The Black Hole Information The Black Hole Information Paradox:Paradox:

Past and FuturePast and Future

Joseph PolchinskiJoseph PolchinskiKavli Institute for Theoretical PhysicsKavli Institute for Theoretical Physics

University of California at Santa BarbaraUniversity of California at Santa Barbara

Institute for Gravitation and the Cosmos Penn State, Aug. 9, 2007Penn State, Aug. 9, 2007

Planck’s challenge:Planck’s challenge:

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lP = hG /c 3 =1.6 ×10−33cmPlanck length,

Planck’s constant , the speed of light c, and Newton’s constant G define a natural system of units:

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h

M. Planck, On Irreversible Radiation Processes (1899).

“These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as natural units.”

Also, G. Stoney (1874), with e2/c instead of .

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h

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lP = hG /c 3 =1.6 ×10−33cm

This is apparently the scale at which quantum mechanics and gravity come together.

The fact that it is so small means that it will likely be very difficult to figure out what is happening, and to test our ideas.

How are we to proceed?

Thought experiments Thought experiments have often played a role in the development of physical theories:

• • EinsteinEinstein

• • GalileoGalileo

• • MaxwellMaxwell

• …• …

• • HawkingHawking

disclaimer

Hawking’s thought experiment (1976):Hawking’s thought experiment (1976):

sing

ular

itysi

ngul

arity

horiz

onho

rizon

1. Initial state: infalling matter1. Initial state: infalling matter

2. Black hole formation2. Black hole formation

3. Black hole evaporation3. Black hole evaporation

4. Final state:4. Final state:Hawking radiationHawking radiation

Thought experiment: repeat this process many times with the same initial state i and make repeated measurements of the final system.

Claim: the final system cannot be described by a pure state f but only by a mixed state (density matrix)

i i i f = A,B SABiii i ≠ f f

The argument: the state on the indicated time slice is pure but entangled, A,B SABiA B . The state behind the horizon is not observable, and after the black hole evaporatesthis information islost.

A

B

Thus the system after evaporation can only be described by the density matrix AC = B SABi SCBi

This thought experiment implies a breakdown of the ordinary rules of quantum mechanics, and this should be happening everywhere, all the time, via virtual black holes.

In addition to this paradox there is a related puzzle: black holes satisfy thermodynamic laws, with a Bekenstein-Hawking entropy S = A/4lP

2. What is the underlying statistical mechanics? What states does this entropy count.

This talk will focus on the paradox.

"Not only does God play dice, but he sometimes throws them where they cannot be seen."

Possible outcomes to black hole evaporation:Possible outcomes to black hole evaporation:

1. The state of the Hawking radiation is actually pure. The information (about what went into the black hole) is encoded in the Hawking radiation.

2. The state is indeed mixed. Information is lost.

3. The evaporation does not proceed to completion, but terminates in a stable remnant with a very large number of internal states.

4. A remnant which (slowly) decays, reemitting the information.

1. The information gets out with the Hawking radiation1. The information gets out with the Hawking radiation

Consider an analogous thought experiment:

0 K coalcoherent light

hot coal

0 K coalIncoherent radiation

0 K coalIncoherent radiation

The final state is similar to the black hole case, but here the state is pure: there are definite phase correlations between the different states of the radiation.

Did Hawking miss such subtle phases in his calculation?

It is easy to confuse oneself:• Coordinate singularity transplanckian energies• Time-reversed evolution unstable.

Quantum Xerox principle (Susskind): take some entangled bit-pairs, and throw half of each pair into the black hole. These pass through the horizon essentially unchanged (the equivalence principle) and reach the interior, which is spacelike separated from the Hawking radiation.

+

There is no quantum xerox machine that can take any state in a single Hilbert space into a state in a pair of Hilbert spaces:

If ˝ ˝ ˝

and ,

then = √ √ ≠

˜ ˝

˝ ˝

˜ ˜

Quantum information thrown into the black hole cannot also be in the Hawking radiation, it is lost. A black hole is not like a lump of coal because it has a horizon.

2. Information is lost2. Information is lost

It is problematic whether there is a consistent theoretical framework. For example, this seems to imply energy non-conservation (Banks, Peskin,

Susskind, 1984). Argument: a local time-evolution law that turns pure into mixed states can be written as evolution with a spacetime-dependent Hamiltonian density (Ellis, Hagelin, Nanopoulos, Srednicki, 1984).

Heff = g(t,x) O (t,x)

Here, O (t,x) are some set of local operators, andg(t,x) is averaged incoherently. That is,

`spacetime is dirty.’

Even if the average distribution of dirt is translation-invariant, any given distribution is not, and so momentum and energy conservation are lost.

