July 25, 2004

This Week's Finds in Mathematical Physics (Week 207)

John Baez

I'm spending the summer in Cambridge, but last week I was in Dublin attending "GR17", which is short for the 17th International Conference on General Relativity and Gravitation:

1) GR17 homepage, http://www.dcu.ie/~nolanb/gr17.htm

This is where Stephen Hawking decided to announce his solution of the black hole information loss problem. Hawking is a media superstar right up there with Einstein and Michael Jackson, so when reporters heard about this, the ensuing hoopla overshadowed everything else in the conference.

As soon I arrived, one of the organizers complained to me that they'd had to spend 4000 pounds on a public relations firm to control the reporters and other riff-raff who would try to attend Hawking's talk. Indeed, there seemed to be more than the usual number of crackpots floating about, though I admit I haven't been to this particular series of conferences before - perhaps general relativity attracts such people? The public lecture by Penrose on the last day of the conference may have helped lure them in. He spoke on "Fashion, Faith and Fantasy in Theoretical Physics", and people by the door sold copies of his brand new thousand-page blockbuster:

2) Roger Penrose, The Road To Reality: A Complete Guide to the Physical Universe, Jonathan Cape, 2004.

(You may enjoy guessing which popular theories he classified under the three categories of fashion, faith and fantasy.) After his talk, all the questions were actually harangues from people propounding idiosyncratic theories of their own, and the question period was drawn to an abrupt halt in the middle of one woman's rant about fractal cosmology. But I bumped into the saddest example when I was having a chat with some colleagues at a local pub. A fellow with long curly grey locks and round horn-rimmed glasses sat down beside me. I'd seen him around the conference, so I said hello. He asked me if I'd like to hear about his theory; at this point my internal alarm bells started ringing. I told him I was busy, but said I'd take a look at his manuscript later.

It turned out to describe an idea I'd never even dreamt of before: a heliocentric cosmology in which the planets move along circular orbits with epicycles a la Ptolemy! And his evidence comes from a neolithic Irish tomb called Newgrange. This tomb may have been aligned to let in the sun on the winter solstice, but some people doubt this, because it seems the alignment would have been slightly off back in 3200 BC when Newgrange was built. However, it's slightly off only if you work out the precession of the equinox using standard astronomy. If you use his theory, it lines up perfectly! Pretty cute. The only problem is that his paper contains no evidence for this claim. Instead, it's only a short note sketching the idea, followed by lengthy attachments containing his correspondence with the Dublin police. In these, he complained that people were trying to block his patent on a refrigerator that produces no waste heat. They were constantly flying airplanes over his house, and playing pranks like boiling water in his teakettle when he was away, trying to drive him insane.

Anyway, on Wednesday the 21st the whole situation built to a head when Hawking gave his talk in the grand concert hall of the Royal Dublin Society. As we had been warned, the PR firm checked our badges at the door. Reporters with press badges were also allowed in, so the aisles were soon lined with cameras and recording equipment. I got there half an hour early to get a good seat, and while I was waiting, Jenny Hogan from the New Scientist asked if she could interview me for my reaction afterwards. In short, a thoroughly atypical physics talk!

But you shouldn't imagine the mood as one of breathless anticipation. At least for the physicists present, a better description would be something like "skeptical curiosity". None of them seemed to believe that Hawking could suddenly shed new light on a problem that has been attacked from many angles for several decades. One reason is that Hawking's best work was done almost 30 years ago. A string theorist I know said that thanks to work relating anti-deSitter space and conformal field theory - the so-called "AdS-CFT" hypothesis - string theorists had become convinced that no information is lost by black holes. Thus, Hawking had been feeling strong pressure to fall in line and renounce his previous position, namely that information is lost. A talk announcing this would come as no big surprise.

After a while Kip Thorne, John Preskill, Petros Florides and Hawking's grad student Christophe Galfard came on stage. Then, amid a burst of flashbulbs, Hawking's wheelchair gradually made its way down the aisle and up a ramp, attended by a nurse - possibly his wife, I don't know. He had been recently sick with pneumonia.

