ExploringtheblackholeinteriorThesurprisingsimplicityofnear-AdS2

JuanMaldacena

Strings2017Israel

Basedon: - Gao,Jafferis andWall- JM,DouglasStanfordandZhenbin Yang.- Ioanna Kourkoulou andJM.

SYKmodel

Emergentreparametrization symmetrywhichisspontaneouslyandexplicitlybroken

S = �C

Zdu{f(u), u} Schwarzian action

MotionoftheboundaryinAdS2

DominatemanyaspectsoftheIRdynamics

- Lowtemperatureentropy- Gravitationalbackreaction- Chaosexponent- Wormholetraversability(locationofhorizon)

NearlyAdS2gravity

VerysimplegravitationaldynamicsinNAdS2

TheSchwarzian àmotionoftheboundary

NearlyAdS2Dynamics

BulkfieldspropagateonarigidAdS2space.

BoundarymovesinarigidAdS2 space,following localdynamicallaws.(likeamassiveparticleinanelectricfield)

UVparticleorUVbraneasinaRandall-Sundrum model

Encodesgravitationaleffects

AdS2

AdS2

ConstraintsLinkedbytheconstraintsthatthetotalSL(2)chargesarezero.

QaL +Qa

matter +QaR = 0 a = 1, 2, 3

Example

Thehorizonmovesoutwhenyoudropinaparticle

Thehorizonmovesoutwhenyoudropinaparticle

Emissionofabulkexcitation

Theboundarytrajectorygetsa“kick”determinedbylocalenergymomentumconservationintheambientAdS2 space.(SL(2)chargeconservation).

Newpositionofthehorizon

Oldpositionofthehorizon

Rindler AdS2coordinates,wormhole

Euclideanblackhole

Euclidean&Lorentzianpictures

H2

|TFDi =X

n

e��En/2|EniL ⇥ |EniR

Interactionbetweenthetwoboundaries

eig�L(tL)�R(tR)

Insertintheevolutionoperator

eigh�L(tL)�R(tR)i

approximate

Impulsiveforcebetweenthetwoboundaries.- Canbeattractivefortherightsignofg.- Kicksthetrajectoriesinwards.

GaoJafferis Wall

Interactionmakesthewormholetraversable

NewpositionofthehorizonWecannowsendasignalfromthelefttotheright.

Thewormholehasbeenrenderedtraversable.

Nocontradictionbecausewehaveanon-localinteractionbetweenthetwoboundaries.

WhatitisNOT!

Thecatfeelsfinegoingthroughthewormhole

Noanimalswereharmedduringthisexperiment

Simplerprotocol

Doingameasurementà Teleportation

Actontherightwith

Fromthepointofviewoftheright,wegetthesame,whetherwemeasureornot.

�L �! sL

sLeigsL�R(tR)

measure

Sendclassicalinformation

Quantumstategoesthroughthewormhole

Minordifferenceswithusualteleportationdiscussion:

- Wedonotneedtomeasuretheleftstatecompletely.

- EncodinganddecodingaredoneviastandardHamiltonianevolution.

Asusual:Thereisaboundontheinformationwecansend:

Nclassical bits � 2Nquits

Originoftheboundingravity

Theinsertionofthemessagealsogivesasmallkicktothetrajectory.

Movestheinsertionpointsofthenon-localoperatorawayfromeachother

h�L(tL)�R(tR)i becomessmaller

Attractiveforceweakensànoopeningofthewormhole.

Backreactionduetoinsertionofthemessage

Preciseformulaforthe2ptfunction

Amountofinformationwecansendisroughlyg hV i = 1 , h�2L(0)i ⇠ 1

- Fouriertransformthesignalswewanttosend.- CorrelatorofVisevaluatedonabackgroundwithmomentumpà pdependentSL(2)transformationonV.

