Recent progress on the black hole

information paradox

Juan Maldacena

Institute for Advanced Study

We will discuss recent progress on the black hole information problem

Review: Almheiri, Hartman, JM, Shaghoulian, Tajdini

Two important papers in 2019:

Almheiri, Engelhardt, Marolf, Maxfield

…many previous and follow up papers...

Also papers by: Saad, Shenker, Stanford

(this contains a more general list of references than this talk)

Penington

Outline

• Black hole entropy.• The fine-grained gravitational entropy

formula.• Compute the entropy of radiation coming out

of black holes. • Get a result consistent with information

conservation (as opposed to information loss).

It will not be historical, but hopefully pedagogical…

Black holes

• Black holes have a temperature.

• 1st Law, dM = TdS, à Entropy

• Area increase theorem à 2nd Law.

Hawking

Hawking

Bekenstein Hawking

Sgen =AreaH4GN

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Semiclassical gravity (as an effective field theory)

• Treat the background geometry as classical. • Quantum fields propagating on a classical

background. (Including perturbative gravitons). • Backreaction due to Hawking radiation treated

using the semiclassical equations.

g2eff / GN

r2s/ 1/S

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Effective coupling:

Energy spacings ⇠ e�S ⇠ e� 1

g2eff

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Microstates seem to requirenon-perturbative accuracy

Entropy, again

Von Neuman entropy, or quantum entropy,of the semiclassical quantum fields in the

region outside ⌃<latexit sha1_base64="989lE/3P2w9zbeyUj5NXmlPK5HE=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAIrspMXehGLLpxWdFeoB1KJs20sUlmSDJCGfoOrgQFceHGN/EBXIhvY3pZaOsPgY//P4ecc8KEM20879tZWFxaXlnNrbnrG5tb2/md3ZqOU0VolcQ8Vo0Qa8qZpFXDDKeNRFEsQk7rYf9ylNfvqdIslrdmkNBA4K5kESPYWKvWumFdgdv5glf0xkLz4E+hcP7hniVvX26lnf9sdWKSCioN4Vjrpu8lJsiwMoxwOnRbqaYJJn3cpU2LEguqg2w87RAdWqeDoljZJw0au787Miy0HojQVgpseno2G5n/Zc3URKdBxmSSGirJ5KMo5cjEaLQ66jBFieEDC5goZmdFpIcVJsYeyHXtFfzZneehVir6x8XStVcoX8BEOdiHAzgCH06gDFdQgSoQuIMHeIJnJ3EenRfndVK64Ex79uCPnPcfVECSVA==</latexit>

Obeys the 2nd law Wall

(entanglement entropy of the quantum fields)

S =AreaH

4GN

+ Ssemi�cl(Outside)

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These results have inspired a very influential hypothesis:

A ``central dogma’’ in the study of quantum aspects of black holes

Black holes as quantum systems

• A black hole seen from the outside can be described as a quantum system with S degrees of freedom (qubits). S = Area/4 (lp =1)

• It evolves according to unitary evolution, as seen from outside.

=

The Central Dogma

Black holes as quantum systems

• A black hole seen from the outside can be described as a quantum system with S degrees of freedom (qubits). S = Area/4 (lp =1)

• It evolves according to unitary evolution, seen from outside.

=

The Central Dogma

Black holes as quantum systemsThe Central Dogma

=

Piece of coal (strongly interacting)

Note:

• These A/4GN qubits, are not very manifest in the gravity description.

It is only a statement about the black hole as seen from the outside !

No statement has been made about the inside (yet).

Evidence

1) Entropy counting

Supersymmetric black holes in supersymmetric string theories can be counted precisely using strings/D-branes à reproduce the Area formula. (+ also corrections to this formula)

Strominger Vafa…..

Sen…. Dabholkar, Gomes, Murthy…

2) Matrix theory

• BFSS matrix model à describes 11 dimensional black hole formation and evaporation. Gives us the S-matrix.

Banks, Fischler, Shenker, Susskind

3) AdS/CFT…

Thermal state in a CFTGravity, StringsBlack hole

JMGubser, Klebanov, PolyakovWitten

but

Hawking 1976 :

This can’t possibly be true!

Geometry of an evaporating black hole made from collapse

The radiation is entangled with partners of radiation.

Since we do not measure the interiorwe get a large entropy for the radiation.

A pure state seems to go a mixed state.

The skeptic’s view:

One universe splits of a ``teenage universe’’ (big baby universe)

The state is pure if you include both universes, but not if you look only at the original universe.

The Hawking curve Compute the fine-grained entropy of the radiation as it comes out of theblack hole (formed by a pure state)

Time

Entropy of outgoing Radiation(qualitative form)

Hawking’scalculation

Thermodynamic entropyof the black hole

tPage

The Hawking curve vs. the Page curveCompute the fine-grained entropy of the radiation as it comes out of theblack hole (formed by a pure state)

Time

Entropy of outgoing radiation

Hawking’scalculation

Expectedfrom unitarity

Thermodynamic entropyof the black hole

tPage

There were other apparent paradoxes with the black hole entropy formula.

