The Black Hole Information Paradox,
Alive and Kicking
Joseph Polchinski
TAMU 10/31/13
A Brief History of the Black Hole Information Paradox:
A Brief History of the Black Hole Information Paradox:
In 1976, Stephen Hawking argued that black holes
destroy information, in a way that requires a
modification of the principles of quantum mechanics.
A Brief History of the Black Hole Information Paradox:
In 1976, Stephen Hawking argued that black holes
destroy information, in a way that requires a
modification of the principles of quantum mechanics.
In 2004, he changed his mind.
A Brief History of the Black Hole Information Paradox:
In 1976, Stephen Hawking argued that black holes
destroy information, in a way that requires a
modification of the principles of quantum mechanics.
In 2004, he changed his mind.
In 2012, some radicals argued that if QM is not
modified, the black hole interior is very different from
what general relativity predicts.
A Brief History of the Black Hole Information Paradox:
In 1976, Stephen Hawking argued that black holes
destroy information, in a way that requires a
modification of the principles of quantum mechanics.
In 2004, he changed his mind.
In 2012, some radicals argued that if QM is not
modified, the black hole interior is very different from
what general relativity predicts.
The information paradox is one of the great thought
experiments in the history of physics.
Thought experiments have played a major role in the
discovery of the laws of physics.
Maxwell inferred the displacement term in part through a
thought experiment, with a
capacitor a time-dependent
current.
Heinrich Hertz observed this directly 25 years later,
using sparks to drive a circuit at nanosecond time
scales.
In quantum gravity the natural time scale is the Planck
time, tP  hG/c 5  5.4 1044 sec, so again thought
experiments will be essential, and black holes have
been a fruitful arena.
singularity
First, consider the
formation of a
classical black hole:
time
Anything behind the
horizon is trapped and
falls into the singularity.
horizon
Hawking’s thought experiment:
The emitted radiation has a
black-body spectrum: the black
hole has an effective
temperature, etc. (Bekenstein,
Hawking, …)
horizon
The horizon is a region of low
curvature, so this calculation
should be reliable.
singularity
Taking into account quantum
mechanics, the spacetime
curvature near the horizon
induces creation of particleantiparticle pairs via tunneling.
As a result, the black hole
eventually emits all of its
energy and disappears,
leaving only the outgoing
Hawking radiation.
time
(Number of quanta
~ M 2/MP2)
time
Hawking’s thought
experiment is to repeat this
process many times with the
same initial state, and
measure the final state in a
very large number of bases.
The outcome:
First, review density matrices. A quantum mechanical
system is described by a state vector
. For
example, the expectation value of an observable O is
But sometimes we use a density matrix r, where the
expectation value is
`mixed state’
We do this when we look at only part of a system, or
at coarse-grained variables: the density matrix reflects
our ignorance of the full system.
The outcome:
For the black hole thought experiment, Hawking argued
that even if the system began in a pure state
, the
final Hawking radiation had to be in a mixed state r.
Ordinary quantum mechanical evolution,
takes pure states into pure states. Hawking was saying
that in quantum gravity this must be generalized to
allow
“God not only plays dice, He sometimes throws the
dice where they cannot be seen.’’
Some negative reactions:
• We’ve seen density matrices before, and they just
reflect our ignorance, not a fundamental property of
physics. Hawking has simply not calculated carefully
enough.
• The black hole is an exotic environment, but through
virtual quantum gravity effects this will feed into
ordinary physics. There are strong limits on this (Ellis,
Hagelin, Nanopoulos, Srednicki ’84).
• This generalized evolution leads to radical energy
nonconservation --- Noether’s theorem doesn’t hold
(Banks, Susskind, Peskin ’84).
Hawking’s argument:
The Hawking process is a
quantum effect, and
produces a superposition,
The two photons are
entangled; the outside
photon by itself is in a
mixed state.
+
Hawking’s argument:
The net result is a highly entangled state, ~
When the evaporation is completed,
the inside (primed) degrees of freedom are gone, leaving the Hawking
radiation in a highly mixed state.
Not sensitive to small corrections.
