecture. He is a master of conjectures, I don't know any other physicist who makes so many deep influential conjectures.

 

[Demystifier]

I did realize several months ago that for a standard Schwarzschild black hole the volume inside the horizon should be infinite since $\partial_t$ is a spacelike vector and also a Killing vector.

Can you elaborate a bit? I don't see how the latter implies the former.

 

[PeterDonis]

I did realize several months ago that for a standard Schwarzschild black hole the volume inside the horizon should be infinite since $\partial_t$ is a spacelike vector and also a Killing vector. Clearly that doesn't translate directly to a collapse spacetime.

Actually, it does. On a Kruskal diagram, an integral curve of $\partial_t$ is a spacelike hyperbola in the upper "wedge" (region II, the black hole region) of the diagram. Even if you replace the left part of the diagram with the collapsing matter region, the spacelike hyperbolas still extend infinitely far in the upper right corner of the diagram.

 

[PeterDonis]

Susskind, of course, makes a conjecture. He is a master of conjectures, I don't know any other physicist who makes so many deep influential conjectures.

"Deep influential" is one thing. "Confirmed by experiments" is another. As far as I can tell, while indeed many of his conjectures are the former (at least in the string theory community), none of them have yet achieved the latter (again, just like string theory).

 

[Dale]

Can you elaborate a bit? I don't see how the latter implies the former.

Since $\partial_t$ is spacelike then together with $\partial_\theta$ and $\partial_\phi$ it forms a spatial volume element. When you integrate $\partial_\theta$ and $\partial_\phi$ over their range you get a finite area. Then since $\partial_t$ is a Killing vector its range is infinite. So integrating a finite area over an infinite length gives an infinite volume. There are some constants, but they are finite over some of the range, so they don't matter too much.

 

[PeterDonis]

since $\partial_t$ is a Killing vector its range is infinite.

Actually a bit more is needed here. $\partial_\phi$ and $\partial_\theta$ are also Killing vectors but their range is not infinite. The key additional item is that $\partial_t$ is a Killing vector in the $R^2$ submanifold of the spacetime, i.e., in a non-compact manifold.

 

[Dale]

Actually a bit more is needed here. $\partial_\phi$ and $\partial_\theta$ are also Killing vectors but their range is not infinite. The key additional item is that $\partial_t$ is a Killing vector in the $R^2$ submanifold of the spacetime, i.e., in a non-compact manifold.

Oh, yes, my logic was wrong there.

Of course, this essentially is assuming the conclusion. It is infinite because the submanifold is non-compact which means it is infinite. But regardless of being tautological, it is tautologically correct.

 

[PeterDonis]

this essentially is assuming the conclusion. It is infinite because the submanifold is non-compact which means it is infinite.

Not necessarily. The complete $R^2$ submanifold is infinite, but that by itself does not imply that the region of that submanifold enclosed by the horizon, which has a finite area, must be infinite. Recognizing the presence of a Killing vector field on the $R^2$ submanifold which has integral curves that are entirely contained in the region inside the horizon is the key additional step, and I don't think that is tautological (except in the trivial sense that, technically, any mathematical theorem is tautological).

 

[PeterDonis]

the region of that submanifold enclosed by the horizon, which has a finite area

Note, btw, that since none of the spacelike integral curves of $\partial_t$ intersect the horizon, it is actually somewhat counterintuitive to view them as being "enclosed" by the horizon in the spatial sense (as opposed to the spacetime sense). Normally we would take "enclosed" to mean that we have a spacelike slice of the spacetime that intersects the horizon, and that the portion of that spacelike slice with $r > 2M$ is "enclosed" by the horizon. In that sense of "enclosed", all such enclosed spacelike regions are finite.

 

[Demystifier]

"Deep influential" is one thing. "Confirmed by experiments" is another. As far as I can tell, while indeed many of his conjectures are the former (at least in the string theory community), none of them have yet achieved the latter (again, just like string theory).

By "confirmed by experiments", did you actually mean "proved mathematically" or something like that? Otherwise the statement is too obvious.

 

[PeterDonis]

By "confirmed by experiments", did you actually mean "proved mathematically" or something like that?

No, I meant exactly what I said. You can't "prove" scientific theories mathematically. You have to do experiments to see if the results match the theory's predictions.

Otherwise the statement is too obvious.

Maybe it is to you. I'm not sure it is to Susskind and other string theory advocates, who keep talking as if string theory is somehow established even though it has not made a single experimental prediction, let alone had one confirmed.

 

[Demystifier]

No, I meant exactly what I said. You can't "prove" scientific theories mathematically. You have to do experiments to see if the results match the theory's predictions.

Maybe it is to you. I'm not sure it is to Susskind and other string theory advocates, who keep talking as if string theory is somehow established even though it has not made a single experimental prediction, let alone had one confirmed.