- Effectisamplifiedbyboosts,orchaos- Getanextraphasefrom hV i

C = he�igV �R(t)eigV �L(�t)i , V = �L(0)�R(0)

Includesgravitationalbackreaction

C ⇠Z

dp(p)2��1eipe�igei g

(1+pGNet)2�

GNet ⇠ 1

Net

Simplifiedlimit

C ⇠Z

dp(p)2��1eipe�igei g

(1+GNpet)2�

C ⇠Z

dp(p)2��1eipe�i(g2�GNet)p

Justasimple``translation’’ (oneoftheoperationsofSL(2)).

Thesignaldoesnot``feel’’anything!Compositeobjectsaresimplytranslatedwhole!

Shockwavesintwodimensionsarenotfelt.(Inhigherdimensionstherearetidalforces).

C = he�igV �R(t)eigV �L(�t)i , V = �L(0)�R(0)

Quantummechanicalmodel

H =X

i1,··· ,i4

Ji1i2i3i4 i1 i2 i3 i4

SameastheactionfortheUVboundaryinthegravitydescription.

Lowenergies.

S[f ] / �N

J

Z{f, ⌧}d⌧

TheSYKmodelSachdev YeKitaevGeorges,Parcollet

H =X

i1,··· ,i4

Ji1i2i3i4 i1 i2 i3 i4

SameastheactionfortheUVboundaryinthegravitydescription.

Lowenergies.

S[f ] / �N

J

Z{f, ⌧}d⌧

G(t, t0) = h i i(t0)i ! 1

|t� t0|2�

G(t, t0)f =

f 0(t)f 0(t0)

(f(t)� f(t0))2

��

DependenceofCorrelatorsontheboundaryposition

MorallyliketheAdS2metric

• Wegotthesameactionfortheboundarydegreeoffreedom(Schwarzian).

• AllthatwesaidaboutthemotionoftheUVboundaryNAdS2 bulk,alsoholdsforSYK.

• Traversability inSYK!• Sameformula!

Convenientoperators

S1 = i 1 2 , S2 = i 3 4 , · · ·

S2k = 1

V =X

SkL(0)SkR(0)

“spinoperators”,+1,-1eigenvalues

Interactionbetweentheboundaries

Definethe``spin”operators:

PurestatesinSYK

Measureallspinsà getthejointeigenstate |BsiSk ! sk

``Diagonal’’correlatorsonthisstate

Sameasinthethermalstate

h |i 1(t) 2(t)| i = h |S1(t)| i = s1G�(t)2 + o(1/N)

Offdiagonalcorrelators:

| i = e�H�/2|Bsi Projectontolowerenergystates

Givenintermsofthermaltwopointfunction

h | i(t) i(t0)| i = Tr[e��H

i(t) i(t0)] + o(1/N q�1)

Thissetgeneratesthefull Hilbertspace

Euclideanconfiguration

|Bsi

hBs|

Shockwave(endofspace)

| i = e�H�/2|Bsi

Euclideanconfiguration

|Bsi

hBs|

Shockwave(endofspace)

| i = e�H�/2|Bsi

Regionoutsidethehorizon

Completemeasurement

Addingbulkparticles

|Bsi

hBs|

Behindhorizon

Euclideanpreparation Lorentzianstate

Howcanweseebehindthehorizon?

Lorentzianstate

HSYK + gX

k

skSk

HSYK

sk

Seethewholeregion

he�igRdtiskSki ⇠ e�i

RdtghBs|iskSk|Bsi ! eiSextra

(f)

Sextra

[f ] = g

Zdt(f 0)2�

S = SSchwarzian

+ Sextra

Givesthismodifiedevolution.

Extratermà newforcepushingtheboundaryparticleinwards

Relationtotheblackholecloningparadox

Relationtotheblackholecloningparadox

• Alicehasanoldblackhole.• BobhasaquantumcomputerentangledwithAlice’sblackhole.

• AlicesendsinaM-qubitmessage.Andwaitsforittoscramblein

• BobonlyneedsafewmorethanMqubits(2Mbits)ofHawkingradiationfromAlice’sholetodecodethemessage.

Susskind-ThorlaciusHayden-Preskill

Bobcouldseetwocopies:- Decodedone- OneAlicesent

Cloningpuzzle

FigurefromthepaperofPreskill andHayden.

