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- TL;DR Summary
- Discussion of a paper entitled "Islands in Schwarzschild Black Holes" which proposes a calculation of entropy for a black hole emitting Hawking radiation.

The following paper appeared earlier this year on arxiv, entitled "Islands in Schwarzschild Black Holes":

https://arxiv.org/pdf/2004.05863.pdf

First, a bit of background: this paper appears to be part of a larger research effort aimed at resolving the black hole information paradox by showing how all of the information that fell into a black hole does end up coming back out in the Hawking radiation emitted by the hole. The general approach seems to be to show how the information is contained in the entanglements between radiation emitted at early and late times, rather than being contained in individual bits of radiation emitted at particular times, taken by themselves.

In this thread, I'm not intending to discuss that general research effort, but just to look at the particular models and claims made in this particular paper. What the impact would be on the larger research effort if those claims turned out not to be justified is a separate question that I do not want to get into in this thread.

The claims made in the paper generally appear to rest on the idea of an "island", which, in somewhat oversimplified terms, is a kind of connection between the Hawking radiation in two "wedges" of the spacetime, the existence of which limits the entanglement entropy between the inside and outside of the hole at late times. (This is relevant for the larger research effort because, if it is true that all of the information that fell into the hole eventually comes out in Hawking radiation before the hole completely evaporates away, then the entanglement entropy, while it might increase for a while once the hole forms, will eventually, after a time called the "Page time", have to start decreasing again, and ultimately must go to zero by the time the hole finally evaporates, since at that point there is no "inside" the hole any longer--all the information is outside, so the outside must be in a zero entropy pure state just as it was before the hole formed in the first place.)

I have not yet completely understood all of the computations in the paper. However, with what I think I have understood so far, I see several potential issues that I am hoping others might be able to comment on:

(1) The background spacetime used in the paper is Schwarzschild spacetime--more precisely, maximally extended Schwarzschild spacetime. However, this spacetime is a vacuum spacetime--there is no stress-energy anywhere--so I don't see how it can be a suitable model for what the paper is trying to do. The paper keeps talking about "matter fields" being present, but if the spacetime is Schwarzschild, there can't be any matter fields present that contain any energy; they must all be in their ground states, which makes them irrelevant to the analysis being done.

Also, the black hole in this spacetime is eternal--it never evaporates away. Strictly speaking, its mass can't change at all. But the whole point of the analysis is supposed to be to evaluate how the entanglement entropy changes as the hole's mass decreases, not just by a little bit, but from some large finite value all the way down to zero. I don't see how Schwarzschild spacetime can possibly be a valid background against which to make such calculations. Something like the outgoing Vaidya spacetime, at least for a portion of the model, would seem to be necessary in order to capture the decreasing mass and the emitted radiation (and there would also need to be a region to the future of the final evaporation of the hole which was, at least to a good approximation, flat Minkowski spacetime, inside the last spherical wave front of outgoing radiation from the final evaporation).

(2) In addition to all the other issues raised above, there is another issue with using maximally extended Schwarzschild spacetime: only one of the two "wedges" being used in the model will actually exist for a real black hole that forms by gravitational collapse of a massive object. In terms of Fig. 1 in the paper, only a portion of the region marked R+, plus a portion of the black hole region at the top, would exist for a real black hole: the rest of that figure would not be present, with that region instead being occupied by the matter that collapses to form the hole, and that matter region would end at an ##r = 0## line on the left, which would meet the singularity at the top. The spacetime diagram in this Insights article gives a rough idea of what I am describing:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-4/

But the paper's whole analysis depends on both the right

(3) There also seems to me to be an issue with the calculation in Appendix B, of what is called the "distance" between the two wedges (as far as I can tell, this is the distance between points a+ and a- in Fig. 1--or possibly between b+ and b-, it's not entirely clear--or rather how that distance changes as the points move upward and to the left/right in the diagram). This calculation uses Schwarzschild coordinates; but it is calculating something that crosses the horizon--actually it crosses

Heuristically, what Appendix B appears to be calculating is a "distance" along the blue line that represents the "island" in Fig. 1, and the purpose of the calculation appears to be to justify the obvious intuition that, as you move "up" the diagram, since the horizons move "further apart", the "distance" along the blue line has to increase. However, this distance is called "geodesic distance" in the paper, but no proof is given that the line along which the "distance" is calculated is actually a spacelike geodesic, and I don't think it is.

All these issues make me skeptical that the calculations in the paper are correct, or that they are relevant to actual evaporating black holes. However, it's quite possible that I am missing something. Any input is appreciated.

https://arxiv.org/pdf/2004.05863.pdf

First, a bit of background: this paper appears to be part of a larger research effort aimed at resolving the black hole information paradox by showing how all of the information that fell into a black hole does end up coming back out in the Hawking radiation emitted by the hole. The general approach seems to be to show how the information is contained in the entanglements between radiation emitted at early and late times, rather than being contained in individual bits of radiation emitted at particular times, taken by themselves.

