Friday, July 22, 2005

Hawking and unitarity

The previous blog article about the very same topic was here.

Stephen Hawking who may currently be the world's most famous applied string theorist among the public has finally submitted a paper that many of us were eagerly expecting for a year or two. The paper is titled "Information Loss in Black Holes" and its preprint number is hep-th/0507171. Because it is less than 5 pages long, I recommend you to read it.

On the first page, he summarizes the history of the information loss puzzle. In 1967 the no-hair theorems started to appear: the black holes are classically more or less unique solutions determined by a few parameters. Because they don't have any hair, they can't wear any haircuts that would distinguish them from other black holes with the same value of conserved quantities; they don't have any features that could give them a large entropy.

This is strange because the black holes seem to be the final outcome of a gravitational collapse, and according to the 2nd law of thermodynamics, the final states should maximize the entropy. The apparently vanishing entropy seems to contradict this law. However, the black holes in the classical theory are eternal and we may envision the information as being stored inside the hole; it is just not accessible to the folks outside.

This argument fails in the quantum theory because of the line of reasoning pioneered by Jacob Bekenstein and Stephen Hawking. Black holes eventually evaporate, via the Hawking process, which eventually uncovers all the details of their interiors. The nonzero temperature may be used to derive their entropy via the equations of thermodynamics; the entropy happens to be proportional to the horizon's area (for large black holes; the extensive progress in determining all the corrections from string theory is discussed elsewhere), as first predicted by Bekenstein.

However, Hawking's semiclassical calculation leads to an exactly (piecewise) thermal final state. Such a mixed state in the far future violates unitarity - pure states cannot evolve into mixed states unitarily - and it destroys the initial information about the collapsed objects which is why we call it "information loss puzzle". A tension with quantum mechanics emerges.

There have been roughly three major groups of answers that people proposed.
  1. One of them is essentially dead today; it is the remnant theory. It argued that the black hole does not evaporate completely. Instead, a small light remnant with a large entropy remains after the evaporation process - and this remnant is what preserves the information. This approach is highly disfavored today because such small seeds simply should not be able to carry large entropy (because it violates holography). Moreover, this approach does not save unitarity anyway because the scenario still assumes the thermal radiation to be in a mixed state.
  2. The other two general answers are obvious. One of them says that the information is lost, indeed. The qualitative features of Hawking's semiclassical calculations - the evolution into mixed states - survive in the exact analysis, too. Such an approach is popular among the General Relativity fundamentalists who believe that the fabric of spacetime is exactly what we think it is classically; causality in particular must be exact and no information can ever get out from a black hole. I formulated the argument in a way that makes it clear that it looks dumb to me - especially today when we know that topology of space may change and that black holes exist in unitary backgrounds of string theory. The Hawking process itself is an example of a violation of the strict rules of locality and causality by black hole physics!
  3. The last answer, the only one that has always respected the principles of the 20th century physics, says that the information is preserved in the same way as in any other process in the world - burning books is an example. (Only later, I noticed that Hawking has independently chosen the very same example.) When we burn books, it looks as though we are destroying information, but of course the information about the letters remains encoded in the correlations between the particles of smoke that remains; it's just hard to read a book from its smoke. The smoke otherwise looks universal much like the thermal radiation of a black hole. But we know that if we look at the situation in detail, using the full many-body Schrödinger equation, the state of the electrons evolves unitarily.

    The same thing must hold for black holes. And the feeling that such a transfer of information is impossible because of the horizon is just an illusion; it is an artifact of the semiclassical approximation that paints the rules of locality and causality as more strict than they are in the full theory. Locality and causality are, in general, approximate emergent concepts that appear in the (semi)classical limit. The power of the full theory of quantum gravity to violate locality and causality in a subtle way is manifested whenever horizons develop, and it is responsible for the conservation of the information.

Note that the conservation of the information is the only answer that can be acceptable for a physicist who treats the postulates of quantum mechanics seriously. No doubt, the postulates of quantum mechanics seem rigid and un-modifiable, while the exact degrees of freedom and terms in the Lagrangian that describe general relativity are flexible. The quantum mechanical postulates have a higher priority, and they tell us that the information must be preserved in the details of the nearly thermal Hawking radiation that remains after the black hole disappears.

While Stephen Hawking has believed that the information was lost - and he has made bets of this kind - he eventually switched to our side in the summer of 2003 or 2004 (I am uncertain now). As you could hear from CNN and other major global new agencies, he officially admitted that his opinion was incorrect. The deep insights in string theory have convinced him that John Preskill was right and the bet is lost; Hawking gave an encyclopedia to Preskill as promised.