Full Schwarzschild solution

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

ER

Eddington, Lemaitre, Einstein, Rosen, Finkelstein, Kruskal

Vacuum solution. No matter. Two exteriors, sharing the interior.

Right exterior

Left exterior

singularity

interior

interior

``Bags of Gold”Initial slice:

Wheeler

Evolves to a black hole as seen from the outsideand a black hole in a closed universe.

Can have arbitrarily large amount of entropy ``inside’’

Counterexample to the statement that theArea entropy counts the entropy “inside” or the entropy of the ``interior’’.

We will see how to resolve these confusions

• These confusions involve the black hole interior.

Wormholes and entangled states

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Connected through the interior

Entangled

= W. IsraelJ.M.JM Suskind

|TFDi =X

n

e��En/2|EniL|EniR

ER = EPR

The rest of the paradoxes involve understanding fine-grained entropy

Two notions of entropy

• Fine grained entropy. (Also called Von Neuman entropy, or quantum entropy, or ``entanglement’’ entropy)

• Coarse grained entropy = thermodynamic entropy. Arises from ``sloppiness”

Subset of observables, ``simple observables”, eg. A = E,Q,..

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Remains constant under unitary time evolution.

Obeys 2nd law.

For the moment we will be talking about the entropy of the black hole as seen from

the outside.

This is the entropy of the quantum system that appeared in the “central dogma”.

The entropy of the ``piece of coal’’.

The horizon area computes thermodynamic entropy

How can we compute the fine grained one ?

Fine grained gravitational entropy

The final surface is called minimal quantum extremal surface.

We are allowed to take the surface to the inside. It depends on the geometry of the interior

Ryu-Takayanagi 2006Hubeny, Rangamani, Takayanagi 2007Faulkner, Lewkowycz, JM 2013Engelhardt, Wall 2014

Follows from AdS/CFT rules: Lewkowycz, JM , Faulkner,Dong,…

Also maxi-min: minimize along a spatial slice (Cauchy slice) and then maximize amongall possible Cauchy slices. Wall; Akers, Engelhardt, Penington, Usatyuk.

S = minX

⇢extX

Area(X)

4GN+ Ssemi-cl(⌃)

��

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We should be surprised by the claim that there is a formula for the fine-

grained entropy

Example

S = Entropy of star , which is zero if it was in a pure state.

New quantum extremal surface

First appears about a scrambling time (rs log S ) after the black hole forms.

PeningtonAlmheiri, Engelhardt, Marolf, Maxfield, 2019

Its entropy is close to the area of the horizon at the time.

PeningtonAlmheiri, Engelhardt, Marolf, Maxfield

Two quantum extremal surfaces

PeningtonAlmheiri, Engelhardt, Marolf, Maxfield

Choose the minimal one

…we get the Page curve for the black hole

but

we really wanted the Page curve for the radiation!

• The radiation lives in a region where quantum gravity effects could be very small. (It could have left the AdS space, or it could be collected into a far away quantum computer).

• Since we obtained the state using gravity àwe should apply the gravitational fine-grained entropy formula!

The “Island” formulaPenington; Almheiri, Mahajan, JM, Zhao

SRad = minX

⇢extX

Area(X)

4GN+ Ssemi-cl[⌃Rad [ ⌃Island]

��

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We should view it as just a special case of the general gravitational fine-grained entropy formula

The “Island” formula

SRad = minX

⇢extX

Area(X)

4GN+ Ssemi-cl[⌃Rad [ ⌃Island]

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Entropy of the exactradiation state Entropy of the state in the semiclassical description

Penington; Almheiri, Mahajan, JM, Zhao

• If the initial matter state is pure, then the quantum extremal surfaces are the same as the ones we discussed before.

• Therefore we get the Page curve for radiation.

The skeptic’s complaint• “This is just an accounting trick!”• I have always said: “If you include the black hole

interior, then the state is pure. The information problem arises because you do not have access to the interior!”

Gravity’s accounting ``trick’’ à ``oracle’’

• It can be derived from the gravitational path integral. (Deriving the Ryu-Takayanagiformula and its generalizations)

• Oracle: It gives us the true fine-grained entropy of the exact state, but using only the semiclassical state.

Deriving the RT formula

• It is conceptually similar to the derivation of the black hole entropy using the Euclidean black hole.

