Wednesday, December 29, 2004

Hawking and information loss

In response to a question (about the recent status of Hawking's 2004 claims) from David Goss, let us start with some well-known history. Hawking was the first person who in 1974 successfully merged (even though just approximately, in what we call the "semiclassical approximation") the laws of general relativity with the laws of quantum field theory to derive a nontrivial quantitative result - namely the Hawking radiation, including its spectrum. Via thermodynamics, it can be also used to derive the black hole entropy.

Hawking's framework to calculate and his insights are beautiful. Many theoretical physicists believe that he deserves a Nobel prize, and we're just unlucky that there are not too many small, radiating black holes around. In one of the unlikely scenarios of "low energy gravity", the LHC could be producing small, radiating black holes - which would be a terrific news for Hawking. But there also exists a confirmation that is the second most convincing one after the experiments: his calculation of the Bekenstein-Hawking entropy has been confirmed by string theory, for very many different non-trivial examples of black holes.

Hawking's and similar calculations have one unpleasant aspect only: they also seem to imply that the initial matter, forming the black hole, is evolving into a mixed thermal state (the radiation that remains after the black hole evaporates) which is completely universal and only depends on some "major" observables describing the initial state such as the total charge and total mass. All the details of the initial state are lost.

This is worrisome: a pure state may evolve into a mixed state. If it were really so, then we would talk about the "information loss". Such a process would violate unitarity - an essential feature of quantum theory that guarantees that a pure state evolves into another pure state, as Schrödinger's equation or the unitary S-matrix imply, and the total probability of all outcomes always equals one. In Hawking's approximation, it is simply true that pure states evolve into mixed states.

Is it true even if you study the black holes with an exact theory? There have been different opinions about this question. Some of them disappeared: for example, the idea that the information is preserved in a "remnant" that is left after the black hole evaporates became very unpopular once it was seen that such a remnant violates the entropy bounds (it carries too much entropy per a very small volume) and nothing like these remnants was seen in string theory.

Well, only two possible answers were left:

  • the information is preserved because of quantum effects that are not visible in the semiclassical approximation, or
  • the information is lost, as Hawking originally thought, and quantum mechanics must be modified. No one knew exactly how such a modification could look like.
Hawking himself thought that the information was lost, indeed, and one needed to modify the rules of quantum mechanics. He has made some bets - with Kip Thorne on his side against John Preskill who's always been an advocate of the information conservatism.

The mainstream approach, I would say, was always that quantum mechanics is preserved in the full theory and the apparent information loss is just an artifact of the semiclassical approximation. This point of view became even stronger when string theory explained the microscopic origin of the black hole entropy, starting with the papers of Strominger and Vafa. In this framework, one obtains the right entropy, and moreover she has a complete control over the quantum states if the "string coupling" is chosen weak. At the weak coupling, the information is manifestly preserved, and it is very natural to think that the same must be true at any coupling. Hawking's own arguments can be circumvented by some sort of stringy non-locality that becomes important in the presence of black holes. Also, the AdS/CFT correspondence and Matrix theory allow the black holes to be described within a completely unitary mathematical formalism. String theory seems to resolve the subtleties connected with the black holes without sacrificing any principles of quantum mechanics - which is one of the reasons why we find this theory so impressive.

Hawking himself realized that these stringy achievements were very strong arguments in favor of the preserved information after all. But he only "switched" to the mainstream opinion in the summer of 2004 when he announced that he had solved the problem and explained why the information is preserved. He also officially declared a defeat in his bet.

Of course, the physicists, much like the laypeople, were interested in Hawking's resolution. Hawking's new answer looks right, and it would be even better if he could really resolve the apparent paradox that appears in the semiclassical approximation. All of us know that Hawking has the capacity to solve such problems. Many people thought that Hawking was inspired by a paper by Maldacena

However, the interviews for media show that Hawking was not quite saying the same things as Maldacena. The immediate predictions of many people who were interested in the subject - and their understanding of Hawking's new insights and the problems with these insights - were the following:

  1. Hawking wants to express the information loss quantitatively in terms of the correlation functions that usually decay exponentially in the presence of the black hole - you may think about the damped "quasinormal ringing modes" that bring the black hole closer to its perfect, e.g. spherical shape
  2. Hawking wants to argue that the correlators also get a contribution from the path integral configurations that don't contain any black holes. You know, this is the standard and key rule of Feynman's approach to quantum mechanics - one must sum over all configurations, including the configurations that the loop quantum gravity people don't find convenient. ;-) These contributions from the "nearly empty spacetime" may be small at the beginning, but at late times they eventually dominate because they are not exponentially supressed. The late time behavior would therefore have the same information properties as an empty spacetime, and the information would therefore be manifestly preserved
  3. Hawking 2004 does not explain how do the "trivial spacetime" configurations conspire in such a way that they "seem" to behave just like dynamics near a black hole. Intuitively, a particular process of formation and evaporation of a black hole is dominated by the black hole intermediate states. By rejecting this assumption, Hawking becomes marginally inconsistent with his old calculations. For example, a possible loophole would be that the dominant "trivial spacetime" contributions to the path integral will "approach" the states that are very close to the black hole (something I've been calling an "almost black hole"), but then the full analysis requires us to understand quantum gravity in the Planckian regime (stretched horizons etc.) which seems as a non-trivial question not answered by Hawking's 2004 interviews
  4. Nearly everyone in the "mainstream" knew that there can be some special quantum gravity phenomena that will resemble the black hole, but avoid the information loss, but Hawking's 2004 argument does not seem to illuminate these new potential phenomena
  5. Because this line of reasoning has been kind of tried, it did not lead to an answer of the question, and an important "missing piece" also seems to be missing in Hawking's interviews, it's reasonable to expect that Hawking did not really solve the information loss paradox in a "final and satisfactory" way, and one would predict that there would not be any technical paper following the interviews that would explain physics behind his interviews in detail