Once Hawking was on stage, the conference organizer Petros Florides made an introduction, joking that while physicists believe no information can travel faster than light, this seems to have been contradicted by the speed with which the announcement of Hawking's talk spread around the globe. Then he recalled the famous bet that Preskill made with Hawking and Thorne. In case you don't know, John Preskill is a leader in quantum computation at Caltech. Kip Thorne is an expert on relativity, also at Caltech, one of the authors of the famous textbook "Gravitation", and now playing a key role in the LIGO project to detect gravitational waves.

The bet went like this:

Whereas Stephen Hawking and Kip Thorne firmly believe that information swallowed by a black hole is forever hidden from the outside universe, and can never be revealed even as the black hole evaporates and completely disappears,

And whereas John Preskill firmly believes that a mechanism for the information to be released by the evaporating black hole must and will be found in the correct theory of quantum gravity,

Therefore Preskill offers, and Hawking/Thorne accept, a wager that:

When an initial pure quantum state undergoes gravitational collapse to form a black hole, the final state at the end of black hole evaporation will always be a pure quantum state.

The loser(s) will reward the winner(s) with an encyclopedia of the winner's choice, from which information can be recovered at will.

Stephen W. Hawking, Kip S. Thorne, John P. Preskill
Pasadena, California, 6 February 1997

After a bit of joking around and an explanation of how the question session would work, Hawking began his talk. Since it's fairly short and not too easy to summarize, I think I'll just quote the whole transcript which I believe Sean Carroll got from the New York Times science reporter Dennis Overbye. I've made a few small corrections.

There were also some slides, but you're not missing a lot by not seeing them. The talk was not easy to understand, so unless quantum gravity is your specialty you may feel like just skimming it to get the flavor, and then reading my attempt at a summary.

The talk began with Hawking's trademark introduction, uttered as usual in his computer-generated voice:

Can you hear me?

I want to report that I think I have solved a major problem in theoretical physics, that has been around since I discovered that black holes radiate thermally, thirty years ago. The question is, is information lost in black hole evaporation? If it is, the evolution is not unitary, and pure quantum states, decay into mixed states.

I'm grateful to my graduate student Christophe Galfard for help in preparing this talk.

The black hole information paradox started in 1967, when Werner Israel showed that the Schwarzschild metric, was the only static vacuum black hole solution. This was then generalized to the no hair theorem: the only stationary rotating black hole solutions of the Einstein-Maxwell equations are the Kerr-Newman metrics. The no hair theorem implied that all information about the collapsing body was lost from the outside region apart from three conserved quantities: the mass, the angular momentum, and the electric charge.

This loss of information wasn't a problem in the classical theory. A classical black hole would last for ever, and the information could be thought of as preserved inside it, but just not very accessible. However, the situation changed when I discovered that quantum effects would cause a black hole to radiate at a steady rate. At least in the approximation I was using, the radiation from the black hole would be completely thermal, and would carry no information. So what would happen to all that information locked inside a black hole, that evaporated away, and disappeared completely? It seemed the only way the information could come out would be if the radiation was not exactly thermal, but had subtle correlations. No one has found a mechanism to produce correlations, but most physicists believe one must exist. If information were lost in black holes, pure quantum states would decay into mixed states, and quantum gravity wouldn't be unitary.

I first raised the question of information loss in '75, and the argument continued for years, without any resolution either way. Finally, it was claimed that the issue was settled in favour of conservation of information, by AdS/CFT. AdS/CFT is a conjectured duality between supergravity in anti-deSitter space and a conformal field theory on the boundary of anti-deSitter space at infinity. Since the conformal field theory is manifestly unitary, the argument is that supergravity must be information preserving. Any information that falls in a black hole in anti-deSitter space, must come out again. But it still wasn't clear how information could get out of a black hole. It is this question I will address.

Black hole formation and evaporation can be thought of as a scattering process. One sends in particles and radiation from infinity, and measures what comes back out to infinity. All measurements are made at infinity, where fields are weak, and one never probes the strong field region in the middle. So one can't be sure a black hole forms, no matter how certain it might be in classical theory. I shall show that this possibility allows information to be preserved and to be returned to infinity.