Lewkowycz, JM ; Faulkner, Lewkowycz, JM; Dong, Lewkowycz, Rangamani; Dong, Lewkowycz

Euclidean black hole

rtE

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GibbonsHawking

``cigar’’

horizon

Igrav / � 1

GN

ZpgR+ · · ·

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S = (1� �@�) logZ =Area

4GN+ Ssemi-cl

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Tr[e��H ] = Zgrav ⇠ e�IgravZsemi-cl<latexit sha1_base64="wrwYPimiB9hQ242/quuNLNzWLeA=">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</latexit>

Fix the temperature far away, gravity chooses the geometry dynamically.

Euclidean black hole

r

GibbonsHawking

``cigar’’

horizon

S = (1� n@n)Tr[⇢n]|n=1 = (1� n@n) logZn|n=1

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Computes the entropy if we can only compute the traces, but the actual density matrix itself

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Other states prepared by a Euclidean functional integral.

S = (1� n@n)Tr[⇢n]|n=1 = (1� n@n) logZn|n=1

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Tr[⇢3]<latexit sha1_base64="uol4KJjWia2YByoUiTyaK+PXqAA=">AAACBnicbVDLSsNAFJ34rPEVdamLwSK4Kkkr6LLoxmWFviCJZTKZtENnJmFmIpTQjSs/xZWgIG79B1f+jdPHQlsPXDiccy/33hNljCrtut/Wyura+sZmacve3tnd23cODtsqzSUmLZyyVHYjpAijgrQ01Yx0M0kQjxjpRMObid95IFLRVDT1KCMhR31BE4qRNlLPOSkCyWFTjqEPA01ZTGAgB+l9DYYQ9pyyW3GngMvEm5MymKPRc76COMU5J0JjhpTyPTfTYYGkppiRsR3kimQID1Gf+IYKxIkKi+kXY3hmlBgmqTQlNJyqvycKxJUa8ch0cqQHatGbiP95fq6Tq7CgIss1EXi2KMkZ1CmcRAJjKgnWbGQIwpKaWyEeIImwNsHZtknBW/x5mbSrFa9Wqd5dlOvX8zxK4BicgnPggUtQB7egAVoAg0fwDF7Bm/VkvVjv1sesdcWazxyBP7A+fwCY8ZcM</latexit> Z3 symmetry

Lewkowycz, JMFaulkner, Lewkowycz, JMDong, Lewkowycz, RangamaniDong, Lewkowycz

Z3 fixed point

= fine grained entropy formula S = min

⇢ext

Area(X)

4GN+ Ssemi�cl(⌃)

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• In the same way that the Euclidean black hole gives us the entropy, this replica trick gives us the gravitational fine-grained entropy formula.

• If the state is prepared by a Euclidean path integral and it has dynamical gravity only in some regions. Then we should allow various topologies in that region.

• Interiors connected by ``replica wormholes’’ à island formula.

Penington, Shenker, Stanford, YangAlmheiri, Hartman, JM, Shaghoulian, Tajdini

Solution that gives Hawking’s result

Replica wormhole, giving the Page answer when it dominatesat late times.

Replica wormholes: n=2

The Hawking curve vs. the Page curve

Time

Entropy of outgoing radiation

tPage

Hawking curve vs Page curve transition àArises from a Hawking/Page transition for the replica geometries

Gets smoothed out by considering the sum over non-replica symmetric geometries

Penington, Shenker, Stanford, Yang; Dong, H. Wang; Marolf, S. Wang, Z. Wang

• The Renyi entropies are given by these other non-trivial geometries. But the von Neuman entropy is given by a computation in the original semiclassical geometry.

So far, we only talked about entropy.

What is this telling us about the interior?

How do we describe the interior?

• The central dogma involves degrees of freedom that describe the black hole from the outside.

• What part of the interior do these degrees of freedom describe?– 1) All of the interior?– 2) None of the interior?– 3) Part of the interior? Which part?

before answering, let us introduce a new concept

Entanglement wedge

• It is the causal domain of dependence of the region that appears in the computation of the entropy.

Czech, Karczmarek, Nogueira, Van Raamsdonk, Wall, Headrick, Hubeny, Lawrence, Rangamani

ExamplesEntanglement wedge of the black hole in green. (black hole = quantum degrees of freedom describe the black hole from the outside)

Entanglement wedge of radiation in blue. (At late times, it includes part of the black hole interior)

Now the answer:

Entanglement wedge reconstruction hypothesis

• The quantum system describes everything that is included in its entanglement wedge.

• We can recover the state of a (probe) qubit inside the entanglement wedge.

• Recovery is state dependent (subspace dependent) and similar to quantum error correction.

Czech, Karczmarek, Nogueira, Van Raamsdonk, Wall, Headrick, Hubeny, Lawrence,Rangamani, Almheiri, Dong, Harlow, Jafferis, Lewkowycz, J.M., Suh, Wall, Faulkner….

Radiation

Black hole

Part of the interior belongs to the black hole degrees of freedom and part to the radiation

Describing the interior

• By performing (a complicated) operation on the radiation à we can extract information from the interior.