Possible loopholes: • Locality --- should be OK for small or virtual

black holes. • Unobservability (Unruh & Wald, 1995).

Instantaneous change in Schwarzschild mass?

3. The information is stored in a stable remnant.3. The information is stored in a stable remnant.

The Hawking calculation breaks down when the black hole mass and curvature approach the Planck scale. Perhaps the evaporation ceases, and we have a `remnant’ which has a large number of internal states (representing the states of the infallen Hawking particles), so that

remnant x Hawking radiation

is in a pure state.

Problem: since the initial black hole could have been arbitrarily large, the necessary number of internal states is unbounded. Divergent effects from virtual remnants?

4. The information is stored in a long-lived remnant, 4. The information is stored in a long-lived remnant, which slowly decays and releases the information.which slowly decays and releases the information.

There are of order M2/MP2 Hawking quanta of energy

MP2/M, where M >> MP is the initial black hole mass.*

If the final decay of the black hole involves a few Planck-scale quanta, there are not enough degrees of freedom to restore purity. However, if the remnant decays via a large number of very low energy quanta (energy of order MP

3/M2).

Problem: still an unbounded number of light states, giving divergent virtual effects.

* h = c = 1.

Possible outcomes to black hole evaporation:Possible outcomes to black hole evaporation:

1. The state of the Hawking radiation is actually pure. The information (about what went into the black hole) is encoded in the Hawking radiation.

2. The state is indeed mixed. Information is lost.

3. The evaporation does not proceed to completion, but terminates in a stable remnant with a very large number of internal states.

4. A remnant which (slowly) decays, reemitting the information.

Other possibilities?

5. Difference between pure and mixed states not 5. Difference between pure and mixed states not observable due to interaction with environment? observable due to interaction with environment? (Srednicki, ) Just put the system in a box, such as AdS.

What about the singularity?What about the singularity?

One possibility: spacetime continues past the singularity, forming a disconnected `baby universe.’

Looks like information loss, but consideration of the quantum state of the baby universe apparently leads to long-lived remnants (JP & Strominger,

Strominger 1994). Long story, -parameters, …

Another possibility (e.g. Ashtekar & Bojowald, 1995):

Planckian region

O

Apparently the information is emitted --- how?

The observer O sees a Planck-sized object with a very large number of internal states: this is a long-lived remnant.

Lesson:Lesson: In order for the information to be in the In order for the information to be in the Hawking radiation, it must be transmitted over Hawking radiation, it must be transmitted over large spacelike distances:large spacelike distances:

Black hole complementarity (Susskind, 1993): these are the same state as seen by two different observers --- radically nonlocal…

From G. ‘t Hooft

Another hint of radical nonlocality: the Bekenstein-Hawking entropy S = A/4lP

2 suggests the holographic principle (‘t Hooft, Susskind, 1993), that quantum gravity in any space can be formulated in terms of degrees of freedom living on the boundary of the space.

coupling

weak strong

black hole or braneD-branes and strings

Strominger and Vafa (1996) argued that by turning down the coupling one could adiabatically turn some supersymmetric black holes into weakly coupled systems whose states can be explicitly counted, giving a statistical interpretation to the Bekenstein-Hawking entropy.

Black hole entropy counting:Black hole entropy counting:

Motivated by the information paradox, various groups studied dynamical properties of this system (scattering amplitudes, decays) and found surprising agreements between very different calculations:

Field theory loop graph Gravitational tree amplitude in black hole background

Maldacena (1997) explained this in terms of a new duality:

coupling

weak strong

black hole or braneD-branes and strings

coupling

weak strongN gauge theory IIB superstring with

AdS5 x S5 b.c.

lowenergylimit

lowenergy

limit

This is another duality; whereas previously known weak-strong duality are

quantum field theory quantum field theory

string theory string theory this is

quantum field theory string theory

Like all such dualities this is not proven, but the evidence is very strong, and growing.

Evidence:Evidence:

• All fock states of graviton and other massless fields can be identified in the gauge theory.

• All trilinear interactions agree.

• Many excited string states can be identified.

• Many semiclassical geometries.

• Many qualitative behaviors (e.g. confinement) appear where expected.

• A growing number of direct numerical studies of the strongly coupled gauge theory.

• The assertion that a ten-dimensional string theory is the same a four-dimensional gauge theory is so counterinuitive that if it were false it should be easy to find counterexamples.

Evidence, page 2:

• Any theory of quantum gravity in AdS spacetime defines a CFT via the Gubser-Klebanov-Polyakov-Witten dictionary. N = 4 gauge theory is the only renormalizable theory with the same symmetries as string theory on AdS5 x S5.