I adopt the Euclidean approach, the only sane way to do quantum gravity non-perturbatively. [He grinned at this point.] In this, the time evolution of an initial state is given by a path integral over all positive definite metrics that go between two surfaces that are a distance T apart at infinity. One then Wick rotates the time interval, T, to the Lorentzian.

The path integral is taken over metrics of all possible topologies that fit in between the surfaces. There is the trivial topology: the initial surface cross the time interval. Then there are the nontrivial topologies: all the other possible topologies. The trivial topology can be foliated by a family of surfaces of constant time. The path integral over all metrics with trivial topology, can be treated canonically by time slicing. In other words, the time evolution (including gravity) will be generated by a Hamiltonian. This will give a unitary mapping from the initial surface to the final.

The nontrivial topologies cannot be foliated by a family of surfaces of constant time. There will be a fixed point in any time evolution vector field on a nontrivial topology. A fixed point in the Euclidean regime corresponds to a horizon in the Lorentzian. A small change in the state on the initial surface would propagate as a linear wave on the background of each metric in the path integral. If the background contained a horizon, the wave would fall through it, and would decay exponentially at late time outside the horizon. For example, correlation functions decay exponentially in black hole metrics. This means the path integral over all topologically nontrivial metrics will be independent of the state on the initial surface. It will not add to the amplitude to go from initial state to final that comes from the path integral over all topologically trivial metrics. So the mapping from initial to final states, given by the path integral over all metrics, will be unitary.

One might question the use in this argument, of the concept of a quantum state for the gravitational field on an initial or final spacelike surface. This would be a functional of the geometries of spacelike surfaces, which is not something that can be measured in weak fields near infinity. One can measure the weak gravitational fields on a timelike tube around the system, but the caps at top and bottom, go through the interior of the system, where the fields may be strong.

One way of getting rid of the difficulties of caps would be to join the final surface back to the initial surface, and integrate over all spatial geometries of the join. If this was an identification under a Lorentzian time interval, T, at infinity, it would introduce closed timelike curves. But if the interval at infinity is the Euclidean distance, beta, the path integral gives the partition function for gravity at temperature 1/β.

The partition function of a system is the trace over all states, weighted with e^{-β H}. One can then integrate β along a contour parallel to the imaginary axis with the factor e^{-β E}. This projects out the states with energy E₀. In a gravitational collapse and evaporation, one is interested in states of definite energy, rather than states of definite temperature.

There is an infrared problem with this idea for asymptotically flat space. The Euclidean path integral with period β is the partition function for space at temperature 1/beta. The partition function is infinite because the volume of space is infinite. This infrared problem can be solved by a small negative cosmological constant. It will not affect the evaporation of a small black hole, but it will change infinity to anti-deSitter space, and make the thermal partition function finite.

The boundary at infinity is then a torus, S¹ cross S². The trivial topology, periodically identified anti-deSitter space, fills in the torus, but so also do nontrivial topologies, the best known of which is Schwarzschild anti-deSitter. Providing that the temperature is small compared to the Hawking-Page temperature, the path integral over all topologically trivial metrics represents self-gravitating radiation in asymptotically anti-deSitter space. The path integral over all metrics of Schwarzschild AdS topology represents a black hole and thermal radiation in asymptotically anti-deSitter.

The boundary at infinity has topology S¹ cross S². The simplest topology that fits inside that boundary is the trivial topology, S¹ cross D³, the three-disk. The next simplest topology, and the first nontrivial topology, is S² cross D². This is the topology of the Schwarzschild anti-deSitter metric. There are other possible topologies that fit inside the boundary, but these two are the important cases: topologically trivial metrics and the black hole. The black hole is eternal. It cannot become topologically trivial at late times.

