I am constantly fighting information loss. Most importantly, there seems to get a lot of information lost in emails I write if those exceed one paragraph. What comes back then is stochastically distributed words that are uncorrelated with what I wrote. Anyway, as I had to realize yesterday, an inbox can also turn into a black hole if IT moves the folders but doesn't tell you how to reconfigure the ssh tunnel.
Not the Paradox
So, what is the black hole information loss paradox? The evolution laws in quantum mechanics are time-reversal invariant. (That does not include the measurement process, which does set limits to our knowledge). Initial states evolve into final states, the evolution is given by a Hamiltonian and is unitary. You can turn it back around. If you start with something it will go into something with probability one. The evolution is a one-to-one map. Unitarity is a fundamental property of quantum mechanics.
Now consider you have some matter distribution (e.g. a pressureless gas) and let it collapse (for simplicity assume it is spherically symmetric). That what you need to specify the precise state I will call information. The collapsing matter forms a horizon and becomes a black hole. The black hole no-hair theorem says that a black hole can carry only three parameters: mass, angular momentum, and electric charge. After the collapsing matter has settled down, this is the only information you can get from examining it. What happened to all the other information of your gas? All the details of that initial state?
Well, you could say, it's inside the black hole. So, no, collapse and formation of a horizon is not the information loss problem. You could say, the information still exists, but we are just disconnected from it. What's the problem with that? As long as my inbox still exists at least somewhere, that's okay, even if I can't access it.
The Paradox
But Hawking tells us black holes emit radiation, and this radiation is thermal. It is purely thermal, completely random, does not contain any information except its temperature. That is in contrast to e.g. the radiation of the sun. Which is for all practical purposes also thermal, but it does contain information 'in principle'. If you'd throw your bag of gas into the sun and you waited long enough you could 'in principle' extract its details again from the sun's radiation. Not so for the black hole. There is nothing to learn from the black hole's radiation.
Still you could say, well, if no information comes out, then it just stays inside.
But if Hawking is right, and the black hole radiates, then it loses mass. And eventually it is completely evaporated. There is nowhere left for the information to remain. The only thing that you have in the final state is that thermal radiation distributed over space. One could say then, well, black hole formation just is not time-reversal invariant. A singularity forms. A singularity is an attractor, whatever you started with it is always equally singular. There is no one-to-one map. Why is that a problem?
Well, the black hole formation could have happened for anything that forms or later falls into the black hole, and we know stuff behaves according to quantum mechanics. We have tested that experimentally uncountably many times. But if you consider an initial quantum state for the black hole, then no matter what precisely it was, the result is always thermal radiation, determined solely by the total mass, charge, and angular momentum. If you look at the final state, you can't trace it back to the initial state. It's not a one-to-one map. The evolution is not unitary, in conflict with the laws of quantum mechanics.
And that's the problem. The paradox. The apparent disagreement between general relativity and the laws of quantum mechanics. Hundreds, if not thousands of papers have been written about it since the solution to this paradox can be a key to our understanding of quantum gravity - whatever that theory looks like it should be able to resolve the problem.
Solution Attempts
A crucial aspect of this problem is that evaporating black holes are to excellent approximation classical objects for a very long time. Quantum gravity only can become important in the very late stages of the evaporation, then when the curvature comes in the Planckian regime, which happens only when the black hole's mass is about Planck mass (or its diameter of the order Planck length respectively.) Quantum gravitational effects thus can only influence the radiation in the last stages.
Various approaches have been tried to solve the problem, all have advantages and disadvantages (this is likely an incomplete list):
- Solution: The evolution just is not unitary and information is indeed lost.
Problem: This is not only unappealing because it requires us to rethink how quantum mechanics works, it also leads to violations of energy conservation.
Reading:
Unitary Rules for Black Hole Evaporation
Andrew Strominger, hep-th/9410187 - Solution: Black hole evaporation is modified in the late stages and black holes do not completely radiate but leave behind a stable remnant of approximately Planck mass that keeps the information.
Problem: Since the initial state that collapsed could have been anything, if the information is kept in the remnant that remnant must be able to carry an arbitrarily high amount of information. This leads you to conclude there must be an in principle infinitely large amount of black hole remnants with the same mass that are however different since they have different information content. This in turn results in the possibility to pair produce these objects infinitely in any arbitrarily complicated process where the energy is high enough. Even if the probability for the production of a single remnant is arbitrarily small, if there are infinitely many of them, you will still produce them. You could also emit them in black hole radiation itself: The high energy tail of the black hole spectrum might be exponentially suppressed, but if multiplied with infinity, black holes of arbitrary mass would decay instantaneously.
Reading:
Comments on information loss and remnants
S.B. Giddings, hep-th/9310101
Constraints on Black Hole Remnants
S.B. Giddings, hep-th/9304027
Trouble For Remnants
Leonard Susskind, hep-th/9501106 - Solution: The information comes out with the radiation in the very late stages.