I think this objection is a bit unfair. First, we on this thread talk about black hole interior, even though there is no any experimental evidence that a black hole interior even exists. Second, many of the modern "string theory" conjectures, including those by Susskind, are often formulated without directly referring to strings. Third, we would all want to understand something about quantum gravity (or would we?), but there is no any experimental evidence for quantum gravity, so we all must make some theoretical speculations, conjectures and educated guesses.

What can be irritating is the over-confidence in the tone and rhetoric of some string theorists, but when one filters out the tone and concentrates on the content, the ideas they present may be very interesting.

 

[PeterDonis]

we on this thread talk about black hole interior, even though there is no any experimental evidence that a black hole interior even exists

That is true, but a black hole interior, if we just talk about extrapolating to inside the horizon, is at least an extrapolation of a theory which has extensive experimental support, into a regime in which none of the physical invariants involved take values that are very different what has been experimentally observed. When you get close to the singularity, that's a different matter--then you are dealing with curvature invariants much, much larger than any we have experimental evidence for. But that is the very regime in which most relativity experts will concede that GR breaks down and will be replaced by a better theory.

What can be irritating is the over-confidence in the tone and rhetoric of some string theorists

It's not just overconfidence, it's presenting speculations and hypotheses as though they were established facts. It's bad enough when they do it with audiences of other experts, who can at least filter out the overstatements. When they do it with the general public, which they do, it's much worse, because the general public has no way of knowing that what they are saying is just speculations and hypotheses, and believes that these way-out conjectures have the same level of confidence as, say, predictions by astronomers about which asteroids will pass close to the Earth and when. That skews the public's view of science as a whole.

 

[Demystifier]

That is true, but a black hole interior, if we just talk about extrapolating to inside the horizon, is at least an extrapolation of a theory which has extensive experimental support, into a regime in which none of the physical invariants involved take values that are very different what has been experimentally observed. When you get close to the singularity, that's a different matter--then you are dealing with curvature invariants much, much larger than any we have experimental evidence for. But that is the very regime in which most relativity experts will concede that GR breaks down and will be replaced by a better theory.

May I ask what's your personal opinion (if you have any) about the ideas that GR breaks down already at the black hole horizon?

 

[Dale]

May I ask what's your personal opinion (if you have any) about the ideas that GR breaks down already at the black hole horizon?

I am very skeptical of that. For a sufficiently large BH the density and curvature are arbitrary small at the horizon. And locally a BH horizon is just a Rindler horizon.

 

[Demystifier]

I am very skeptical of that. For a sufficiently large BH the density and curvature are arbitrary small at the horizon. And locally a BH horizon is just a Rindler horizon.

This argument is based on classical relativity which is based on a principle of locality. But maybe quantum gravity, or some more fundamental theory that incorporates quantum gravity as its part, should be based on some global principles which violate the locality principle we are used to. Perhaps in this theory there is a preferred system of coordinates, e.g. Minkowski coordinates in flat spacetime and Schwarzschild coordinates in the black hole case, suggesting that the Schwarzschild horizon could be a kind of a physical boundary with properties very different from the Rindler horizon. Of course it's a speculation, but recently I've constructed some toy models suggesting me that such a speculation is not completely ungrounded: https://arxiv.org/abs/2301.04448

 

[Dale]

But maybe …

As I said, I am very skeptical of any such “but maybe”. I would need to see some solid experimental evidence. A “toy model” would not be sufficient to address my skepticism.

 

[martinbn]

This argument is based on classical relativity which is based on a principle of locality. But maybe quantum gravity, or some more fundamental theory that incorporates quantum gravity as its part, should be based on some global principles which violate the locality principle we are used to. Perhaps in this theory there is a preferred system of coordinates, e.g. Minkowski coordinates in flat spacetime and Schwarzschild coordinates in the black hole case, suggesting that the Schwarzschild horizon could be a kind of a physical boundary with properties very different from the Rindler horizon. Of course it's a speculation, but recently I've constructed some toy models suggesting me that such a speculation is not completely ungrounded: https://arxiv.org/abs/2301.04448

May be there is a black knight at the horizon that stops anything from passing in the black hole.

 

[PeterDonis]

what's your personal opinion (if you have any) about the ideas that GR breaks down already at the black hole horizon?

I don't think it's plausible that GR would break down at the horizon of a black hole of stellar mass or larger; the spacetime curvature is too small.

I think it's much more plausible that we will end up finding that the things we now call "black holes" actually have only apparent horizons, not event horizons, and that quantum effects inside the apparent horizons change the equation of state so that such an object eventually (on very long time scales comparable to the standard Hawking radiation time) radiates away. The "Bardeen black hole" and models based on it are simple models of such objects. These models can be constructed entirely with standard GR; the only "new" assumption is that quantum effects can produce the required equation of state in the deep interior of such objects, and since we already know that quantum fields can violate energy conditions, that doesn't seem like a very extravagant assumption.