Purity:
If the Hawking radiation is to be in a
pure state (information is not lost, but
carried away by the radiation) it
seems that somehow information
must travel faster than light…
String theory didn’t seem to help…
. 
Remnants:
Once the black hole is Planck-sized,
we no longer know what happens.
Maybe the evaporations stops,
leaving a pure state where the
Hawking radiation is entangled with
the remnant.
A small object with an arbitrarily
large number of internal states…
Problems: violates BekensteinHawking entropy, infinite pair
production.
Going around in circles (1976-97):
Information
loss
Information carried
away by the
Hawking radiation
Remnants
In 1996-7, a revolution: Matrix Theory (Banks, Fischler,
Shenker, Susskind) and AdS/CFT duality (Maldacena)
Duality: when two seemingly distinct systems are
actually the same, usually under some non-obvious
change of variables (e.g. bose/fermi equivalence, Ising
high/low temperature duality in 1+1 dimensions).
Equivalently, a quantum system with multiple classical
limits.
More and more examples have been discovered over
time, but BFSS and Maldacena’s were the first in
which one description involved quantum gravity and
the other just `ordinary’ physics.’
I. Quantum gravity (actually
string theory) in an anti-de
Sitter box.
II. A quantum field theory of
gauge fields, fermions, and
scalars living on the surface
of the box.
`Holographic’
One equation:
What does this say about the information paradox?
We can consider the Hawking
experiment in an AdS box.
Since the dual quantum field
theory is described by ordinary
QM, pure states must evolve
to pure states.
Moreover, as anticipated, the
dual description is highly
nonlocal, and holographic.
The winner!
Information
loss
Information carried
away by the
Hawking radiation
Remnants
A black hole is actually dual to an ordinary thermal system.
So what is left to do?
• The answer is not fully satisfying: it appeals to
AdS/CFT duality (which is not fully proven), and
doesn’t directly explain where Hawking went wrong.
• How does spacetime emerge in AdS/CFT?
• AdS/CFT duality gives us a construction of quantum
gravity in an AdS box, but cosmology doesn’t happen
in a box. How does holography work in other
spacetimes? (example: the black hole interior)
So what is left to do?
• The answer is not fully satisfying: it appeals to
AdS/CFT duality (which is not fully proven), and
doesn’t directly explain where Hawking went wrong.
• How does spacetime emerge in AdS/CFT?
• AdS/CFT duality gives us a construction of quantum
gravity in an AdS box, but cosmology doesn’t happen
in a box. How does holography work in other
spacetimes? (example: the black hole interior)
Good news: a new paradox! Ahmed Almheiri, Don
Marolf, JP, James Sully, arXiv 1207.3123
Black hole complementarity. A proposal for a new
relativity principle (Susskind ’93).
Observer who
falls into the
black hole sees
an infalling bit:
Observer who
stays outside
sees the same
bit encoded in
the later
radiation:
No observer can see both copies (important!)
A radical breakdown of spacetime locality.
The postulates of black hole complementarity:
I. Purity: the Hawking radiation is in a pure
state.
II. No drama: an infalling observer
experiences nothing unusual at the horizon.
III. Effective field theory (EFT): Semiclassical
gravity is valid outside the horizon. (The
horizon acts like an effective membrane as
seen by the outside observer.)
IV. SBH counts the states of the black hole.
The first three of these cannot all be true.
cf. Mathur, Giddings, Braunstein
b = Aa + Ba†
Consequences of
No Drama + EFT
a = Cb + Db† + C’b’ + D’b’
b
b’
a
Creation/annihilation operators:
a: Inertial observer near horizon
b: Outgoing Hawking modes
b’: Ingoing Hawking modes
Adiabatic principle/no drama:
a|y= 0 so b|y≠ 0
This implies:
• Hawking radiation
• b and b’ are entangled.
Consequences of Purity
Separate Hawking radiation into early
(~first 2/3) and late (last 1/3), where the
mode b is late. Expanding in a basis Li
The early Hilbert space is much larger
than the late Hilbert space, so to very
good accuracy the Ei are distinct. So the
late radiation is fully entangled with the
early radiation. There is some bit bE in the
early Hilbert space such that b and bE are
in a pure state. (cf Page, Hayden and
Preskill).
b
b’
E
A contradiction:
Purity: b + bE are in a pure state.