Analysis

Bob’scomputer

Maximallyentangled

Alice’sblackhole

Bobà producesasecondblackhole,maximallyentangledwiththefirst.

Maximallyentangled

(ThisishardtodoHarlowHayden)

Alice’sblackhole

Bob’sblackhole

Maximallyentangled

Alice’sblackhole

SaytheyarenearlyAdS2 blackholes…

Alice’smessage

Bob’sblackhole Alice’sblackhole

Alice’smessage

Bobgetssomeradiationandfeedsittohiscomputer(blackhole).

Bobgettingradiation

Feedinghiscomputer=blackhole.

TrajectoriesoftheboundariesafterBobcatchestheHawkingmode

Bob’sblackhole

Alice’sblackhole

BobnowgetsthemessageatP

P

Onlyonecopyofthemessageinthebulk!

Alice’smessage

Themessageswitchedsides!

P

BackwardextrapolationOfthestateofAlice’sboundaryafterBob’sextraction,usingtheunperturbedHamiltonian.

Newhorizon

Bob’sblackholeAlice’sblackhole

Alice’smessage

MoreliketheHPfigure…

PBobwillnotgetithere.

Bobgetstheemptymachinery

Bob’sblackholeAlice’sblackhole

Alice’smessage

MoreliketheHPfigure…

PBobextractsthemachinery

Bob’sblackholeAlice’sblackhole

Alice’smessage

Bob

P

Alice’sblackhole

Alice’smessage

Bobevolvesbackwardsintime

Bob’sblackhole Alice’sblackhole

Bobgetsthemessage

• Themessageisneverduplicatedinthebulkpicture.

• TheprocessofextractingthemessageputsitoutofreachfromAlice.

• Noneedtoinvokeunkown newtransplanckianphysicstosolvetheno-cloningproblem.

• Allunderstandablefromstandardrulesofgravityonthewormholegeometry.

• AssumesER=EPR.

Conclusions

• Simplepictureforthegravitationaldynamicsofnearly-AdS2.

• Traversability hasasimpleorigin.• Nothingspecialisfeltbythetraveler.• Traversability andteleportation.• Traversability inSYKhasthesamedescription.• Consistentwithinformationtransferbounds.• Weconstructedafullsetofpurestateswhichappeartohavesmoothhorizons.AndgeneratetheHilbertspace.

• Weshowedhowtolookatthewholeregionbehindthesehorizons.

ExtraSlides

NearlyAdS2KeeptheleadingeffectsthatperturbawayfromAdS2 Jackiw Teitelboim

Almheiri Polchinski

Groundstateentropy

Comesfromtheareaoftheadditionaldimensions,ifwearegettingthisfrom4dgravityforanearextremalblackhole.GeneralactionforanysituationwithanAdS2 region.

�0

Zd

2x

pgR+

Zd

2x

pg�(R+ 2)

Equationofmotionfor à metricisAdS2.Rigidgeometry!

Onlydynamicalinformationà locationoftheboundary.

�

�b

Z

BdyK Localactionontheboundary

Z

Bulk

pg�(R+ 2) + �b

Z

BdyK + Smatter[g,�]

Onesolution

ADMmassà size

RestofsolutionsRelatedbyAdS2isometries

• Generalrelativityhaswormhole-likesolutions.• SimplestversionisthemaximallyextendedSchwarzschildsolution.

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

L R

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

RightexteriorLeftexterior

singularity

Thesearenottraversable.Evenquantummechanicallyà Integratednullenergycondition

Zdx

+T++ � 0

Observercannotgetout

• Thisisgood,otherwisegeneralrelativitywouldleadtoviolationsoftheprincipleonwhichitisbased:amaximumpropagationspeedforsignals.

• Wewilltalkaboutsomespecialsituationswhereitmakessensetotalkabouttraversablewormholes.Thesedonotviolateanyoftheaboveprinciples.Buttheytellusinterestingthingsaboutblackholes.