Problem: Then you have only very little energy left to carry all that information, which could have been arbitrarily much. This means per each unit of information you have a very small amount of energy to emit it which takes a long time, and you can't emit it simultaneously because if the wavefunctions overlap they'd be correlated. So that last stages would last very long (the more information needs to go out the longer) leading to quasi-stable black holes which cause essentially the same problems as the remnants of the previous point (their mass spectrum is not exactly degenerated but almost). Besides this, one would like to have an exact mechanism for how that happens.
Reading:
Do Black Holes Destroy Information?
John Preskill, hep-th/9209058 - Solution: Black holes evaporate completely but while so have formed a causally completely disconnected baby universe in which the information survives.
Problem: Requires you to believe in a multiverse in whose totality information is conserved, though locally, in our universe, it isn't.
Reading:
A Possible Resolution of the Black Hole Information Puzzle
Joseph Polchinski and Andrew Strominger, hep-th/9407008 - Solution: Since the AdS/CFT conjecture relates black holes (in AdS space) to a quantum theory one knows is unitary on the boundary, this would mean the evolution in the bulk is also unitary.
Problem: As long as the conjecture is unproved one could equally well consider the information loss problem, if real, as a counter-example for the validity of the conjecture. (Also, I personally would find it unsatisfactory would this only work in AdS space).
Watching:
The Black Hole Information Paradox, Past and Future
Joe Polchinski, PIRSA: 08040001 - Solution: Black holes have quantum hair that is not taken into account in the no-hair theorem.
Problem: Since the black hole can carry arbitrarily much information, one needs arbitrarily many quantum numbers to do that and a modification of our theories that can accommodate them. Why haven't we yet observed any of that? (Also, it's not clear to me how one would know that the black hole always carries enough of these new quantum numbers.)
Reading:
Quantum Hair on Black Holes
Sidney Coleman, John Preskill and Frank Wilczek, hep-th/9201059 - Solution: Other - There are no black holes / There is no Hawking radiation / There is no spoon, ie we live in a virtual reality and this paradox was created just for the amusement of our programmer.
Problem: Possible but far fetched.
Reading:
The bleakness of reality.
Why I voted "No for other reasons"
From how I stated the problem above, it hopefully became clear why I think the problem is the singularity, not the horizon. The horizon is where information becomes inaccessible, but the singularity is where it gets lost. That's more or less by definition what a singularity is all about. The singularity is where the evolution becomes non-deterministic, where it can't be uniquely continued, it's where all initial states are crunched into the same divergence - already classically. Unlike the classical case however, here we have a scenario where the final state is without singularity again. Thus, the slicing has somehow to pass the singularity*.
But it is generally expected that some version of quantum gravity resolves the singularity and smoothens it out. Then, there is as far as I can see no reason why the evolution should be non-deterministic if the black hole eventually completely evaporates. If there is no slice on which initially different states run together, evolution from the initial to the final state has to be a one-to-one map. Determinism isn't sufficient, but necessary for unitarity. Either way, even if avoiding the singularity would imply unitarity, the problem then was not why, but how the information comes out.
Vaguely related reading:
Black hole evaporation: A paradigm
Abhay Ashtekar and Martin Bojowald, gr-qc/0504029
The paper mentioned in the previos post by Dr. Who (Horowitz and Maldacena, hep-th/0310281) tackles this problem with the uniqueness of the state by imposing a final state boundary condition at the singularity which effectively transfers the information in the outgoing radiation. It's an interesting paper (thanks for pointing it out!), but the solution seems to me very ad hoc.
Bottomline
The black hole information loss paradox makes for an excellent topic over which to argue, because everybody has a different favourite solution. Of course I don't believe any of the above offered solutions, and of course nobody agrees with me. Anyway, here are the preliminary results of our poll "Do Black Holes destroy Information?":
From presently 146 people who voted, the majority, 37.7%, said "No, it comes out in the radiation.". A for me surprising 25.3% said yes, black holes destroy information. 16.5% including me voted "No, for other reasons.", documenting the mentioned plurality of opinions. Again surprising for me a full 8.9% voted for the remnant solution. (Surprising because whenever I say 'remnant' I get shouted down immediately.) Also 8.9% said 'Other' which includes the quantum hair option that I forgot for the poll, and 2.7% think the information survives in a baby universe.
If you didn't yet vote, vote now! If you did, has this post change your mind?
Epilogue
The one-hour phone call to Canada to figure out how to reconfigure my email client and tunnel through to my inbox goes on my mom's bill. I had to re-download 12,442 emails, but no information got lost.
* Andrew: In the paper you mentioned by Zeh: gr-qc/0507051, he has an endstate with singularity, see Fig 1. In this case you can draw surfaces up to arbitrarily late - but finite! - times that do not reach the singularity, but this is not the relevant scenario in which the black hole is eventually completely evaporated.
TAGS: BLACK HOLE, INFORMATION LOSS, PHYSICS
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