This Week's Finds in Mathematical Physics (Week 207)
From: John Baez (baez_at_math.removethis.ucr.andthis.edu)
Date: 07/26/04
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Date: 26 Jul 2004 14:22:19 -0400
Also available at http://math.ucr.edu/home/baez/week207.html
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
British 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
It's signed by Thorne and Preskill, with a thumbprint of Hawking's.
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/beta.
The partition function of a system is the trace over all states,
weighted with e^{-beta H}. One can then integrate beta along
a contour parallel to the imaginary axis with the factor e^{-beta E_0}.
This projects out the states with energy E_0. 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 beta 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, S^1 cross S^2. 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 S^1 cross S^2. The simplest
topology that fits inside that boundary is the trivial topology,
S^1 cross D^3, the three-disk. The next simplest topology, and
the first nontrivial topology, is S^2 cross D^2. 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 R^4, 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.
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 that while the talk seemed convincing to
him, he still wanted to see the details. He explained to 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 calculate 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 by saying "I'll be honest - I didn't understand the
talk", adding that 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 much better explanations of unitarity, but not very
quick ones.
At this point Kip Thorne solicited more questions from the press but
said they should confine themselves to questions with yes-or-no answers.
Jenny Hogan got off the first one, asking what Hawking would do now
that he's solved this problem. Kip Thorne pointed out that this was
not a yes-or-no question, but in the midst of the ensuing conversation
Hawking shot off an unexpectedly rapid reply: "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.
Now, the way Hawking likes to calculate things in this sort of
problem is using a "Euclidean path integral". This is a rather
controversial approach - hence his grin when he said it's the
"only sane way" to do these calculation - but let's not worry about
that. Suffice it to say that we replace the time variable "T"
in all our calculations by "iT", do a bunch of calculations, and
then replace "iT" by "T" again at the end. This trick is called
"Wick rotation". In the middle of this process, we hope all our
formulas involving the geometry of 4d *spacetime* have magically
become formulas involving the geometry of 4d *space*. The answers
to physical questions are then expressed as integrals over all
geometries of 4d space that satisfy some conditions depending on
the problem we're studying. This integral over geometries also
includes a sum over topologies.
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.
Unfortunately, nobody knows how to define these integrals. However,
physicists like Hawking are usually content to compute them in a
"semiclassical approximation". This means integrating not over all
geometries, but only those that are close to some solution of the
classical equations of general relativity. We can then do a clever
approximation to get a closed-form answer.
(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 R^4, 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.
The mysterious part is why the geometries near the classical solution
where there's a black hole don't contribute at all to information loss,
even though they do contribute to other important things, like the
Hawking radiation. Here I'd need to see an actual calculation. Hawking
gives a nice hand-wavy topological argument, but that's not enough for
me.
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 R^4, to anti-deSitter spacetime, which is
topologically R x D^3 after we add the "boundary at infinity".
Here R is time and the 3-disk D^3 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 S^1 x D^3.
This is what Hawking uses to handle the geometries without a black
hole. The boundary of this manifold is S^1 x S^2. But there's
another obvious manifold with this boundary, namely D^2 x S^2. 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 S^1 x D^3 has a nontrivial 3-cycle
in it if we use relative cohomology and work relative to the boundary
S^1 x S^2. But, D^2 x S^2 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 S^1 x D^3 - but not when there is, namely for D^2 x S^2.
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 the original BTZ paper, try:
3) Maximo Banados, Marc Henneaux, Claudio Teitelboim, and Jorge Zanelli,
Geometry of the 2+1 black hole, available as gr-qc/9302012.
Maldacena's papers can also be found on the physics arXiv, but I'm
not sure which one Hawking is referring to, so I'll wait until someone
tells me before adding a link to that one. Sometime I will also add links
to a bunch of photos taken at this conference - including photos of the
plaque under the bridge where Hamilton wrote his defining relations for
the quaternions!
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