In view of this, one can understand why information is preserved in topologically trivial metrics, but exponentially decays in topologically non trivial metrics. A final state of empty space without a black hole would be topologically trivial, and be foliated by surfaces of constant time. These would form a 3-cycle modulo the boundary at infinity. Any global symmetry would lead to conserved global charges on that 3-cycle. These would prevent correlation functions from decaying exponentially in topologically trivial metrics. Indeed, one can regard the unitary Hamiltonian evolution of a topologically trivial metric as the conservation of information through a 3-cycle.

On the other hand, a nontrivial topology, like a black hole, will not have a final 3-cycle. It will not therefore have any conserved quantity that will prevent correlation functions from exponentially decaying. One is thus led to the remarkable result that late time amplitudes of the path integral over a topologically non trivial metric, are independent of the initial state. This was noticed by Maldacena in the case of asymptotically anti-deSitter in 3d, and interpreted as implying that information is lost in the BTZ black hole metric. Maldacena was able to show that topologically trivial metrics have correlation functions that do not decay, and have amplitudes of the right order to be compatible with a unitary evolution. Maldacena did not realize, however that it follows from a canonical treatment that the evolution of a topologically trivial metric, will be unitary.

So in the end, everyone was right, in a way. Information is lost in topologically nontrivial metrics, like the eternal black hole. On the other hand, information is preserved in topologically trivial metrics. The confusion and paradox arose because people thought classically, in terms of a single topology for spacetime. It was either R⁴, or a black hole. But the Feynman sum over histories allows it to be both at once. One can not tell which topology contributed the observation, any more than one can tell which slit the electron went through, in the two slits experiment. All that observation at infinity can determine is that there is a unitary mapping from initial states to final, and that information is not lost.

My work with Hartle showed the radiation could be thought of as tunnelling out from inside the black hole. It was therefore not unreasonable to suppose that it could carry information out of the black hole. This explains how a black hole can form, and then give out the information about what is inside it, while remaining topologically trivial. There is no baby universe branching off, as I once thought. The information remains firmly in our universe. I'm sorry to disappoint science fiction fans, but if information is preserved, there is no possibility of using black holes to travel to other universes. If you jump into a black hole, your mass-energy will be returned to our universe, but in a mangled form, which contains the information about what you were like, but in an unrecognisable state.

There is a problem describing what happens, because strictly speaking the only observables in quantum gravity are the values of the field at infinity. One cannot define the field at some point in the middle, because there is quantum uncertainty in where the measurement is done. However, in cases in which there are a large number, N, of light matter fields, coupled to gravity, one can neglect the gravitational fluctuations, because they are only one among N quantum loops. One can then do the path integral over all matter fields, in a given metric, to obtain the effective action, which will be a functional of the metric.

One can add the classical Einstein-Hilbert action of the metric to this quantum effective action of the matter fields. If one integrated this combined action over all metrics, one would obtain the full quantum theory. However, the semiclassical approximation is to represent the integral over metrics by its saddle point. This will obey the Einstein equations, where the source is the expectation value of the energy momentum tensor, of the matter fields in their vacuum state.

The only way to calculate the effective action of the matter fields, used to be perturbation theory. This is not likely to work in the case of gravitational collapse. However, fortunately we now have a non-perturbative method in AdS/CFT. The Maldacena conjecture says that the effective action of a CFT on a background metric is equal to the supergravity effective action of anti-deSitter space with that background metric at infinity. In the large N limit, the supergravity effective action is just the classical action. Thus the calculation of the quantum effective action of the matter fields, is equivalent to solving the classical Einstein equations.

The action of an anti-deSitter-like space with a boundary at infinity would be infinite, so one has to regularize. One introduces subtractions that depend only on the metric of the boundary. The first counter-term is proportional to the volume of the boundary. The second counter-term is proportional to the Einstein-Hilbert action of the boundary. There is a third counter-term, but it is not covariantly defined. One now adds the Einstein-Hilbert action of the boundary and looks for a saddle point of the total action. This will involve solving the coupled four- and five-dimensional Einstein equations. It will probably have to be done numerically.

