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EDIT: I edited the question to reflect Moshe's objections. Please, look at it again.


It's apparently a black hole time around here so I decided to ask a question of my own.

After a few generic questions about black holes, I am wondering whether string theory is able to provide something beyond the usual semiclassical Hawking radiation talk. Feel free to provide an answer from the standpoint of other theories of quantum gravity but AFAIK none of the other theories has yet come close to dealing with these questions. That's why I focus on string theory.

So let's talk about micro black holes. They have extreme temperature, extreme curvature, and I guess they must be exceptional in other senses too. At some point the gravitational description of these objects breaks down and I imagine this kind of black hole could be more properly modeled like a condensate of some stringy stuff. So let's talk about fuzzballs instead of black holes.

  1. What does that microscopic fuzzball model look like?
  2. What does string theory tell us about the evaporation of those fuzzballs? Is the Hawking radiation still the main effect (as for the regular black holes) or do other phenomena take over at some point?
  3. Also feel free to add any other established results regarding black hole decay (as Jeff did with information preservation).
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I'm also under the impression that black holes in string theory are constrained to be extremal Nordstrom holes, which are thermodynamically very strange objects. (of course, I could be wrong, too). Also, this calculation has been done explicitly in LQG, with a result that gets the Hawking entropy, as well as higher order (in $\hbar$) corrections to the law. And LQG naturally has a smallest possible area governed by the Hubble length. –  Jerry Schirmer Jan 7 '11 at 3:09
    
I think your impression is not really correct. Also, you must mean the Planck length. –  user566 Jan 7 '11 at 3:52
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Q1. is analogous to asking what is the smallest droplet of water allowed by hydrodynamics? Q2. is answered once you frame Q1 correctly. I think a better question might be: "What does ST tell us about the atoms of space. By what mechanism do these assemble together to yield Black Holes (and other states of geometry) in a thermodynamic limit?" –  user346 Jan 7 '11 at 4:14
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It would be great if someone could spell out the actual steps involved in a string-theory calculation of Black Hole entropy, for the simplest case. –  user346 Jan 7 '11 at 7:14
    
@Jerry: I think you are not talking about the black hole decay? I thought LQG can't even start to approach that kind of questions because it doesn't incorporate Standard Model (or has something changed in the recent years?). Same way black holes are created from dense matter, dense matter should be the final products of the black hole decay. Or is this assumption of mine wrong? –  Marek Jan 7 '11 at 8:49
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4 Answers 4

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Moshe's answer gets the main point across. If you're interested in trying to learn something more detailed, here's a thought about where you might start. (I'm not even close to expert on these matters, but this is something that I've found readable and worthwhile.)

There's an interesting thread of literature including http://arxiv.org/abs/hep-th/9309145 by Susskind and http://arxiv.org/abs/hep-th/9612146 by Horowitz and Polchinski, along with other papers that those might lead you to. The idea is that free strings have a certain entropy, in the sense that for a given mass, there are many different combinations of oscillator modes you can turn on to find a string state of that mass. The string energy is proportional to its length, and a typical string of a given energy looks roughly like a random walk of a particular length. Once you turn on a coupling, at some point some of these states will become black holes, because their energy is contained in a region smaller than their Schwarzschild radius. For a given mass, a black hole state also has some entropy. There are various consistency checks you can do on the way these things scale, to see that it's sensible to talk about a sort of smooth transition between string states and black hole states if you fix the energy and vary the coupling.

There are more technical and more precise papers in the literature on black hole entropy and microstate counting, but for a non-expert like me this particular thread of literature seems interesting because it relies on parametric scaling laws that are pretty readily comprehensible, and paints a relatively clear picture of the physics.

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Thank you, this is the line of thought I was interested in. –  Marek Jan 7 '11 at 9:16
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I think there is an interesting question in there, but you are not quite there yet.

The mental picture of black holes in string theory is as collection of many microscopic ingredients in some generic complicated state, which looks for almost all intents and purposes as thermal state made of these constituents. If you think about an analogy to a gas made of many molecules, your questions are analogous to asking what is the smallest possible gas and what is the fate of such a gas. Roughly speaking, when the system becomes smaller the thermodynamic description becomes less accurate, and at some stage you'd have to discuss the dynamics of the individual constituents, none of which is of course described by thermodynamics.

Similarly, when black holes become smaller, they become highly curved and are described less and less accurately by their geometry alone, ultimately you'd need to discuss the microscopic physics of their constituents. I personally don't expect that individual constituents are usefully described as black holes, or any other classical geometry for that matter, at least not in general. However, the fuzzball program of Mathur's seeks exactly such description, at least partially and under favourable conditions, so who knows.

To paraphrase slightly what I am saying is that black holes in GR should be thought of as coarse grained (thermodynamic) description of a complicated system (whose details, amounting to a statistical mechanical description of the same system, are exactly known in string theory in some special cases). When the black hole gets smaller, corresponding to your micro-black holes question, you are starting to talk about a small collection of those constituents. Those objects are no longer black holes, as those are defined by some aspect of their geometry, which is no longer a useful description of the system.