• Even the most severe critics acknowledge that the gauge theory contains both classical GR and quantum mechanics. It is therefore some theory of quantum gravity, and there are many indications that it is the same theory that was originally discovered by quantizing strings in a classical spacetime background.

This duality provides an This duality provides an algorithmicalgorithmic nonperturbative constuction of string nonperturbative constuction of string theory with AdS boundary conditions.theory with AdS boundary conditions.

Issue: lattice breaks SUSY, but the continuum limit can be understood analytically in 2+1, 1+1, 0+1.

AdS (covering space)

Almost background independent:the asymptotic geometry is fixed, but in the interior the geometry and even topology can fluctuate freely, and pass through nongeometric Planckian states.

The interior spacetime, and its coordinate group are emergent. The coordinate invariance acts trivially on the gauge theory variables.

We can repeat our thought experiment in an AdS box. The dual process is described explicitly as an ordinary coherent system: information is preserved (option 1).

The gauge theory variables are indeed strongly nonlocal, and holographic.

With a duality, information flows in both directions:With a duality, information flows in both directions:

gauge theory gravity/string theory

gauge theory gravity/string theory

The application to the information problem is an example of the first. The application to RHIC (Relativistic Heavy Ion Collider) physics is an example of the second, but it also casts an interesting light on the nature of gravity.

The dualities allow us to solve exactly some strongly coupled gauge theories - not yet QCD, but getting closer. In fact, there appear to be surprising universalities in strongly coupled gauge theories, so that some results are applicable to QCD.

• A black hole is dual to a thermal state in the gauge theory, generalizing the black hole mechanics thermodynamics connection. The Bekenstein-Hawking entropy gives a good (~10%) approximation to the gauge theory entropy.

• The black hole membrane paradigm says that the horizon has properties such as viscosity and conductivity. Under the duality, these become the viscosity and conductivity of the dual gauge theory plasma. The numbers match the observations at RHIC better than QCD-based calculations.

• Other properties, such as drag, also seem to match.

But is RHIC really making a black hole?

Obvious answer: of course not, it’s just quarks and gluons!

If we could increase Nc and add superpartners, we would reach a point where the gravitational, and black hole, description is the only natural one (and LHC may find such a theory). There is essentially a continuum:

quarks & gluons black hole

The surprise is that the RHIC plasma appears to be around the middle! Refrain

The point is that duality erases distinctions that we make with our classical experience - between particle and wave, and black hole and plasma - because a single quantum theory has many classical limits.

Any theory of quantum gravity in AdS should define a boundary CFT, by the GKP-Witten dictionary (emphasized by Shenker). Pure gravity in 2+1 may be a non-string example.

More generally…More generally…

Seems to imply that information is never lost - isomorphic to Banks-Peskin-Susskind argument?

The future…The future…

A few of the big questions:

How do we extend the construct of string theory via AdS/CFT to other boundary conditions (i.e. realistic compactifications, and interesting cosmologies). This probably requires some conceptual leaps: while AdS/CFT describes emergent space, other boundary conditions seem to require emergent time. And, how does the understanding of black hole horizons extend to cosmologicial horizons…

The future…The future…

One question has been answered, but more are opened up:

• If the information is fact emitted with the Hawking radiation, where was Hawking’s mistake?

• How exactly does locality emerge, and how does it break down?

Where was Hawking’s mistake - where does his argument for information loss break down?

1. At the level of effective field theory. E.g., a breakdown of perturbation theory was overlooked, or a saddle point was missed (Maldacena, Hawking):

AdS black hole thermal AdS

2. At the level of perturbative string theory.

3. Only in fully nonperturbative quantum gravity.

How exactly does locality emerge, and how does it break down?

If the black hole has an S-matrix, how do we calculate it?

In AdS/CFT:

initial bulk state initial CFT stateduality

final bulk stateduality

final CFT state

gauge theory evolution

Can we short-circuit this?

Can we learn more from AdS/CFT?

Problem: the large-Nc theories that we can solve are too simple to have interesting to have interesting spacetime duals, and vice-versa.

Recent (esp. Festuccia & Liu): even the weakly coupled N gauge theory have analogs of the black hole singularity and the information problem. Interesting to study this and further examples (Iizuka & JP)

ConclusionsConclusions

Don Page (1993) showed that if the information is emitted with the Hawking radiation, there is essentially none in the first half of the photons, and then it gradually begins to emerge.

This is a good metaphor for the information problem itself. Hawking found the problem in 1976, and for a while much effort went in and nothing came out. The first information emerged in 1993 (complementarity, holography), and more since, but I don’t think that the problem has decayed completely…