In this talk, I have argued that quantum gravity is unitary, and information is preserved in black hole formation and evaporation. I assume the evolution is given by a Euclidean path integral over metrics of all topologies. The integral over topologically trivial metrics can be done by dividing the time interval into thin slices and using a linear interpolation to the metric in each slice. The integral over each slice will be unitary, and so the whole path integral will be unitary.

On the other hand, the path integral over topologically nontrivial metrics, will lose information, and will be asymptotically independent of its initial conditions. Thus the total path integral will be unitary, and quantum mechanics is safe.

It is great to solve a problem that has been troubling me for nearly thirty years, even though the answer is less exciting than the alternative I suggested. This result is not all negative however, because it indicates that a black hole evaporates, while remaining topologically trivial. However, the large N solution is likely to be a black hole that shrinks to zero. This is what I suggested in 1975.

In 1997, Kip Thorne and I bet John Preskill that information was lost in black holes. The loser or losers of the bet are to provide the winner or winners with an encyclopaedia of their own choice, from which information can be recovered with ease. I'm now ready to concede the bet, but Kip Thorne isn't convinced just yet. I will give John Preskill the encyclopaedia he has requested. John is all-American, so naturally he wants an encyclopaedia of baseball. I had great difficulty in finding one over here, so I offered him an encyclopaedia of cricket, as an alternative, but John wouldn't be persuaded of the superiority of cricket. Fortunately, my assistant, Andrew Dunn, persuaded the publishers Sportclassic Books to fly a copy of "Total Baseball: The Ultimate Baseball Encyclopedia" to Dublin. I will give John the encyclopaedia now. If Kip agrees to concede the bet later, he can pay me back.

At this point the encyclopedia was brought on stage and given to John Preskill, who waved it over his head in a parody of athletic triumph. The order of events is a bit fuzzy in my mind, but sometime around then he said "I always hoped that when Stephen conceded, there would be a witness - this really exceeds my expectations."

After this, Kip Thorne ran a question and answer period, saying that he would alternate between questions from conference participants, which Hawking's grad student would answer, and questions from the press, which Hawking would answer - after Thorne checked Hawking's facial expressions to see whether he felt they were worth answering.

First, a correspondent from the BBC asked Stephen Hawking what the significance of this result was for "life, the universe and everything". (Here I'm using John Preskill's humorous paraphrase.) Hawking agreed to answer this, and while he began laboriously composing a reply using the computer system on his wheelchair, his grad student Christophe Galfard fielded three questions from experts: Bill Unruh, Gary Horowitz and Robb Mann. I didn't find the replies terribly illuminating, except that when asked if information would be lost if we kept feeding the black hole matter to keep it from evaporating away, Galfard said "yes". Everyone afterwards commented on what a tough job it would be for a student to field questions in front of about 800 physicists and the international press.

At this point Kip Thorne checked to see if Hawking was done composing his reply. He was not. To fill time, Thorne explained why he hadn't yet conceded the bet, saying "This looks to me, on the face of it, to be a lovely argument. But I haven't seen all the details." He took this opportunity to tell the reporters a bit about how science was done: we don't just listen to Hawking and take his word for everything, we have to go off and check things ourselves. He told a nice story about how when Hawking first showed that black holes radiate, everyone with their own approach to quantum field theory on curved spacetime needed to redo this calculation their own way to be convinced - with Yakov Zeldovich, who'd gotten the game started by showing that energy could be extracted from rotating black holes in the form of radiation, being one of the very last to agree! Preskill chimed in, saying "I'll be honest - I didn't understand the talk", and that he too would need to see more details.

After a bit more of this sort of thing, Hawking was ready to answer the BBC reporter's question. His answer was surprisingly short, and it went something like this (I can't find an exact quote): "This result shows that everything in the universe is governed by the laws of physics." A suitably grandiose answer for a grandiose question! One can imagine better explanations of unitarity, but not quicker ones.