Among these insights that have convinced Hawking, you find Matrix theory and especially the AdS/CFT correspondence. Gravity in asymptotically AdS spaces has an equivalent description in terms of a conformal field theory living on its boundary. This conformal field theory is manifestly unitary and has no room for destruction of the information. This answers an equivalent question about gravity, too.

This brings most sane physicists to the opinion that the information is preserved and gravitational physics is not that special after all. But it does not give us a quantitative, calculable framework that would explain how does the information get out of the black holes and what do these subtle correlations that remember the initial state look like.

Hawking's recent solution

Hawking has announced that he had solved the problem. The main ideas of his solutions are the following ones:

  • The scattering S-matrix is the main "nice" observable that should be calculated in a theory of quantum gravity. (I fully agree.)
  • The scattering does not prevent a black hole from being formed, but such a black hole is just like any other intermediate state or resonance. (I fully agree.)
  • The thermal nature of the resulting radiation is a consequence of an approximation (that becomes accurate for large black holes) but there is no qualitative difference between black hole intermediate states and other intermediate states; the transition if smooth. (It was actually just me who formulated this point in this way.)
  • Just like in quantum field theory, the Euclidean setup combined with the Wick rotation is an essential technical tool to do the calculations; Hawking refers to Euclidean gravity as the "only sane way" to do quantum gravity. In the gravitational context, this approach was promoted and improved by Hawking and Gibbons. In fact, the Euclidean approach may be even more important in quantum gravity than it is in quantum field theory and its procedures may represent am even larger fraction of the derivations in the gravitational context. (I agree, and as far as I know, the people who disagree - such as Jacques Distler - have not offered any rational and valid arguments so far.)

OK, so Hawking tells you to calculate the S-matrix by a Euclidean path integral over topologically trivial configurations (spacetimes) - those that are continuously connected to the empty spacetime. Such a process may involve a production of a large number of particles in the final state which is a hallmark of an intermediate black hole. Once you calculate the Euclidean S-matrix, you Wick rotate the results to get the amplitudes for the Minkowski signature.

Note that we have only included the topologically trivial spacetimes and this is a good choice that preserves unitarity.

On the second page, Hawking proceeds with some technical subtleties. He wants to allow strong gravitational fields to occur even in the initial and final states, it seems. (It does not seem necessary when one talks about the generic S-matrix elements but it is conceivable that these strong fields appear in the Euclidean spacetime anyway.) With strong gravitational fields in place, one can't meaningfully define the wavefunction at time "t" because there is no preferred diff-invariant way of slicing the spacetime.

Hawking solves this by a seemingly bizarre operation. He calculates a partition sum with periodic Euclidean time instead of the transition amplitude; it is not 100% clear at this point how will he introduce the initial and final states to this setup. (Note that the Euclidean time is spacelike and it should therefore not be interpreted as a source of the usual violation of causality.) Moreover, this partition sum has a volume-extensive divergent factor. Hawking regulates this infrared problem by introducing a small negative (anti-de-Sitter-like) cosmological constant that does not change local physics of small black holes much.

He obviously deforms the picture into an AdS one in order to get a background that is as well-defined as the usual AdS/CFT backgrounds in string theory. Hawking states that because we are making all measurements at infinity, we can never be sure whether a black hole is present inside or not.

This looks like cheating to me; equivalently, it suggests that no true solution is being looked for. Of course that if we only work with the boundary degrees of freedom, we will see no unitarity violations and no problems associated with the black hole dynamics. It's simply because all these things are encoded in the CFT which is unitary. The true surviving question is how is this unitary description reconciled with the bulk interpretation in which a macroscopic black hole is demonstrably present and has the potential to cause information loss headaches.

Hawking does not have a working convergent path integral beyond the semiclassical approximation, but let us join Hawking and pretend that this problem is absent. He computes the partition sum over geometries whose boundaries are topologically S^2 (the sphere at infinity) times an S^1 (the periodic Euclidean time) at infinity; he works in four spacetime dimensions. There are two simple spacetimes with this boundary: B^3 times S^1 is the empty flat (or anti-de-Sitter) spacetime while S^2 times D^2 is the anti-de-Sitter Schwarzschild topology.

While the empty spacetime can be foliated, the S^2 times D^2 cannot because it has no S^1 factor, roughly speaking. Because it can't be foliated, you can't even define what the conservation of the information should mean in this topologically non-trivial case. The contribution to the correlators coming from the topologically trivial case are conserved as the Lorentzian time T grows; the contributions from the topologically non-trivial backgrounds decay.