So far, these predictions seem to hold, don't they? I'm happy that Hawking joined the information consevatives and all of us still face more or less the same remaining puzzles.


  1. Lumo - A couple of questions: You say ...his calculation of the Bekenstein-Hawking entropy has been confirmed by string theory, for very many different non-trivial examples of black holes. Last I heard, these were all extremal black holes. Is that still true? Can (BH)^2 entropy be calculated for any of the kind of BH we think we see in galaxy centers and collapsed stars? If not can you explain to a non-string theorist what the impediment is to extending the calculation to non-extremal BH?

    OK, that was more than two questions, but at least my error was less than two orders of magnitude.



  2. Dear P.I.G.,

    what you say was never really true, and I doubt that your sources were well-informed. Many non-extremal black holes have been calculated already in early 1996, a few months after the Strominger-Vafa pioneering paper, like

    In this case, the calculated formula works arbitrarily far from extremality, although the reason why it does work so well is slightly mysterious because missing supersymmetry may be expected to give you corrections. These corrections don't arise, however.

    There are also many cases in which the agreement is under control and it works up to the first non-leading order away from extremality (in a perturbative expansion in "Q-M", so to say), for classes of black holes that can have as many as 7 parameters, see e.g. the first paper in this direction

    Also, up to a numerical coefficient, one can calculate the entropy of all other black holes, including the neutral ones, following a correspondence principle

    Note that all these references are from 1996. Today there is much more known; these 1996 papers have hundreds of followups. There exist no reasonable doubts that the agreement works for all black holes in string theory, including the realistic ones that you will find in realistic models of string theory and that can be compared with the large BH at our Gallactic center, for example.

    The very question "whether the BH formula works in string theory" is certainly not a hot topic anymore because everyone has been assured that it does. There are some active topics calculating the subleading corrections to the entropy, and relating them to topological string theory, D0-D4 branes with lowest Landau levels, attractor mechanisms etc. etc. I could give you a lot of recent references, there are many very active directions in this ballpark.

    Note that all of these checks of the BH entropy are internal consistency checks between microscopic string theory and the semiclassical approximation of gravity: the black hole entropy was never measured experimentally, of course. Nevertheless, these agreements assure us that string theory is a consistent theory of quantum gravity.

    All the best

  3. I don't understand much of the Horowitz-Polchinski paper, but it looks to me from more recent papers like those of Fursaev and Mohaupt that many people still consider that only extremal and near-extremal black holes can be computed from string theory. Also, both those authors suggest that the details of the microscopic (string) theory don't seem to be fundamental to the calculations. Any comment? Any contemporary review article you would recommend?

  4. Hi pig,

    yes, as explained in my previous comment, it's true that the full and exact and controlled entropy calculation, including the numerical coefficient, has only been done for extremal and near-extremal black holes.

    But of course it does not mean that the string theory's prediction for highly non-extremal ones is wrong! On the contrary, everyone is pretty sure that it's correct. We just don't know how to calculate the full result in the complete absence of supersymmetry.

    There are cases where the agreement holds arbitrarily far from extremality, even though no one can show that the calculation is complete away from extremality. A straightforward, but incomplete, string theory calculation "works better than one would expect" in these cases.

    And there are heuristic arguments that the agreement must work for any black holes, but these arguments don't allow one to calculate the numerical coefficient (Horowitz Polchinski).

    For a specific black hole that you calculate, the details of string theory are of course extremely important if you want to calculate the entropy from string theory.

    On the other hand, I fully agree that the statement that large black holes have BH entropy which is A/4G does not depend on any short distance details, and must be generally true in any consistent quantum theory of gravity. I've been trying to make this argument rigorous, too - and I believe that there is a rather rigorous proof of this statement.

    When I say "all consistent quantum theories of gravity", it is actually equivalent to "all formalisms, descriptions, and vacua of string/M-theory", but this equivalence is not necessary for the hypothetical general argument showing S=A/4G itself.

    Well, A/4G is not the only interesting thing in physics. There are many other properties and features of the black holes for which the stringy details are very important.


  5. Conventional measurements of a quantum system result in a pure state collapsing to a mixed state. Presumably this is a semi-classical approximation; in the full theory, with a quantum theory of the measurement apparatus, unitarity should return. Presumably, apart from string theory, no one can compute blackhole formation from any kind of fundamental stuff either.

  6. I'm obviously not the one to dis anonymous posting, but why not use a handle so we can keep who said what straight?

  7. OK, Pig, call me Snowball.