The entropy of Hawking radiation

Juan Maldacena

Institute for Advanced Study

Talk at QGSC , Bariloche/Valdivia, December, 2019

Outline

• Black holes as quantum systems.• Black hole interiors.• The RT-HRT-EW entropy formula. • Entanglement wedge reconstruction. • Applications to black holes. • Entropy of radiation coming out of black holes. • Page curve. • Island rule.• Replica wormholes.

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 (lp =1)

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

=

Central hypothesis

Evidence: Entropy counting

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

Strominger Vafa…Sen…

Another reason is AdS/CFT…

Hot fluid made out of very stronglyinteracting particles. Gravity,

StringsBlack hole

Black hole in a box. Evolving unitarity.

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

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

What about the inside?

Taxonomy of insidesSpecimen collection (stamp collecting)

Geometry of a Black Hole made from collapse

interior

star

SingularityOppenheimer Snyder 1939

horizon

One exterior, one interior.

Other geometries allowed by the equations

``Bags of Gold”Initial slice:

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 Area entropy counts the entropy “inside”

Wheeler

The holographic principle resolves the black hole information paradox within the framework of string theory.[4] However, there exist classical solutions... "Wheeler's bags of gold". …. are not yet fully understood.[5]

Wikipedia article:

Geometry of an evaporating black hole made from collapse

Partners of radiation

Singularityradiation

A pure state seems to go a mixed state.

The radiation is entangled with partners of radiation.

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

Geometry of an evaporating black hole made from collapse

Singularityradiation

Entropy on green slice (nice slice), inside the black hole, could be much bigger than the area at T,for a black hole that has evaporated fora long time.

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

T

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

Time

Entropy of outgoing radiation Hawking’s prediction

Expected from unitarity

What computation reproduces this ?

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. Two exteriors, sharing the interior.

Right exterior

Left exterior

singularity

interior

interior

The full Schwarzschild wormhole

horizonRight exterior

left exterior

AdS asymptotics.

Central hypothesis applies to each exterior separately ! à two quantum systems

Wormhole 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.

|TFDi =X

n

e��En/2|EniL|EniRIn a particular entangled state

Inside ?

• Black holes from the outside look like quantum systems.

• How is that system related to the inside ?

• Before addressing this question we will have to talk about entropy and gravitational entropy.

Two notions of entropy for general physical systems

Two notions of entropy

• Coarse grained entropy = thermodynamic entropy. Obeys 2nd law.

• Fine grained entropy. Remains constant under unitary time evolution. (sometimes called ``entanglement’’ entropy)

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Subset of observables, ``simple observables”

Thermodynamic black hole entropy

Is coarse grained entropy. Increases under time evolution (Hawking’s area theorem + Wall 2011 )

Is the coarse grained entropy of the quantum system that describes the black hole as seen from the outside.

Note: It increases under Hawking radiation.

S =Area

4l2p+ Smatter

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Area of the horizon + entropy of fields outside, including the quantum entropy of the fields.

Bekenstein 70’s ; Bombelli, Koul, Lee,Sorkin 1986

Fine grained gravitational entropy

It turns out that there is also an ``area’’ formula for the fine grained entropy, but it is the area of another surface!

Fine grained gravitational entropy

We need to find an extremal area. The ``smallest’’ extremal area.

Start with a surface going around the horizon and shrink it as much as possible.

We are allowed to take the surface tothe inside.

Ryu-Takayanagi 2006Hubeny, Rangamani, Takayanagi 2007Engelhardt, Wall 2014

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

It is supposed to be the fine grained entropy of the quantum system that describes the black hole as seen from the outside.

S = min

⇢ext

Area

4l2p+ Smatter

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More precise: minimal area along a spatial slice (Cauchy slice) and maximal amongall possible Cauchy slices

interior

star

Singularity

horizon

Zero areaEntropy = entropy of matter that makes up the star

Extremal surface for a black hole formed from collapse

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

The fine grained entropy for one side of the Schwarzschild wormhole is equal to the Area.

Right exterior

Left exterior

interior

interior

There are intermediate cases…

Extremal surface for the Schwarzschild wormhole

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

grained entropy

Entanglement wedge

• We keep track of the region swept by the surface as it tries to extremize the entropy.

• That region outside the quantum extremal surface is called the ``entanglement wedge’’.

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

A

As we move the surface inwards,we keep track of the entropy in the fields outside

horizonRight exterior

left exterior

Here the extremal surfaceIs the bifurcation surface

The entanglement wedge isall of the outside.

interior

star

horizon

Entanglement wedge covers the whole interior.

