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In one of the messenger lectures at Cornell in 2013, Leonard Susskind gave a heuristic argument for black hole complementarity. Suppose Alice is stationed far away from a Schwarzschild black hole and a particle Bob falls freely through the event horizon. There is a conflict of principles in the sense that Alice would believe that the event horizon is very hot for Bob, and think that Bob will be burnt, while Bob will theoretically experience nothing special, given the equivalence principle.

Here is Susskind's argument to resolve the apparent conflict: When Bob is a distance $\lambda$ away from the event horizon, Alice wants to tell if Bob has been burnt by doing an experiment(i.e. by shining photons at Bob and measuring its position). However, in order to resolve Bob (to reduce the measurement uncertainty to below $\lambda$), Alice has to shine photons of short enough wavelength and thus high enough energy (due to the diffraction limit of optics, used in Heisenberg's heuristic demonstration of measurement uncertainty principle). It turns out that by trying to measure the position of Bob and detecting whether or not it has been burnt, the photons sent by Alice would have burnt Bob anyways. So there is some kind of complementarity in place.

However, the argument seems to break down given that the diffraction limit has been broken in recent years by quantum measurement techniques (for example, see this paper by Mankei Tsang). So is it possible to modify Susskind's argument to make it still work? Or is there a more technical and fundamental argument that will circumvent this problem?

Any suggestions for readings related to this topic would also be nice!

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    $\begingroup$ Could you tell me which Messenger Lecture this argument can be found in (and, if possible, roughly where it is located in it)? I'd like to listen to it to make sure I didn't miss any subtleties of the argument. $\endgroup$ Commented Jul 24, 2016 at 18:06
  • $\begingroup$ see this youtube link: youtube.com/watch?v=B2ksDczJOAs the argument starts at about 56:45. But based on your answer, you have found it already presumably :) $\endgroup$ Commented Jul 24, 2016 at 23:45
  • $\begingroup$ I found it, but this will help other people who want to watch it. $\endgroup$ Commented Jul 25, 2016 at 0:10

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Susskind's original argument doesn't work. Alice just needs Bob to send a message to her saying "I'm still alive!" She doesn't have to illuminate Bob.

Of course, it's hard to get a message out from the near-horizon region of a black hole because of the redshift, but there's no theoretical reason that this shouldn't work. Suppose you don't have a illumination source that's bright enough to make it out from the black hole near-horizon. Just send a sequence of Bobs in, one following another. The $k$th Bob picks up the signal from the $(k-1)$st Bob and relays it to the $(k+1)$st Bob. This system will get the signal to Alice no matter how much it is redshifted1.

One could even replace all the Bobs by automated probes if one has moral objections to suicide missions.

1This is the way that optical fiber works; there are repeaters stationed every 100 km or so to keep the signal from fading too much to be detectable. A signal could never make it 5000 km over an optical fiber without repeaters, but people regularly call California from New York with very little noise.

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  • $\begingroup$ I agree with the falsity of Susskind's argument. But, the optical fiber is not affected by redshift, simply intensity, i.e. number of photons. The repeaters regenerate them before the intensity is too low. $\endgroup$
    – Bob Bee
    Commented Jul 25, 2016 at 1:41
  • $\begingroup$ @PeterShor. Thanks for the answer. However, I am still skeptical of the scheme you proposed. Sure, if Bob was a classical observer with a definite position, then we could perform the series of Bob relay process that you described and convey the information "I am alive at $\lambda$ away from BH". But as a quantum entity, Bob would not have a definite position until someone measures it, so I don't see how Bob can even convey any meaningful information before Alice (or whoever else) shoots photons at it. $\endgroup$ Commented Jul 25, 2016 at 7:54
  • $\begingroup$ Of course Bob can get the information out with a relay. From the Bob point of view, he's in a convoy of spaceships that are following each other through relatively flat space ... there are tidal forces pulling the ships in the convoy further apart from each other, but for a large enough black hole these are very small. $\endgroup$ Commented Jul 25, 2016 at 12:16
  • $\begingroup$ You can try using the uncertainty principle to argue that Bob doesn't know exactly where the horizon is. But I don't think that matters. All we really need is for Bob to tell that it's getting hotter, and then send the information to Alice via the relay before the Hawking radiation destroys him. $\endgroup$ Commented Jul 25, 2016 at 12:17
  • $\begingroup$ One potential problem I see is whether Bob has enough time to decide that it's getting hotter and then relay the information to Alice before he gets destroyed. I think Bob can arrange the experiment so he does, but I haven't checked the calculations in detail. $\endgroup$ Commented Jul 25, 2016 at 12:18

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