[math-fun] black hole paradox (Hawking radiation)
Joe jumps into a Schwarzschild black hole. Mary stays behind and watches him fall. The standard stories are then as follows: 1. Joe, in finite time as measured by his wristwatch (say 1 hour for concreteness), falls into the center of the black hole and is crushed into a point, albeit if it is a large enough black hole then tides will be small enough that he will still be alive for some time after entering the horizon. The horizon crossing for Joe is just a normal moment, nothing special happens at that moment. 2. Mary, however, sees Joe getting closer and closer to the horizon (and redder and redder) but according to her view Joe never enters the horizon, never is crushed, never dies. 3. These two stories are supposedly noncontradictory because of time distortion approaching infinity as Joe approaches horizon from outside, his rate of time is such that 1 second for Joe is a lifetime for Mary, and so on, the ratio becomes infinite. 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?
http://en.wikipedia.org/wiki/Thorne-Hawking-Preskill_bet On Sun, May 27, 2012 at 12:25 PM, 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: 1. Joe, in finite time as measured by his wristwatch (say 1 hour for concreteness), falls into the center of the black hole and is crushed into a point, albeit if it is a large enough black hole then tides will be small enough that he will still be alive for some time after entering the horizon. The horizon crossing for Joe is just a normal moment, nothing special happens at that moment.
2. Mary, however, sees Joe getting closer and closer to the horizon (and redder and redder) but according to her view Joe never enters the horizon, never is crushed, never dies.
3. These two stories are supposedly noncontradictory because of time distortion approaching infinity as Joe approaches horizon from outside, his rate of time is such that 1 second for Joe is a lifetime for Mary, and so on, the ratio becomes infinite.
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?
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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 (-: 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 ∆E∆t > h) prevents Mary from distinguishing when Joe's red-shifted image became invisible. So the "paradox" created by quantum effects (i.e. Hawking evaporation) is also resolved by quantum effects. 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?
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