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Skarp
Christmas Island
135 Posts |
 Posted - 22 Jul 2004 : 08:32:23
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From what I could gather - This is his speech. Darned if I can figer what he's saying. Glad I'm not him ..... cuz then I'd have to understand what I'm saying. :-)
quote:
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 de Sitter space, and a conformal
field theory on the boundary of anti de Sitter 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 de Sitter 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-perturbative. 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 non
trivial 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 non trivial topologies, can not be foliated by a family of
surfaces of constant time. There will be a fixed point in any time
evolution vector field on a non trivial 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
non-trivial 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 space like
surface. This would be a functional of the geometries of space like
surfaces, which is not something that can be measured in weak fields
near infinity. One can measure the weak gravitational fields, on a
time like 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
time like curves. But if the interval at infinity is the Euclidean
distance, beta, the path integral gives the partition function for
gravity at temperature, one over beta.
The partition function of a system, is the trace over all states,
weighted with e to the minus beta H. One can then integrate beta along
a contour parallel to the imaginary axis, with the factor, e to the
beta E0. 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 beta, is the partition
function for space at temperature, one over 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 de Sitter 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 de Sitter space, fills in the
torus, but so also do non-trivial topologies, the best known of which
is Schwarzschild anti de Sitter. 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 de Sitter space. The path integral
over all metrics of Schwarzschild ADS topology, represents a black
hole and thermal radiation, in asymptotically anti de Sitter.
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
non-trivial topology, is S2 cross D2. This is the topology of the
Schwarzschild anti de Sitter 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 can not 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 three cycle, modulo
the boundary at infinity. Any global symmetry, would lead to
conserved global charges on that three 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 three cycle.
On the other hand, a non trivial topology, like a black hole, will not
have a final three 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 de Sitter3, 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 non trivial 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 can not 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 semi-classical 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 de Sitter 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 de Sitter 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 non trivial
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.
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Stephen hawking "revamps" black hole theory