We saw that there was a quantum system that describe the black hole from the outside.

How much of the spacetime does this quantum system describe ?

- Only the outside ? - A portion of the inside ? - Which portion ?

We will introduce a new hypothesis

Entanglement wedge reconstruction hypothesis

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

• The reconstruction depends on the state. (or on a code subspace)

• There are some arguments for why this is the case, but I will not go over them. But there is not explicit construction.

• Notice that if we change anything within the entanglement wedge we change the fine grained entropy.

• The fine grained entropy is a notion that involves the full quantum system that describe the black hole, as seen from outside.

Almheiri, Dong, Harlow, Jafferis, Lewkowycz, J.M., Suh, Dong, Wall, Faulkner

We will now argue that this removes some apparent paradoxes

``Bags of Gold”Two possibilities: A) Bag has little entropy due to matter (compared with the area of the black hole horizon)B) Bag has lots of entropy due to matter

Little entropy. Entanglement wedge covers all. Could still be very big, but since the numberof states is small, and the map can be state dependent, this is not a problem

Large entropy. Entanglement wedge at the neck.

A B

Size of entanglement wedge depends on the entropy (fine grained), not on the energy, of the matter inside.

Wall

Old evaporating black hole

Old evaporating black hole

SingularityradiationThere is a second quantum extremal surface.

The entanglement wedge Includes on a small part of the interior, just behind the horizon.

It is crucial to include the quantum correction due to the fields.

The situation is somewhat similar to the bag of gold example.

T

Two implications: 1) Entropy is close to the old black hole entropy.

Entropy is close to the coarse grained black hole entropy of the old black hole

Penington , Almheiri, Engelhardt, Marolf, Maxfield ; 2019

Fine grained entropy of the black hole (of the quantum system describing the black hole )

Time

Black hole fine grained entropy

From the trivial extremal surface

From new extremal surface

Almheiri, Engelhardt, Marolf, Maxfield ; Penington 2019

What about the entropy of the radiation ?

• Since the initial system was in a pure state, the entropy of the rest, namely the radiation, should be equal.

• Then we get the Page curve.

Entropy of radiation?

• The radiation appears to be in a mixed state. • Why? • Because it was entangled with the fields that

were inside the black hole.

• How do we know this? • Through the evolution of gravity. • We should use the gravity rules to compute the

fine grained entropy.

Entanglement wedge of radiation

Matter entropy is small. Total entropy is the same as the black hole system

Singularity

T

Island

radiation

We will argue for this:

New rule

S(Rad) = min

S(Rad [ Islands) +

Area(Islands)

4GN

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Exact entropyEntropy in the effective field theory description

Singularity

T

Island

radiation

Derivation

It is really contained in the standard arguments, once you realize that when you use the replica trick, you are allowed to fill in the gravitational part with different topologies. These can look like adding extra twist fields in the bulk quantum field theory. And the n à 1 limit, then leads to the above prescription.

Dong, Lewkowycz 2017. Penington, Shenker, Stanford, Yang ; Almheiri, Hartman, J.M., Shaghoulian, Taj, 2019

A simple set up

• Black hole coupled to a bath with no gravity

Two copies in the TFD state

Replica trick• Replica trick à introduce n copies, glued

together in the regions whose entropy we are attempting to compute.

• In the rest of the spacetime, they can connect however they like.

• If extra connections appear à islands appear.

Solution that gives Hawking’s result

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

n=2

n =3

Extracting information• Should be possible according to general quantum information

ideas.

• Petz map (Penington’s talk)

• For a free fermion bulk theory (in 1+1) dimensions à use the expression for the modular Hamiltonian for the radiation

K =Area

4l2p+Kbulk

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Jafferis, Lewkowycz, JM, SuhDong, Lewkowycz

This is the modular Hamiltonian for two intervals in a free fermiontheory à has an explicit bi-local expression Casini, Fosco, Huerta , 2005

Using the flow generated by this modular Hamiltonian we can extract theinformation from the island

Yiming Chen, 2019

Conclusions

• We reviewed the gravitational fine grained entropy formula.

• We reviewed the concept of ``entanglement wedge’’

• We discussed how to compute the fine grained entropy of the black hole or the radiation, both describing the Page curve.

• We discussed the ``island rule’’ and its origin in terms of Renyi wormholes.

Future

• Can we understand better what happens to spacetime dynamically as we perform the operations that extract the information ?

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

Thank you !