In this thread, I'm not intending to discuss that general research effort, but just to look at the particular models and claims made in this particular paper. What the impact would be on the larger research effort if those claims turned out not to be justified is a separate question that I do not want to get into in this thread.

The claims made in the paper generally appear to rest on the idea of an "island", which, in somewhat oversimplified terms, is a kind of connection between the Hawking radiation in two "wedges" of the spacetime, the existence of which limits the entanglement entropy between the inside and outside of the hole at late times. (This is relevant for the larger research effort because, if it is true that all of the information that fell into the hole eventually comes out in Hawking radiation before the hole completely evaporates away, then the entanglement entropy, while it might increase for a while once the hole forms, will eventually, after a time called the "Page time", have to start decreasing again, and ultimately must go to zero by the time the hole finally evaporates, since at that point there is no "inside" the hole any longer--all the information is outside, so the outside must be in a zero entropy pure state just as it was before the hole formed in the first place.)

I have not yet completely understood all of the computations in the paper. However, with what I think I have understood so far, I see several potential issues that I am hoping others might be able to comment on:

(1) The background spacetime used in the paper is Schwarzschild spacetime--more precisely, maximally extended Schwarzschild spacetime. However, this spacetime is a vacuum spacetime--there is no stress-energy anywhere--so I don't see how it can be a suitable model for what the paper is trying to do. The paper keeps talking about "matter fields" being present, but if the spacetime is Schwarzschild, there can't be any matter fields present that contain any energy; they must all be in their ground states, which makes them irrelevant to the analysis being done.

Also, the black hole in this spacetime is eternal--it never evaporates away. Strictly speaking, its mass can't change at all. But the whole point of the analysis is supposed to be to evaluate how the entanglement entropy changes as the hole's mass decreases, not just by a little bit, but from some large finite value all the way down to zero. I don't see how Schwarzschild spacetime can possibly be a valid background against which to make such calculations. Something like the outgoing Vaidya spacetime, at least for a portion of the model, would seem to be necessary in order to capture the decreasing mass and the emitted radiation (and there would also need to be a region to the future of the final evaporation of the hole which was, at least to a good approximation, flat Minkowski spacetime, inside the last spherical wave front of outgoing radiation from the final evaporation).

(2) In addition to all the other issues raised above, there is another issue with using maximally extended Schwarzschild spacetime: only one of the two "wedges" being used in the model will actually exist for a real black hole that forms by gravitational collapse of a massive object. In terms of Fig. 1 in the paper, only a portion of the region marked R+, plus a portion of the black hole region at the top, would exist for a real black hole: the rest of that figure would not be present, with that region instead being occupied by the matter that collapses to form the hole, and that matter region would end at an ##r = 0## line on the left, which would meet the singularity at the top. The spacetime diagram in this Insights article gives a rough idea of what I am describing:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-4/

But the paper's whole analysis depends on both the right

*and*the left "wedges" being present; otherwise the whole construction with the "island" doesn't work. So this analysis, even if it is valid for the case considered (i.e., if the issues raised in #1 above don't invalidate it by themselves), would still seem to me to be irrelevant to a real black hole, since the analysis depends on a region of spacetime being present that is not present for a real black hole.(3) There also seems to me to be an issue with the calculation in Appendix B, of what is called the "distance" between the two wedges (as far as I can tell, this is the distance between points a+ and a- in Fig. 1--or possibly between b+ and b-, it's not entirely clear--or rather how that distance changes as the points move upward and to the left/right in the diagram). This calculation uses Schwarzschild coordinates; but it is calculating something that crosses the horizon--actually it crosses

*two*horizons (since it has to go from the right wedge, through the interior of the hole, out to the left wedge). But Schwarzschild coordinates are singular at the horizon, which would mean that either some other chart should be used (the obvious one would be Kruskal, or else the Penrose chart used to draw Fig. 1 itself), or that the calculation should be done in three pieces, one for each segment (right wedge point a+ to horizon, between horizons in the interior, horizon to left wedge point a-), with appropriate limits being taken as the horizons were approached. Nothing like this appears to be done in Appendix B.Heuristically, what Appendix B appears to be calculating is a "distance" along the blue line that represents the "island" in Fig. 1, and the purpose of the calculation appears to be to justify the obvious intuition that, as you move "up" the diagram, since the horizons move "further apart", the "distance" along the blue line has to increase. However, this distance is called "geodesic distance" in the paper, but no proof is given that the line along which the "distance" is calculated is actually a spacelike geodesic, and I don't think it is.

All these issues make me skeptical that the calculations in the paper are correct, or that they are relevant to actual evaporating black holes. However, it's quite possible that I am missing something. Any input is appreciated.