No drama: b + b’ are in a pure state, while b itself is
in a mixed state.
EFT: These are the same b.
Quantum mechanics doesn’t allow this! No Drama
needs state of (b’,b,bE) to be e.g.
(|0>|0> + |1>|1>)|0>
while Purity needs e.g.
|0> (|0>|0> + |1>|1>)
Moreover, a single observer can see all of b, bE and b’,
so complementarity does not save us.
So, what to give up?
Purity?
Absence of drama?
EFT outside the horizon?
Something else, like quantum mechanics for the
infalling observer?
So, what to give up?
Purity?
I still trust AdS/CFT here.
So, what to give up?
Absence of drama?
How bad is it - what energy excitations, and how
many?
Energy is limited only by the assumed cutoff on EFT.
The first argument only applies to low angular
momenta, due to a centrifugal barrier, but a `mining
argument’ applies to all L: the infalling observer
encounters a firewall of Planck-energy particles. A
radical conclusion.
• If firewalls exist, how do they form?
Many people have proposed that the black hole
interior is not as expected, mostly on dubious
grounds. Mathur’s fuzzball seems like most
coherent existing idea, branes tunnel out to horizon:
• If firewalls exist, how do they form?
Intuition: self-entanglement of the horizon
builds up the interior spacetime. As the
entanglement is transferred to the
radiation, the singularity expands and the
interior disappears (Susskind).
From G. ‘t Hooft
So, what to give up?
EFT outside the horizon? Need O(1) violation of
locality to extend a macroscopic distance from the
horizon. Difficult to do in a consistent way.
Trivial resolution/mistake? Perhaps, like Maxwell’s
demon, the necessary measurements are not
possible… So far, the argument has survived scrutiny.
Many suggestions to solve black hole information
paradox weaken/generalize quantum mechanics:
Limits on quantum computation (Harlow & Hayden ‘12)
Final state boundary condition at the black hole
singularity (Horowitz & Maldacena ’03; Preskill & Lloyd
‘13).
EPR = ER (Spacetime from entanglement, Maldacena & Susskind ‘13).
Nonlinear observables (Papadodimas & Raju ‘12, Verlinde2 ‘12).
All of these are preliminary
frameworks, not theories.
Open questions
• Are there any observational effects for black holes?
The argument is consistent with the exterior being
exactly as in the usual picture, except perhaps for
very subtle quantum effects. But who knows?
Open questions
• Are there any consequences for cosmology?
Are cosmological horizons like black hole horizons?
Is there a version of the information problem?
If we just carry over the black hole result, our current
cosmological horizon is very young, but the horizon
during inflation may have been middle-aged,
depending on number of e-foldings vs. ln(R/lP).
Most important, this may give us a new lever on
applying holography to cosmology.
Open questions
• Where is this going?
Trivial resolution? Looking unlikely.
I still trust AdS/CFT, so keep purity, but AdS/CFT may
tell us less about the interior than assumed.
We once again have a sharp paradox, that seems as
puzzling as the original information problem… we can
hope to learn something interesting.
extra slides
Another puzzling properties of black holes is that they
have a thermodynamic entropy proportional to their
horizon area rather than their volume, S = A/4lP2
(Bekenstein, Hawking).
Moreover, they have more entropy than any form of
ordinary matter in the same volume.
This suggests the holographic
principle: that quantum gravity
in any space should be
formulated in terms of degrees
of freedom living on the
boundary of that space
(‘t Hooft, Susskind ’93).
From G. ‘t Hooft
Again, radically nonlocal.
Another version: mining the black hole:
Drop a box near to the horizon, let it fill with Unruh
(acceleration) radiation, and pull it out. Same
conclusion, but sharper.
Open questions
• If firewalls exist, when do they form?
Entanglement argument gives upper bound ~ (black
hole lifetime)/2, but most black hole properties are
expected to come to equilibrium in a much shorter
time, comparable the light-crossing time RS/c or
RS/c (ln R/LP).