At this point Kip Thorne solicited more questions from the press but said they should confine themselves to yes-or-no questions, so Hawking could answer them more efficiently. Jenny Hogan got the first question, asking what Hawking would do now that he'd solved this problem. Kip Thorne pointed out that this was not a yes-or-no question. Hogan replied that it shouldn't take long to reply; Thorne was doubtful, but in the midst of the ensuing conversation Hawking shot off an unexpectedly rapid response: "I don't know." Everyone laughed, and at this point the public question period was called to a close, though reporters were allowed to stay and pester Hawking some more.

At the time Hawking's talk seemed very cryptic to me, but in the process of editing the above transcript it's become a lot clearer, so I'll try to give a quick explanation.

I should start by saying that the jargon used in this talk, while doubtless obscure to most people, is actually quite standard and not very difficult to anyone who has spent some time studying the Euclidean path integral approach to quantum gravity. The problem is not the jargon so much as the lack of detail, which requires some imagination to fill in. When I first heard the talk, this lack of detail had me completely stumped. But now it makes a little more sense....

He's studying the process of creating a black hole and letting it evaporate away. He's imagining studying this in the usual style of particle physics, as a "scattering experiment", where we throw in a bunch of particles and see what comes out. Here we throw in a bunch of particles, let them form a black hole, let the black hole evaporate away, and examine the particles (typically photons for the most part) that shoot out.

The rules of the game in a "scattering experiment" are that we can only talk about what's going on "at infinity", meaning very far from where the black hole forms - or more precisely, where it may or may not form!

The advantage of this is that physics at infinity can be described without the full machinery of quantum gravity: we don't have to worry about quantum fluctuations of the geometry of spacetime messing up our ability to say where things are. The disadvantage is that we can't actually say for sure whether or not a black hole formed. At least this seems like a "disadvantage" at first - but a better term for it might be a "subtlety", since it's crucial for resolving the puzzle:

Black hole formation and evaporation can be thought of as a scattering process. One sends in particles and radiation from infinity, and measures what comes back out to infinity. All measurements are made at infinity, where fields are weak, and one never probes the strong field region in the middle. So one can't be sure a black hole forms, no matter how certain it might be in classical theory. I shall show that this possibility allows information to be preserved and to be returned to infinity.

That's what Hawking means by this:

I adopt the Euclidean approach, the only sane way to do quantum gravity non-perturbatively. In this, the time evolution of an initial state is given by a path integral over all positive definite metrics that go between two surfaces that are a distance T apart at infinity. One then Wick rotates the time interval, T, to the Lorentzian. The path integral is taken over metrics of all possible topologies that fit in between the surfaces.

(Nota bene: here I'm talking about the equations of general relativity on 4d space, not 4d spacetime. That's because we're in the middle of this Wick rotation trick.)

Actually, I'm oversimplifying a bit. We don't get "the answer" to our physics question this way: we get one answer for each solution of the equations of general relativity that we deem relevant to the problem at hand. To finish the job, we should add up all these partial answers to get the total answer. But in practice this last step is always too hard: there are too many topologies, and too many classical solutions, to keep track of them all.

So what do we do? We just add up a few of the answers, cross our fingers, and hope for the best! If this procedure offends you, go do something easy like math.

In the problem at hand here, Hawking focuses on two classical solutions, or more precisely two classes of them. One describes a spacetime with no black hole, the other describes a spacetime with a black hole which lasts forever. Each one gives a contribution to the semiclassical approximation of the integral over all geometries. To get answers to physical questions, he needs to sum over both. In principle he should sum over infinitely many others, too, but nobody knows how, so he's probably hoping the crux of the problem can be understood by considering just these two.

He says that if you just do the integral over geometries near the classical solution where there's no black hole, you'll find - unsurprisingly - that no information is lost as time passes.

He also says that if you do the integral over geometries near the classical solution where there is a black hole, you'll find - surprisingly - that the answer is zero for a lot of questions you can measure the answers to far from the black hole. In physics jargon, this is because a bunch of "correlation functions decay exponentially".

So, when you add up both answers to see if information is lost in the real problem, where you can't be sure if there's a black hole or not, you get the same answer as if there were no black hole!

