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Paradox unexplained
On 4/3/2016 11:32 PM, Steven Carlip wrote:
On 4/1/16 4:47 PM, Jos Bergervoet wrote: For the black hole information paradox I would expect two seemingly conflicting results (as is necessary for a paradox by definition,) e.g. like: 1) We know that information is lost because [...] We believe that matter in a pure quantum state can collapse to form a black hole. This is not quite clear if we cannot provide the solution of the GR equations that show the formation (so indeed we now can only say that we "believe" it). In particular, the often shown Kruskal-Szekeres diagram like this one he http://i.stack.imgur.com/U2EFS.gif is certainly *not* the correct description of formation, temporary existence and final complete evaporation! What it may describe correctly is an eternally existing black hole evaporating at the same rate as accreting new mass (in homogeneous form, GW150914 is not a good example!) Any mass in Region II (positive V axis) would never come out and would not contribute to any paradox in this case. The mass falling in would linger very long (in Schwarzschild time) above the horizon and would obviously have ample time to imprint its information on the Hawking radiation being emitted there. In fact the in-falling information is just bouncing of the horizon for this "eternal equilibrium". Such a black hole will then evaporate by Hawking radiation, which is thermal. This again is not so clear, as the Polchinsky-paper http://arxiv.org/abs/1207.3123 states immediately in the abstract: we want it to be a pure state! You are now claiming something that is not granted. When the black hole has completely evaporated, the net result will be the conversion of a pure state to a mixed state. For that, we clearly cannot use the static Schwarzschild solution, and just using Kruskal-Szekeres coordinates won't help since they still describe the same solution. Is there any closed form solution for the "transient black hole"? In any case I cannot quickly find it (so I can't even rule out it might fit in the margin of this post..) But we can look at Rindler coordinates for a uniformly accelerating frame. The Unruh effect corresponds to Hawking radiation and the Rindler horizon is the correct event horizon for the accelerating observer. Now, for the non-eternal case, we only have to make the observer reduce its acceleration, corresponding to the black hole losing its mass. For the observer, the Rindler horizon fades away as acceleration is reduced (of course we get perturbed Rindler coordinates by the "variation of constant" method). So whatever matter was forever behind the horizon in the eternal case, will simply come into view again with non- eternal acceleration. This is not yet the correct solution for the non-eternal black hole, but it suggests that the paradox will be resolved just by allowing the black hole mass to decrease, which the correct solution must do. In any case, the event horizon cannot have the simple light- like infinite wedge shape as it has for the eternal case. ... 2) But also that it is not lost because [...] Quantum mechanics is unitary. This means that the "fine-grained" information in a quantum state is never lost; pure states evolve to pure states. Yes, this part we all want to keep. (It might even be true!) ... There are some obvious places to look for loopholes. I suspect that at least the obvious ones have all been analyzed to death, and all have unpleasant consequences. For instance, Hawking radiation might not really be thermal; but to get back enough correlations seems to require some very nonlocal interactions. That's not clear. if in-falling information just bounces back at the horizon then that's local (as could happen in the eternal equilibrium hole). And if information in region II comes back into view by some curved shape of the non-eternal horizon, then also nothing non-local is needed.. [...] The problem only appears quantum mechanically, when you allow black holes to evaporate thermally via Hawking radiation. And claiming that Hawking radiation is a pure state, statement i) in the above Polchinsky paper, does not avoid the paradox either. I would really like to see a diagram/coordinate system describing the non-eternal black hole for a better judgement! -- Jos |
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