Kruskal-Schwarzschild-AdS blackhole

Entangled state in two non-interacting quantum systems.

IsraelJM

two interior regions. It is important not to confuse the future interior with the left exterior.

Sometimes the left exterior is referred colloquially as the “interior” of the right black hole,

but we think it is important not to do that. Note that no signal from the future interior

can travel to either of the two exteriors.

InteriorFuture

Past Interior

L R

Left

ExteriorRight

Exterior

Figure 1: Penrose diagram of the eternal black hole in AdS. 1 and 2, or Left and Right,denote the two boundaries and the two CFT’s that the system is dual to.

The system is described by two identical uncoupled CFTs defined on disconnected

boundary spheres. We’ll call them the Left and Right sectors. The energy levels of the

QFT’s En are discrete. The corresponding eigenstates are denoted |n⇤L, |n⇤R. To simplify

the notation the tensor product state |n⇤L ⇥ |m⇤R will be called |n, m⇤.The eternal black hole is described by the entangled state,

|�⇤ =�

n

e��En/2|n, n⇤ (2.1)

where � is the inverse temperature of the black hole. The density matrix of each side is a

pure thermal density matrix.

This state can be interpreted in two ways. The first is that it represents the thermofield

description of a single black hole in thermal equilibrium [6]. In this context the evolution of

the state is usually defined by a fictitious thermofield Hamiltonian which is the di⇥erence

of Hamiltonians of the two CFTs.

Htf = HR �HL. (2.2)

The thermofield hamiltonian (2.2) generates boosts which are translations of the usual

hyperbolic angle ⇥. One can think of the boost as propagating upward on the right side

4

ER

| i =X

n

e��En/2|EniL ⇥ |EniR

Geometric connectionfrom entanglement

Traversablewormholestwo interior regions. It is important not to confuse the future interior with the left exterior.

Sometimes the left exterior is referred colloquially as the “interior” of the right black hole,

but we think it is important not to do that. Note that no signal from the future interior

can travel to either of the two exteriors.

InteriorFuture

Past Interior

L R

Left

ExteriorRight

Exterior

Figure 1: Penrose diagram of the eternal black hole in AdS. 1 and 2, or Left and Right,denote the two boundaries and the two CFT’s that the system is dual to.

The system is described by two identical uncoupled CFTs defined on disconnected

boundary spheres. We’ll call them the Left and Right sectors. The energy levels of the

QFT’s En are discrete. The corresponding eigenstates are denoted |n⇤L, |n⇤R. To simplify

the notation the tensor product state |n⇤L ⇥ |m⇤R will be called |n, m⇤.The eternal black hole is described by the entangled state,

|�⇤ =�

n

e��En/2|n, n⇤ (2.1)

where � is the inverse temperature of the black hole. The density matrix of each side is a

pure thermal density matrix.

This state can be interpreted in two ways. The first is that it represents the thermofield

description of a single black hole in thermal equilibrium [6]. In this context the evolution of

the state is usually defined by a fictitious thermofield Hamiltonian which is the di⇥erence

of Hamiltonians of the two CFTs.

Htf = HR �HL. (2.2)

The thermofield hamiltonian (2.2) generates boosts which are translations of the usual

hyperbolic angle ⇥. One can think of the boost as propagating upward on the right side

4

ER

Gao,Jafferis,Wall

Couplethetwofieldtheories.Directinteractionbetweenfieldsneareachboundary.

Cancreatenegativeenergyintheinteriorà

Gravitationalscatteringpushestheparticlethrough.

Negativeenergy

Sint = g�L(0)�R(0)

• Wewillstudythisphenomenoninmoredetailfortheparticularcaseofnearly-AdS2 ,wheretheeffectisparticularlysimple

• WewillshowthattheSYKquantummechanicaltheorydisplaysthesamephenomenon.

• Wewillshowthatwecanusethistoanalyzesomeaspectsaboutcloningofquantuminformationinblackholes.

InClassicalmechanics?

• TwoclassicalsystemsintheanalogoftheTFD(samepositionsandoppositemomenta)

• Twocupsofwater(classical)intheTFD.

