Black Holes:
Spacetime vs. Quantum Mechanics
Joseph Polchinski
CCGRRA, Winnipeg, 5/21/14
The black hole information paradox revealed a
conflict between QM and spacetime locality:
Hawking (1976): QM must be modified, replacing the S-matrix
with a $-matrix that takes pure states to mixed states.
‘t Hooft, Susskind, Maldacena, … (1993-97): QM is
unmodified, but spacetime is fundamentally nonlocal,
holographic. However, no single observer sees any
nonlocality (black hole complementarity).
AMPS (2012): If QM is to be preserved, an infalling observer
sees something radically different from GR, a firewall.
Many attempts to avoid the firewall modify QM, in new ways.
Review of the Information problem: the Page curve for an
evaporating black hole
S
Hawking result
t
S = von Neumann entropy of the Hawking radiation
= entanglement entropy of the radiation and black hole
= von Neumann entropy of the black hole
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.
+
S
Hawking result
t
S = von Neumann entropy of the Hawking radiation
= entanglement entropy of the radiation and black hole
= von Neumann entropy of the black hole
Information loss:
S
Hawking result
t
While the black hole is there, the total system is in a pure
state. When it disappears, the radiation is all that is left,
and it is in a mixed state. Pure mixed evolution.
Remnants
S
Hawking result
t
Alternative: black hole evaporation ends in a remnant
with a large number of internal states.
Remnants
S
Hawking result
SBekenstein-Hawking
t
Alternative: black hole evaporation ends in a remnant
with a large number of internal states.
S
Hawking result
Page curve
t
Page: the fine-grained entropy of the black hole exceeds
the coarse-grained (BH) entropy around the midpoint.
If the information is to escape with the Hawking radiation, it
must begin to emerge then, when the black hole is still
large. O(1) correction.
Somehow, information about the
quantum state must travel in a
spacelike direction:
Going around in circles (1976-97):
Information
loss
Information carried
away by the
Hawking radiation
Remnants
BFSS Matrix Theory/AdS/CFT duality
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
Brief history:
black hole entropy puzzle
 D-brane state counting
D-brane vs. black brane dynamics
duality
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.
The winner!
Information
loss
Information carried
away by the
Hawking radiation
Remnants
A black hole is actually dual to an ordinary thermal system.
Level of trust in AdS/CFT?
Quantum theory, with the spectrum of
massless spin-2 in AdS, which
couples to energy…
… Hanada, Hyakatuke, Ishiki, and
Nishimura 1311.5607 numerically
simulate the field theory, obtain E(T)
for the thermal system. Agrees with
black hole result to order
Einstein-Hilbert action +  correction
+ gravity loop correction
Many open questions:
• 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)
Information
loss
Information carried
away by the
Hawking radiation
Remnants
The most conservative alternative, but also the most
radical.
Black hole complementarity. A proposal for a new
relativity principle (‘t Hooft, Susskind, Preskill ’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 (information-free horizon),
Giddings, Braunstein (energetic curtains!)
AMPS I: Consequences of No Drama + EFT
b
b’
Creation/annihilation operators:
a: Inertial observer near horizon
b: Outgoing Hawking modes
b’: Ingoing Hawking modes
b = Aa + Ba†
a = Cb + Db† + C’b’ + D’b’
a
Adiabatic principle/no drama:
a|y= 0 so b|y≠ 0
This implies:
• Hawking radiation
• b and b’ are entangled.
AMPS II: Consequences of purity
SvN(radiation)
info loss
b
b’
info conserved
b
t
If information is not lost, b must be entangled
with the earlier radiation. (Page, Hayden &
Preskill)
E
A contradiction:
If b and b’ are in a pure state, then b cannot be
entangled with anything else, like E.
Strong subadditivity (Mathur): Sb’b + SbE ≥ Sb + Sb’bE
Here Sb’b = 0  Sb’bE = SE
 SbE ≥ Sb + SE
Moreover, a single observer
can see all of b, E and b’, so
complementarity does not
save us.
b
b’
E
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: a|y>≠ 0?
