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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.

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.

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 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

P  h

5  5.4

 10  44 sec experiments will be essential, and

black holes

have been a fruitful arena.

Hawking’s thought experiment: First, consider the formation of a

classical

black hole: Anything behind the horizon is trapped and falls into the singularity.

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 temperature, etc. (Bekenstein, Hawking, …)

As a result, the black hole eventually emits all of its energy and disappears, leaving only the outgoing Hawking radiation.

(Number of quanta ~

2 /

P 2 )

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

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 free dom 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 Bekenstein Hawking 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.

BH counts the states of the black hole.

The first three of these cannot all be true. cf. Mathur, Giddings, Braunstein

Consequences of No Drama + EFT

b ’ a b

Aa + Ba

†

Cb + Db

† +

C ’ b ’

D ’ b ’ b

Creation/annihilation operators:

: Inertial observer near horizon

: Outgoing Hawking modes

b ’

: Ingoing Hawking modes Adiabatic principle/no drama:

| y = 0 so

| 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

is late. Expanding in a basis

L i

The early Hilbert space is much larger than the late Hilbert space, so to very good accuracy the

E i

are distinct. So the late radiation is fully entangled with the early radiation. There is some bit

E early Hilbert space such that

and

E in the are in a pure state. (cf Page, Hayden and Preskill).

b ’ b E

A contradiction: Purity:

E are in a pure state.

No drama:

b ’

in a mixed state.

are in a pure state, while

itself is EFT: These are the same

b .

Quantum mechanics doesn’t allow this! No Drama needs state of (

b’,b,b

E ) 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, b E so complementarity does not save us.

and b ’ ,

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

: 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

(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 entan glement, Maldacena & Susskind ‘13).

Nonlinear observables (Papado dimas & Raju ‘12, Verlinde 2 ‘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

-foldings vs. ln(

R/l

P ) .

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,

P 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.

From G. ‘t Hooft

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

/c R

(ln

R/L

P ).

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The Black Hole Information Paradox, Alive and Kicking

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