1. The Black Hole Information Paradox,The Black Hole Information Paradox,
Alive and KickingAlive and Kicking
Joseph PolchinskiJoseph Polchinski
Caltech, 5/31/13Caltech, 5/31/13

2. A Brief History of the Black Hole Information Paradox:

3. 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.

4. 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.

5. 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.
Actually, one of the great thought experiments in the
history of physics.

6. Thought experiments have played a major role in the
discovery of the laws of physics.
Maxwell inferred the displace-
ment 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, , so again thought
experiments will be essential.
tP = hG /c5
= 5.4 ×10−44
sec

7. Hawking’s thought experiment:
singularitysingularity
horizonhorizon
First, consider the
formation of a
classical black hole:
time
Anything behind the
horizon is trapped and
falls into the singularity.

8. singularitysingularity
horizonhorizon
Taking into account quantum
mechanics, the spacetime
curvature near the horizon
induces creation of particle-
antiparticle pairs via tunneling.
The horizon is a region of low
curvature, so this calculation
should be reliable.
The emitted radiation has a
black-body spectrum: the black
hole has an effective temper-
ature, etc. (Bekenstein,
Hawking, …)

9. As a result, the black hole
eventually emits all of its
energy and disappears,
leaving only the outgoing
Hawking radiation.
time
(Number of quanta
~ M2
/MP
2
)

10. 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.
time

11. 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 ρ, where the
expectation value is
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.
`mixed state’

12. 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 ρ.
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.’’

13. 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).

14. 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.

15. Hawking’s argument:
The net result is a highly entangled state, ~
When the evaporation is completed,
the inside (primed) degrees of free-
dom are gone, leaving the Hawking
radiation in a highly mixed state.
Not sensitive to small corrections.

16. 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…
. →

17. 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 Bekenstein-
Hawking entropy, infinite pair
production.

18. Going around in circles (1976-97):
Information
loss
Information carried
away by the
Hawking radiation
Remnants

19. In 1997, a revolution: 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 Maldacena’s was the first in which one
description involved quantum gravity and the other just
`ordinary’ physics.’

20. 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’

21. 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: holographic.

22. The winner!
Information
loss
Information carried
away by the
Hawking radiation
Remnants
A black hole is actually dual to an ordinary thermal system.
QM vs. spacetime.

23. 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)

24. 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

25. 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, ~ AdS/CFT

26. 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. 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

27. b
b’
a
b = Aa + Ba†
a = Cb + Db†
+ C’b’ + D’b’
Creation/annihilation operators:
a: Inertial observer near horizon
b: Outgoing Hawking modes
b’: Ingoing Hawking modes
Adiabatic principle/no drama:
a|ψ = 0 so b|ψ 0≠
This implies:
• Hawking radiation
• b and b’ are entangled.
Consequences of
No Drama + EFT

28. 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

29. 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.
A contradiction:
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.

30. So, what to give up?
Purity?
Absence of drama?
EFT outside the horizon?
Something else, like quantum mechanics for the
infalling observer?

31. So, what to give up?
Purity?
I still trust AdS/CFT here.

32. 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.

33. • 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:

34. • 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

35. So, what to give up?
EFT outside the horizon? Need O(1) violation of locality
to extend a macroscopic distance from the horizon.
Expanded version of complementarity:
if b’ and bE are both entangled with b, then
b’ = bE. Problems:
I. One observer can see both, leading to
illegal cloning (limits from quantum comp?).
II. Any interaction with one early bit bE will
perturb all the late bits b’ and create a
firewall.
Nonlocal interactions (Giddings): prob-
lematic (information can be pumped
in either direction.
b
b’
E

36. Modification of quantum mechanics for the infalling
observer? Several ideas, no coherent story yet:
• Final state boundary condition at singularity (Horowitz
& Maldacena; Kitaev & Preskill)
• Limits on quantum computation (Harlow & Hayden)
• Strong complementarity – each observer has their
own Hilbert space (Banks & Fischler, Bousso, Harlow &
Hayden, …)
• Quantum donkey – nonlinear mapping of observables
(Verlinde & Verlinde)
• EPR = ER: every entangled pair of particles is
connected by a spacetime wormhole (Maldacena &
Susskind) ↓
↓

37. Doesn’t AdS/CFT imply that QM is unmodified?
Perhaps only for the observer at the boundary

38. • 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

39. • 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

40. • 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.
Open questions

42. From G. ‘t Hooft
Another puzzling properties of black holes is that they
have a thermodynamic entropy proportional to their
horizon area rather than their volume, S = A/4lP
2
(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).
Again, radically nonlocal.

43. 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.

44. • 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).
Open questions