Black Holes: Spacetime vs. Quantum Mechanics

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Black HolesSpacetime vs. Quantum Mechanics

• Joseph Polchinski

CCGRRA, Winnipeg, 5/21/14
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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.
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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
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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.
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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
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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.
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Remnants
S
Hawking result
t
Alternative black hole evaporation ends in a
remnant with a large number of internal states.
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Remnants
S
Hawking result
SBekenstein-Hawking
t
Alternative black hole evaporation ends in a
remnant with a large number of internal states.
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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.
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Somehow, information about the quantum state must
travel in a spacelike direction
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Going around in circles (1976-97)
Information loss
Information carried away by the Hawking radiation
Remnants
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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
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• Brief history
• black hole entropy puzzle
• D-brane state counting
• D-brane vs. black brane dynamics
• duality

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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.
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The winner!
Information loss
Information carried away by the Hawking radiation
Remnants
A black hole is actually dual to an ordinary
thermal system.
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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 a correction
gravity loop correction
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Many open questions
The answer is not fully satisfying it appeals
to AdS/CFT duality (which is not fully proven),
and doesnt 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 doesnt happen in a box. How does
holography work in other spacetimes? (example
the black hole interior)
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Information loss
Information carried away by the Hawking radiation
Remnants
The most conservative alternative, but also the
most radical.
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Black hole complementarity. A proposal for a new
relativity principle (t Hooft, Susskind,
Preskill 93).
Observer who stays outside sees the same bit
encoded in the later radiation
Observer who falls into the black hole sees an
infalling bit
No observer can see both copies (important!) A
radical breakdown of spacetime locality.
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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!)
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AMPS I Consequences of No Drama EFT
Creation/annihilation operators a Inertial
observer near horizon b Outgoing Hawking modes
b Ingoing Hawking modes
b Aa Ba a Cb Db Cb Db
Adiabatic principle/no drama ay 0 so
by ? 0 This implies Hawking radiation
b and b are entangled.
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AMPS II Consequences of purity
If information is not lost, b must be
entangled with the earlier radiation. (Page,
Hayden Preskill)
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A contradiction
If b and b are in a pure state, then b cannot be
entangled with anything else, like E. Strong
subadditivity (Mathur) Sbb SbE Sb
SbbE Here Sbb 0 SbbE SE SbE Sb
SE
Moreover, a single observer can see all of b, E
and b, so complementarity does not save us.
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So, what to give up?
Purity? Absence of drama? EFT outside the
horizon? Something else, like quantum mechanics
for the infalling observer?
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So, what to give up?
Purity? I still trust AdS/CFT here.
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So, what to give up?
Absence of drama aygt ? 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.
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If firewalls exist, how do they form? Many
people have proposed that the black hole interior
is not as expected, mostly on dubious grounds.
Mathurs fuzzball seems like most coherent
existing idea, branes tunnel out to horizon
Or, string creation at horizon? (Silverstein 14)
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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
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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 Maxwells
demon, the necessary measurements are not
possible So far, the argument has survived
scrutiny.
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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 singu-larity
(Horowitz Maldacena 03 Preskill Lloyd
13) EPR ER (Spacetime from entanglement,
Maldacena Susskind 13) Nonlinear observables
(Papadodimas Raju 12, Verlinde2 12) However,
unlike Hawkings original proposal, they do not
affect the observations of an outside observer.
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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 singu-larity
(Horowitz Maldacena 03 Preskill Lloyd
13) EPR ER (Spacetime from entanglement,
Maldacena Susskind 13) Nonlinear observables
(Papadodimas Raju 12, Verlinde2 12) However,
unlike Hawkings original proposal, they do not
affect the observations of an outside observer.
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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 Î E, also known as A Î
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 Î E is a
modification of the rules of quantum mechanics.
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General idea (PR 1211.6767,1310.6335, VV
1211.6913). Consider a typical black hole state
ytgt. The distribution of the mode b is
thermal ytgt Z-1/2 ?n e-wb/2T ngtb
yt,ngtB where B is the complement to b.
Compare 0gta Z-1/2 ?n e-wb/2T ngtb
ngtb Thus identify the internal Hilbert
space, ngtb yt,ngtB Problem given a black
hole in some specified state ygt, which ytgt do
we use to identify the internal Hilbert space?
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PR solution single out a small space of
operators A. Then ytgt Uygt where U is in A,
and the expectation values of A in ytgt are
thermal. Problem observables behind the horizon
are now functions of ygt 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.
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Ordinary QM The system is in a state Ygt. We
have a basis igt. The probability of finding
the system in a given basis state is The
probability of finding a given excitation
is 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 ygt. This is the Born rule, and PR
modify it PS depend on ygt.
ltiygt2 ltyPiygt
?i ÎS ltiygt2 ltyPSygt
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For observables outside the black hole (e.g. the
occupation number for b), we have the usual
rules PS does not depend on ygt. For
observables behind the horizon (occupation number
for b ), PS depends on ygt. Is this a bug or a
feature? Not well-defined in current form,
assigns multiple interpretations to same ygt
(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).
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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.
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Limits on quantum computation (Harlow Hayden
1301.4504) perhaps there is not time to verify
the b-E entanglement. Doesnt 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?
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Final state boundary condition at the black hole
singu-larity (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)
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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?
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extra slides
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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.