• Tapontheleftoneatsomeearlytime,-t.• Att=0weletthemtoucheachotherandtransfervibrations• Attimetontherightwefeelthebumpontherightcup.

Similareffectinclassicalmechanics

• TwoclassicalsystemsintheanalogoftheTFD(samepositionsandoppositemomenta)

• Atsomeearlytime,-t,weperturbx2L ontheleftside.• Att=0wecouplex1Lx1R (isanothercoordinate).

• Attimetontherightwemeasurep2Randfinditdisplacedinamannercorrelatedwiththeinitialdisplacementofx2L

Sint = g(x1L(0)� x1R(0))2

Changeinx1LFromchangingx2L

Interactionà kicktop1R.

@x1L(0)

@x2L(�t)

@p2R(t)

@p1R(0)⇠ {x1L(0)p1L(�t)}{p2R(t)x1R(0)} ⇠ (· · · )2

Changeinp2R duetochangeinp1R

Areequal

Changeinp2R iscorrelatedwithchangeofx2L

Sint = g(x1L(0)� x1R(0))2

�p2R(t) / e

�Lt�x2L(�t)

Chaosfueledgrowth

Dynamics

Theboundarytrajectorygetsa“kick”determinedbylocalenergymomentumconservation.

Newpositionofthehorizon

Oldpositionofthehorizon

Absoption ofabulkexcitation

TheSYKmodelSachdev YeKitaevGeorges,Parcollet

NMajorana fermions { i, j} = �ij

Randomcouplings,gaussian distribution.

Toleadingorderà treatJijkl asanadditionalfield

J=dimensionful coupling.Wewillbeinterestedinthestrongcouplingregion

hJ2i1i2i3i4i = J2/N3

H =X

i1,··· ,i4

Ji1i2i3i4 i1 i2 i3 i4

1 ⌧ �J, ⌧J ⌧ N

G(⌧, ⌧ 0) =1

N

X

i

h i(⌧) i(⌧0)i

S = Nf [G(⌧, ⌧ 0)]

Gf = (f 0(⌧)f 0(⌧ 0))�G(f(⌧), f(⌧ 0))

Definenewvariable

IntegrateoutfermionsandgetanactionintermsofanewfieldG

Isanalogoustothefullbulkgravity+matteraction.

ThereisaparticularGthatminimizestheaction.ItisSL(2)invariant.(analogoustothevacuumAdS geometry)

Setoflowactionfluctuationsofthissolution.Parametrizedbyafunctionofasinglevariable.Reparametrization mode.

Lowenergyaction:

S =N

J

Z{f, ⌧}d⌧

SameastheactionfortheUVboundaryinthegravitydescription.

H =X

i1,··· ,i4

Ji1i2i3i4 i1 i2 i3 i4

SameastheactionfortheUVboundaryinthegravitydescription.

Lowenergies.

G(t, t0) =1

N

X

i

i(t) i(t0)

ZDGe�S[G] !

ZDfe�S[f ]

S[f ] / �N

J

Z{f, ⌧}d⌧

ZDGG(t, t0)e�S[G] !

ZDfe�S[f ](f 0(t)f 0(t0))�Gc(f(t), f(t

0))

hG(t, t0)i = Gc(t, t0) / 1

|t� t0|2�

Samepreciseformulaforthe2ptfunction

Amountofinformationwecansendisroughlyg

C = he�igV �R(t)eigV �L(�t)i , V = g�L(0)�R(0)

C = he�igV j R(t)eigV j L(�t)i , V = g

1

K

KX

j=1

L(0) R(0)

C ⇠Z

dp(p)2��1eipe�igei g

(1+pet)2�

Alice’smessage

Bob

Alice’sblackhole

Beforetransfer:AlicehasthemessagebutBobdoesnotAftertransfer:BobhasitbutAlicedoesnot!

Bobgettingradiation

P

BackwardextrapolationOfthestateofAlice’sboundaryafterBob’sextraction,usingtheunperturbedHamiltonian.