[math-fun] black hole paradox (Hawking radiation)
From: Mike Stay <metaweta@gmail.com> http://en.wikipedia.org/wiki/Thorne-Hawking-Preskill_bet
--I fail to see the relevance. I ask "did Joe fall in, or not?"
From: Henry Baker <hbaker1@pipeline.com> Story 3: Scenario 1. I'm looking out at the universe from the Earth, and in every direction I see a red shift which is larger, the "further away" it is. I conclude that I am in an expanding universe.
Scenario 2. I'm looking out at the universe from the Earth, but at some very large distance from Earth, there is a gigantic black hole, which bends _all_ the light that I see, so that no matter in which direction from Earth I look, I'm actually seeing the same black hole, but from different directions. Furthermore, objects & radiation going into this black hole become red-shifted due to the enormous gravity of the black hole.
How do I tell these two scenarios apart? Does it even matter; perhaps they are equivalent?
I got this idea by considering what our universe looks like to Joe as he approaches the black hole; the black hole actually fills up _more_ than 50% of the sky, due to the bending of the light rays. Just before Joe falls in, the black hole fills up 99.9% of Joe's sky, and the universe that he's exiting looks like a very distant black hole to him.
--Interesting claims, but I doubt true. First, the "universe Joe is exiting" is blue shifted in his view not redshifted. Second, light rays from the black hole horizon surface cannot reach Joe (while outside hole) at all. What Joe sees is not immediately obvious to me, though I think somebody has computed a movie and there is known math on this. I think Joe's view will be very distorted like thru a funhouse mirror but he'll see the outside-the-hole universe in over 50% of his perceptual sky.
From: Robert Munafo <mrob27@gmail.com> Since I am totally unable to understand Hawking, I suppose I am qualified to give the obvious answer, and not feel too bad about being wrong (-:
--you don't need to understand Hawking, you just need to know the general claimed properties of Hawking radiation (I think).
Mary, while watching the black hole shrink, sees the red-shifted image of Joe disappear. This happens after a sufficient length of time that Joe's image has already been redshifted so far that the uncertainty principle (viewed as the energy-time relation deltaEdeltat > h) prevents Mary from distinguishing when Joe's red-shifted image became invisible.
--true. Joe gets exponentially red shifted in Mary's view, which soon overcomes even Mary's ability to wait 10^70 years. So Mary's "view" of Joe "never falling in" effectively gets blacked out, she can't see him with normal light. If however Joe were transmitting magic nonphoton information-particles which just carry info not energy, at lightspeed, then Mary would continually "see" Joe never falling in.
So the "paradox" created by quantum effects (i.e. Hawking evaporation) is also resolved by quantum effects.
--not really. I mean, Joe still never falls in, even though Mary has trouble seeing him. --A better objection is this: in Mary's view, Joe gets ultra-near the horizon without fall in. But as the black hole Hawking-shrinks, the horizon shrinks, ultimately to far smaller than an atomic nucleus, with Joe still incredibly close to (but outside) it. Hence Joe really is crushed, even in Mary's view where he never falls in. But even that does not really resolve the paradox. Here is why: Joe's component particles (neutrons, etc) stay outside, in Mary's view. Their characteristic counts (e.g. "lepton number") hence are preserved. But in Joe's view, those particles all fell in. The Hawking radiation then knows only about the mass and charge of the hole, and will NOT preserve lepton number, etc. So Joe still is preserved (Mary's view) versus not (Joe's view) for some weaker notion of "preserved." This still seems a contradiction.
On 5/27/12, Warren Smith <warren.wds@gmail.com> wrote:
Joe jumps into a Schwarzschild black hole. Mary stays behind and watches him fall.
The standard stories are then as follows: [...] 4. NOW, new addition to above stories. The black hole gradually is shrinking due to "Hawking radiation." It will vanish completely in about 10^70 years (let's say).
So Mary, who is very long-lived, will then say "the new state of the universe is: there is no black hole anymore, Joe never quite fell in, and so Joe is still with us. In fact, I'm going to go meet him now, since he's only 1 hour older by his personal clock."
But Joe will say "I fell in, I was crushed into a point, and my mass-energy was Hawking-radiated away. I'm gone."
How do you resolve this paradox?
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
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On Sun, May 27, 2012 at 1:55 PM, Warren Smith <warren.wds@gmail.com> wrote:
From: Mike Stay <metaweta@gmail.com> http://en.wikipedia.org/wiki/Thorne-Hawking-Preskill_bet
--I fail to see the relevance. I ask "did Joe fall in, or not?"
That's exactly the subject of the bet. Hawking's resolution was to say this: "So in the end, everyone was right, in a way. Information is lost in topologically nontrivial metrics, like the eternal black hole. On the other hand, information is preserved in topologically trivial metrics. The confusion and paradox arose because people thought classically, in terms of a single topology for spacetime. It was either R^4, or a black hole. But the Feynman sum over histories allows it to be both at once. One can not tell which topology contributed the observation, any more than one can tell which slit the electron went through, in the two slits experiment. All that observation at infinity can determine is that there is a unitary mapping from initial states to final, and that information is not lost." So whether there was a black hole or not, the information about Joe does not (according to Hawking) get lost in the black hole, but is recoverable from all the Hawking radiation that eventually leads to the evaporation of the black hole. John Baez wrote this in his summary of Hawking's presentation [my notes in brackets]: ====== He's studying the process of creating a black hole and letting it evaporate away. He's imagining studying this in the usual style of particle physics, as a "scattering experiment", where we throw in a bunch of particles and see what comes out. Here we throw in a bunch of particles [Joe], let them form a black hole, let the black hole evaporate away, and [Mary] examine[s] the particles (typically photons for the most part) that shoot out. The rules of the game in a "scattering experiment" are that we can only talk about what's going on "at infinity", meaning very far from where the black hole forms - or more precisely, where it may or may not form! The advantage of this is that physics at infinity can be described without the full machinery of quantum gravity: we don't have to worry about quantum fluctuations of the geometry of spacetime messing up our ability to say where things are. The disadvantage is that we can't actually say for sure whether or not a black hole formed. At least this seems like a "disadvantage" at first - but a better term for it might be a "subtlety", since it's crucial for resolving the puzzle: [Hawking:] Black hole formation and evaporation can be thought of as a scattering process. One sends in particles and radiation from infinity, and measures what comes back out to infinity. All measurements are made at infinity, where fields are weak, and one never probes the strong field region in the middle. So one can't be sure a black hole forms, no matter how certain it might be in classical theory. I shall show that this possibility allows information to be preserved and to be returned to infinity. [snip some stuff on Euclidean path integrals...] He says that if you just do the integral over geometries near the classical solution where there's no black hole, you'll find - unsurprisingly - that no information is lost as time passes. He also says that if you do the integral over geometries near the classical solution where there is a black hole, you'll find - surprisingly - that the answer is zero for a lot of questions you can measure the answers to far from the black hole. In physics jargon, this is because a bunch of "correlation functions decay exponentially". So, when you add up both answers to see if information is lost in the real problem, where you can't be sure if there's a black hole or not, you get the same answer as if there were no black hole! ====== -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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