Black Holes:Black Holes:Spacetime vs. Quantum MechanicsSpacetime vs. Quantum Mechanics

Joseph PolchinskiJoseph 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 anevaporating black hole

S

t

Hawking result

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

t

Hawking result

S = von Neumann entropy of the Hawking radiation = entanglement entropy of the radiation and black hole = von Neumann entropy of the black hole

S

t

Hawking result

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.

Information loss:

Remnants

S

t

Hawking result

Alternative: black hole evaporation ends in a remnant with a large number of internal states.

Remnants

S

t

Hawking result

Alternative: black hole evaporation ends in a remnant with a large number of internal states.

SBekenstein-Hawking

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.

S

t

Hawking result

Page curve

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

I. Quantum gravity (actuallystring 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

BFSS Matrix Theory/AdS/CFT duality

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 orderEinstein-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!)

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= 0 so b≠ 0

This implies:

• Hawking radiation

• b and b’ are entangled.

AMPS I: Consequences of No Drama + EFT

AMPS II: Consequences of purity

b

b’

E

SvN(radiation)

tb

info loss

info conserved

If information is not lost, b must be entangledwith the earlier radiation. (Page, Hayden & Preskill)

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

A contradiction:

Moreover, a single observer can see all of b, E and b’, so complementarity does not save us.

bb’

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≠?

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 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 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 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 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.

bb’

E

General idea (PR 1211.6767,1310.6335, VV 1211.6913). Consider a typical black hole state |t. The distribution of the mode b is thermal:

|t= Z-1/2 ∑n eb/2T |nb|t,nB*

where B* is the complement to b. Compare

|a= Z-1/2 ∑n eb/2T |nb|nb’

Thus identify the internal Hilbert space,

|nb = |t,nB*

Problem: given a black hole in some specified state |,

which |tdo we use to identify the internal Hilbert space?

PR solution: single out a small space of operators A. Then: |t = U| where U is in A, and the expectation

values of A in |t are thermal.

Problem: observables behind the horizon are now functions of |: 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 |>. We have a basis |i>. 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 |>.

This is the Born rule, and PR modify it: PS depend on |>.

<i|>= <|Pi|>

∑i S<i|>= <|PS|>

For observables outside the black hole (e.g. the occupation number for b), we have the usual rules: PS

does not depend on |>. For observables behind the horizon (occupation number for b’ ), PS depends on |>.

Is this a bug or a feature?

• Not well-defined in current form, assigns multiple interpretations to same |> (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-(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 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)

• 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?

Open questions

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.