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:
Or, string creation at horizon? (Silverstein ‘14)
• 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 (Marolf, 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. (but see:
Giddings).
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:
Strong complementarity (no global Hilbert space)
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)
However, unlike Hawking’s original proposal, they do
not affect the observations of an outside observer.
Many suggestions to solve black hole information
paradox weaken/generalize quantum mechanics:
Strong complementarity (no global Hilbert space)
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)
However, unlike Hawking’s original proposal, they do
not affect the observations of an outside observer.
A common idea (Nomura, Varela & Weinberg,
Papadodimas & Raju, Verlinde & Verlinde, Maldacena
& Susskind): Since the problem is a double
entanglement of b with b’ and E, then these are the
same, b’ E, also known as A RB.
This is similar to the original idea of black
hole complementarity. However, that was
supposed to be a breakdown of locality
within the usual framework of QM. b’ E
is a modification of the rules of quantum
mechanics.
b
b’
E
General idea (PR 1211.6767,1310.6335, VV 1211.6913).
Consider a typical black hole state |yt>. The distribution
of the mode b is thermal:
|yt>= Z-1/2 ∑n e-wb/2T |n>b |yt,n>B*
where B* is the complement to b. Compare
|0>a = Z-1/2 ∑n e-wb/2T |n>b |n>b’
Thus identify the internal Hilbert space,
|n>b = |yt,n>B*
Problem: given a black hole in some specified state |y>,
which |yt>do we use to identify the internal Hilbert
space?
PR solution: single out a small space of operators A.
Then: |yt> = U|y> where U is in A, and the expectation
values of A in |yt> are thermal.
Problem: observables behind the horizon are now
functions of |y>: state-dependent. This is often
confused with the background-dependence that is
inevitable in a theory of gravity, but it is different, it is a
modification of quantum mechanics.
Ordinary QM: The system is in a state |Y>. We have a
basis |i>. The probability of finding the system in a
given basis state is
|<i|y>|2 = <y|Pi|y>
The probability of finding a given excitation is
i S |<i|y>|2 = <y|PS|y>
where S is the set of all states with the given excitation
and background. The `background-dependence’, i.e. the
black hole or whatever is being excited, is all built into i
and S. PS is a linear operator, which does not depend on
|y>.
This is the Born rule, and PR modify it: PS depend on |y>.
For observables outside the black hole (e.g. the
occupation number for b), we have the usual rules: PS
does not depend on |y>. For observables behind the
horizon (occupation number for b’ ), PS depends on |y>.
Is this a bug or a feature?
• Not well-defined in current form, assigns multiple
interpretations to same |y> (Harlow 1405.1995).
• Even if the above is repaired, states that are physically
orthogonal (e.g. 0 or 1 b’ excitations) are not
orthogonal, but can have inner product 1-e(Marolf &
JP).
• Current form works, if at all, only for a black hole in a
box (depends on properties of equilibrium states), not
for one that is decaying (Bousso).
EPR = ER (Maldacena & Susskind 1306.0533):
Spacetimes connected by an Einstein-Rosen bridge
are entangled (Israel ‘76, Maldacena hep-th/0106112), so
ER  EPR.
Is the reverse true, are entangled systems necessarily
connected by bridges, EPR  ER?
Seems to reduce to PR for observables.
Limits on quantum computation (Harlow & Hayden
1301.4504): perhaps there is not time to verify the b-E
entanglement.
• Doesn’t apply to AdS black holes (AMPSS 1304.6483).
• Can be evaded by pre-computing (Oppenheim &
Unruh 1401.1523).
• What would it mean – an uncertainty principle for the
wavefunction?
Final state boundary condition at the black hole singularity (Horowitz & Maldacena hep-th/0310281; Preskill &
Lloyd 1308.4209)
Conditioning on a final state at the singularity gives
necessary entanglements, but does not lead to a consistent
description of the interior (Bousso & Stanford 1310.7457)
Open questions
• Are there any observational effects for black holes?
• Are there any consequences for cosmological
horizons?
• Do I really believe in firewalls?
• Where is this going?
extra slides
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.