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Huh? I don't get the analogy with gas. I am not asking for the description of the micro-BH by GR (which would be an analogue to thermodynamics here and indeed must break for those BH). I am asking for description by string theory, which is the ultimate microscopic theory of those BH. If the complete picture is not known (or you are not aware of it), then just say so. –  Marek Jan 7 '11 at 1:35
    
Certainly the complete picture is not known, but I think enough is known to address your question. I tried to paraphrase what I mean in the original answer. –  user566 Jan 7 '11 at 2:06
    
thanks, I got your point now; I guess I was half asleep as I read your answer yesterday. I'll try to reformulate my question. –  Marek Jan 7 '11 at 8:45
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One can argue formally using the AdS/CFT correspondence that black hole formation and evaporation does not destroy information, contrary to earlier claims by Hawking that it must. This was the basis for Hawking's payoff of a bet he made with John Preskill. The bet, and a brief discussion of some of the issues involved can be found here http://www.theory.caltech.edu/~preskill/jp_24jul04.html

Although Hawking claimed to have solved the problem in a press conference, credit is really due to J. Maldacena for AdS/CFT and for his later work using this to argue for the unitarity of BH formation and evaporation.

Important caveats are that there is no proof of AdS/CFT (although at this point it is very hard to believe it is not correct) and that the argument really doesn't tell one very explicitly just how the information manages to be encoded in the outgoing Hawking radiation.

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Hi Jeff, I was not sure that the question was about information loss, though the thought crossed my mind. While I am at it, I am curious if you believe Hawking's argument for his conclusion. It was a variant of Juan's argument, but that argument applies to large eternal black holes which do not evaporate, and for which the original paradox did not apply. –  user566 Jan 7 '11 at 2:12
    
I think even in Juan's context it was later argued that sum over topologies is not sufficient to restore unitarity. Anyhow, maybe not the right forum for this type of conversation (maybe research oriented site, one of these days...). –  user566 Jan 7 '11 at 2:35
    
To leave absolutely no room for doubt the solution would have to be mapped backed to the gravity language, which is the intent of those papers. How much they succeeded is a matter of opinion. –  user566 Jan 7 '11 at 3:44
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these are good questions but already assume some fallacies in their very formulation. In particular, it is misleading to mix "fuzzballs" and "Hawking evaporation" in the same sentence.

If you're talking about fuzzballs, then you are talking about very particular microstates of the black holes. There are many of them - $\exp(S)$ of them where $S$ is the entropy.

The Hawking evaporation - as a radiation of a temperature - was derived in the semiclassical approximation for an object that macroscopically behaves as a black hole with certain parameters. To obtain it from the accurate theory - string theory - you have to compute averages over the ensemble of all the microstates, and if you want to be 100% accurate, you have to tell us what your ensemble of states is and what are your weights.

At any rate, this task can be done in full detail, at least in principle, for many black holes in particular descriptions of string theory such as Matrix theory or AdS/CFT. A result is that the low-energy approximation of course fully agrees with the conclusions derived by Hawking semiclassically. Moreover, it may be showed that the new physics of string theory guarantees that the information is preserved.

If you talk about a particular fuzzball, you may also calculate its decay from string theory, at least in principle, but it will be a decay of a very generic and messy object and the decay shouldn't be called Hawking radiation. In particular, a fuzzball is a single microstate, so it should be said to have no entropy. By the relationship between entropy and area of event horizons, it also follows that fuzzballs don't have an event horizon.

Again, by averaging over the microstates or fuzzballs, one may derive the semiclassical results that look like temperature-T radiations. I won't write that these things can't be calculated in string theory because such a statement would be untrue. It is exactly a class of problems that string/M-theory has "fully mastered", at least in principle - but also, in many case, in practice - with an accuracy that is often amazing. The open questions about "quantum gravity" effects in string theory are completely elsewhere than in the decay of black holes which is understood (including many of the stringy corrections to the Hawking approximation, and other things).

Best wishes Lubos

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Thank you Luboš. I didn't realize that fuzzball means a single micro-state; I used it in a sense of "quantum gravity black hole" (because black hole is often understood just as a GR object) as someone (I think Moshe) suggested. So should I just talk about black holes even at the quantum gravity level? –  Marek Jan 14 '11 at 12:34
    
Sorry I caused a confusion. All I was trying to say, in response to the original form of the questions, is that large objects (objects with lots of entropy) can be treated as thermal, you can talk about their temperatures etc., when they become small, they have no longer a thermodynamic description. In particular Lubos is right that a single microstate does not have such a description. There is a programme to realize some such states as "fuzzball" geometries, but I don't think it is useful to get into that specific issue before you understand the general story of black holes in string theory. –  user566 Jan 14 '11 at 15:55
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