So in the end, everyone was right, in a way. Information is lost in topologically nontrivial metrics, like the eternal black hole. On the other hand, information is preserved in topologically trivial metrics. The confusion and paradox arose because people thought classically, in terms of a single topology for spacetime. It was either R⁴, or a black hole. But the Feynman sum over histories allows it to be both at once. One can not tell which topology contributed the observation, any more than one can tell which slit the electron went through, in the two slits experiment. All that observation at infinity can determine is that there is a unitary mapping from initial states to final, and that information is not lost.

Since this issue is long enough already and I want to get it out soon, I won't talk about other things that happened at this conference - nor will I talk about the conference on n-categories earlier this summer! I just want to say a few elementary things about the topology lurking in Hawking's talk... since some mathematicians may enjoy it.

As he points out, the answers to a bunch of questions diverge unless we put our black hole in a box of finite size. A convenient way to do this is to introduce a small negative cosmological constant, which changes our default picture of spacetime from Minkowski spacetime, which is topologically R⁴, to anti-deSitter spacetime, which is topologically R x D³ after we add the "boundary at infinity". Here R is time and the 3-disk D³ is space. This is a Lorentzian manifold with boundary, but when we do Wick rotation we get a Riemannian manifold with boundary having the same topology.

However, when we are doing Euclidean path integrals at nonzero temperature, we should replace the time line R here by a circle whose radius is the reciprocal of that temperature. (Take my word for it!) So now our Riemannian manifold with boundary is S¹ x D³. This is what Hawking uses to handle the geometries without a black hole. The boundary of this manifold is S¹ x S². But there's another obvious manifold with this boundary, namely D² x S2. And this corresponds to the geometries with a black hole! This is cute because we see it all the time in surgery theory. In fact I commented on Hawking's use of this idea a long time ago, in "week67".

In his talk, Hawking points out that S¹ x D³ has a nontrivial 3-cycle in it if we use relative cohomology and work relative to the boundary S¹ x S². But, D² x S² does not. When spacetime is n-dimensional, conservation laws usually come from integrating closed (n-1)-forms over cycles that correspond to "space", so we get interesting conservation laws when there are nontrivial (n-1)-cycles. Here Hawking is using this to argue for conservation of information when there's no black hole - namely for S¹ x D³ - but not when there is, namely for D² x S².

All this is fine and dandy; the hard part is to see why the case when there is a black hole doesn't screw things up! This is where his allusions to "exponentially decaying correlation functions come in" - and this is where I'd like to see more details. I guess a good place to start is Maldacena's papers on the black hole in 3d spacetime - the so-called Banados-Teitelboim-Zanelli or "BTZ" black hole. This is a baby version of the problem, one dimension down from the real thing, where everything should get much simpler. For a bunch about the BTZ black hole, try:

3) Maximo Banados, Marc Henneaux, Claudio Teitelboim, and Jorge Zanelli, Geometry of the 2+1 black hole, Phys. Rev. D48 (1993) 1506-1525, also available as gr-qc/9302012.

The relevant paper by Maldacena seems to be:

4) Juan Maldacena, Eternal Black Holes in AdS, JHEP 0304 (2003) 021, also available as hep-th/0106122.

You can also see a talk he gave on this at the Institute for Theoretical Physics at U. C. Santa Barbara:

5) Juan Maldacena, Eternal Black Holes in AdS, http://online.itp.ucsb.edu/online/mtheory_c01/maldacena/.

By the way, here are some photos of the conference...

6) John Baez, Dublin, http://math.ucr.edu/home/baez/dublin/

... and also photos of the plaque on the bridge where Hamilton carved his defining relations for the quaternions!

Addendum: My friend Ted Bunn filled a gap in my understanding of the history of astronomy. I had written:

It turned out to describe an idea I'd never even dreamt of before: 
a heliocentric cosmology in which the planets move along circular orbits 
with epicycles a la Ptolemy!  

There is nothing new under (or orbiting) the Sun.  This idea is
originally due to Copernicus.  Thomas Kuhn's book "The Copernican
Revolution" has a nice discussion.

© 2004 John Baez
baez@math.removethis.ucr.andthis.edu