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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!