• How does this happen?

• The complicated operation creates a wormhole connection that connects to the interior and pulls the information from there.

• Use the modular Hamiltonian, which can be non-local in the bulk.

Gao Jafferis WallPetz map: Almheiri, Dong, Harlow; Cotler, Hayden, Penington, Salton, Swingle, Walter; Chen, Penington, Salton; Penington, Stanford, Shenker, Yang

Jafferis, Lewkowycz, JM, Suh; Almheiri, Anous, Lewkowycz;…, Y. Chen

The AMPS paradox

• The AMPS paradox involved the central dogma + one further implicit hypothesis:

• The interior operators act on the Hilbert space of the black hole, the same Hilbert space that appears in the central dogma.

Mathur; Marolf, Wall; Almheiri, Marolf, Polchinski, Sully

The AMPS paradox

• The AMPS paradox involved the central dogma + one further implicit hypothesis:

• The interior operators act on the Hilbert space of the black hole, the same Hilbert space that appears in the central dogma.

Mathur; Marolf, Wall; Almheiri, Marolf, Polchinski, Sully

NO!

The AMPS paradox

• After the Page time, the troublesome interior mode is actually part of the radiation. It is in the entanglement wedge of radiation.

b

b’

a=b’

a

b’

Don’t confuse the following two things

• 1) Quantum degrees of freedom that are necessary to describe the black hole from the outside. Sometimes also called ``black hole degrees of freedom’’. (= all the dof of the ``piece of coal’’).

• 2) Quantum degrees of freedom in the black hole interior. Also sometimes called black hole degrees of freedom. Not the same as the degrees of freedom inside the piece of coal!

=

Ideas of quantum error correction, quantum complexity, are necessary to explain why a simple operation in the radiation does now affect a qubit in the interior.

Swingle; Almheiri, Dong, Harlow; Brown, Susskind, Kim, Tang, Preskill, + many others…..

comment

Radiation

Black hole

The entanglement wedge is a property of the state at some time t.

Horizons depend on the Hamiltonian and on what will happen in the future.

Suppose we decouple the two systems at the cutoff surface at time t. By evolving the black hole part differently we could recover the information behind the horizon that is in the green region. But we cannot recover the information that is in the blue region in the interior. Even if we decouple the systems, the blue and green regions would still be connected at the quantum extremal surface.

Decouple

``Bags of Gold”If there is a lot of entropy inside à the entanglement wedge of the black hole ends near the neck

Wall

Interior of the bag belongs to somebody else…

Bag of gold vs old black hole

The geometry and entropy on the orange slice is somewhat similar to the bag of gold.

Conclusions

• We reviewed the gravitational fine-grained entropy formula.

• We applied it to the computation of the entropy of radiation and obtained results consistent with unitarity.

• At late times, most of the interior is part of the radiation. It is not part of the ``black hole degrees of freedom’’.

What was Hawking’s mistake?

• Not to use the fine-grained entropy formula.

• A lot of what we discussed was derived by thinking about aspects of AdS/CFT, which itself involves string theory.

• But you only need gravity as an effective theory to apply these formulas.

There is an amazingly deep connection between gravity and quantum mechanics!

Is the information puzzle solved?• One aspect: computing the entropy, yes.

• Another aspect: Understanding what the state is, no.

• Replica wormholes compute the entropy, but do not tell us what the actual state is.

• As with black hole entropy, it is an accounting ``oracle’’. The explicit gravitational representation of the states is still mysterious. But the semiclassical solution is representing some aspects of the state.

Another aspect of the information problem

• There is a form of the information paradox that involves corrections to the long time behavior of correlators.

• Corrections due to the discreteness of the energy eigenvalues and the finiteness of the entropy.

Long times• A particular feature, ``the ramp’’, was

understood in terms of a wormhole geometry.• Averages (over couplings) are important for

this result.

Saad, Shenker, Stanford

JT gravity from random Hamiltonians

• JT gravity was interpreted as arising when we average over Hamiltonians.

• A new view on the random matrix/string theory connection. A new interpretation for the matrix.

Saad, Shenker, Stanford

Random couplings

• The idea that random couplings give rise to geometric connections and wormholes was suggested in the past.

• These ideas are being revisited.

• What role do they play in AdS/CFT ?

Coleman; Giddings, StromingerPolchinski, Strominger

Marolf, Maxfield; Giddings, Turiaci; …

Future

• What further lessons is this teaching us about the interior? The singularity ?

• Implications for cosmology ?

Thank you !

The Hawking curve vs. the Page curveCompute the fine-grained entropy of the radiation as it comes out of theblack hole (formed by a pure state)

Time

Entropy of outgoing radiation

Hawking’scalculation

Expectedfrom unitarity

Thermodynamic entropyof the black hole

tPage