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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.
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