On page 3, Hawking confirms that he was inspired by Maldacena's hep-th/0106112 about the eternal black holes in anti de Sitter space. In that case, you also have two - actually three - geometries that fit into the S^1 times S^2 boundary: empty space, small black holes, large black holes (compared to the radius of curvature). The large black holes dominate the ensemble; they have a large negative action. Nevertheless, using the bulk techniques you may calculate that a correlator of O(x)O(y) on the boundary decays for large separations (while it has the usual flat-space behavior if x,y are nearby).

Such a decrease looks much like other cases of information loss; nevertheless in this case you may argue that there is a unitary CFT behind it and the exponential decrease may be in principle reduced to repeated scattering. Maldacena also showed that the contribution of the empty spacetime does not decay and it has the right magnitude to be consistent with unitarity; Hawking argues that he strengthened this observation by having showed that the path integral over topologically trivial spacetimes only is unitary. (Again, it is not obvious whether his formal argument holds in reality because of the usual loop UV problems of general relativity.)

The large black holes are not too interesting because they don't evaporate. Instead, we want to look at the small black holes. Hawking has been trying to find a Euclidean geometry corresponding to an evaporating Lorentzian black hole for years. Now he says that he failed because there is no such geometry. In the Euclidean setup, only the metrics that can be foliated - empty space and eternal black holes - should be added to the path integral.

One of the main question that you must certainly ask is: Why does dynamics over topologically trivial spacetime look like the creation of a long-lived black hole with horizons in the Lorentzian signature? I believe that Hawking does not fully answer this question; he only says that "thermal fluctuations may occasionally be large enough to cause a gravitational collapse that creates a small black hole". Let me re-iterate that such a short comment is deeply unsatisfactory. What we want to understand in the first place is the bulk description of the process in which we can see that the usual long-lived black hole is there; we want to see how are the concepts of locality and causality corrected so that the information can escape.

Hawking only says that this solution of the information loss puzzle is possible. We could have said the same thing just because there is a dual unitary CFT description. But the local bulk dynamical mechanisms that make these things possible remain nearly as cloudy as before.

Some of Hawking's conclusions say:

  • There are no baby universes branching off - which is what Hawking used to think. The information is preserved purely in our Universe.
  • The black hole can form while remaining topological trivial because its evaporation may be viewed as a tunnelling process (Hartle-Hawking). Although this comment can't be considered to be a quantitative answer to my main question, I like it, and let me describe an analogy.

Imagine quantum mechanics of a particle on a line. The classically inaccessible regions (E smaller than V) may be compared to the black hole interior. Classically, these are qualitatively different regions from the rest. However, quantum mechanically, the qualitative difference disappears because of tunnelling. All points on the line are qualitatively on equal footing. You can get there. This is why the black hole should be thought of as having a trivial topology quantum mechanically. The situation would change for an infinite inaccessible region (infinite black hole) where you can't tunnel.

Let me summarize: Hawking's argument why the evolution is unitary probably works and The Reference Frame agrees with virtually all of Hawking's broader opinions, but such a solution is not much different from the observation that the dual CFT is unitary. The question why these unitary processes look like a small long-lived black hole and how the necessary correlations are created remains mostly unanswered.

Hawking has lost a bet but he seems to think that he has made the critical steps to solve the information loss puzzle. While he has given the encyclopedia of baseball to John Preskill, next time he will give him the ashes from a burned book (or the nearly thermal Hawking radiation) because John Preskill can always reconstruct the information out of them.


  1. Lubos said:
    "next time he will give him the ashes from a burned book (or the nearly thermal Hawking radiation) because John Preskill can always reconstruct the information out of them."

    Not so easily, Hawking would have to give not just the ashes, but also every little bit of debrises and that could be flying away, and every single photons emitted during the burning, as well as preserve the exact direction and time that the photons fly away. You need every little bit of quantum information to reconstruct the book. Hawking might as well could be easier give Preskill a time machine to allow him travel back to the moment right before the book was burned, or, just give the book.

  2. Lubos, I believe there is an hidden variable contained within the end-script by Hawking. The Ashes:
    may be with reference to Cricket, rather than Baseball?

    There is also a saying in the English Language:Its Just not Cricket!..which has the meaning of, Rule's have been Broken?

    So in the context of say, Baseball, one can state that if a player has gained an unfair advantage, by say bending the Rule Book, English people tend to shout:It's just not Cricket! the hawking paragraph again , and one can see a little bit of Hawking 'handwaving', tinged with a famous sense of humour?

  3. Thanks for an accessable explaination of this situation. I caught part of the recent Discovery Channel program on this tonight in a bar with poorly-written subtitles, and needed a quick catch-up on the current state of things in this world. I can't wait for a book to come out that is as comprehensible to a casual user of high science like myself as Gleik's Chaos was back in high school, but covering more recent events.