BlackHolesand

QuantumMechanics

JuanMaldacena

InstituteforAdvancedStudy

KarlSchwarzschildMeetingFrankfurtInstituteforAdvancedStudy,2017

22 December 1915,LetterfromSchwarzschildtoEinstein.

Itasbeenveryconfusingeversince…

Classically…

Extremeslowdownoftime

Thecoordinate``singularity”Eddington1924 findsanonsingularcoordinatesystembutdidnot

recognize(orcommentonitssignificance).

Lemaitre1933Firstpublishedstatementthatthehorizonisnotsingular.

EinsteinRosen1935(Stillcallita``singularity’’)

Szekeres,Kruskal 59-60coordinatescoverthefullspacetime

WheelerFulling62Itisawormhole!

G.Lemaˆıtre (1933).Annales delaSoci«et«e Scientifique deBruxelles A53:51-85Figure 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

Futuresingularity

horizonstar

OppenheimerSnyder1939

Observeronthesurfacedoesnotfeelanythingspecialatthehorizon.Finitetimetothesingularity

Ittook45yearstounderstandaclassicalsolution…Why?

Thesymmetriesarerealizedinafunnyway.Thetimetranslationsymmetryà boostsymmetryatthe

bifurcationsurface

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

FuturesingularityInterioristimedependentforanobserverfallingin.Itlookslikeabigcrunch.

Blackholesinastrophysics

• Quasars(mostefficientenergysources)

• Stellarmassblackholes

• Sourcesforgravitywaves!

• OurownsupermassiveblackholeintheMilkyWay…

Goldenage!

Interiorandsingularity

Quantumgravityisnecessaryatthesingularity!

Whatsignals?

interior

star

SingularitySingularityisbehindthehorizon.

ItisshieldedbehindtheblackholehorizonthatactsasaSchwarz-Shild.

Blackholesandquantummechanics

• Blackholesareoneofthemostsurprisingpredictionsofgeneralrelativity.

• Incorporatingquantummechanicsleadstoanewsurprise:

Blackholesarenotblack!

Hotblackholes

• Blackholeshaveatemperature

• Anacceleratedexpandinguniversealsohasatemperature

Veryrelevantforus!

Wecanhavewhite ``blackholes’’T ⇠ ~

rH

T ⇠ ~H =~RH

Hawking

Chernikov,Tagirov,Figari,Hoegh-Krohn,Nappi,Gibbons,Hawking,Bunch,Davies,….

Quantummechanicsiscrucialforunderstandingthelargescalegeometryoftheuniverse.

Whyatemperature?

• Consequenceofspecialrelativity+quantummechanics.

Flatspacefirst

Whyatemperature?t

x

θ

Acceleratedobserverà energy=boostgenerator.Timetranslationà boosttransformationContinuetoEuclideanspaceà boostbecomesrotation.

Lorentzian Euclidean

Whyatemperature?t

x

θ

ContinuetoEuclideanspaceà boostbecomesrotation.

Angleisperiodicà temperature

r

Bisognano Weichman,UnruhOrdinaryaccelerationsareverysmall,g=9.8m/s2à β=1lightyear

� =1

T= 2⇡r =

2⇡

aThermische Unruhe

Entanglement&temperature

Horizon:acceleratedobserveronlyhasaccesstotherightwedge.

Ifweonlymakeobservationsontherightwedgeà donotseethewholesystemà getamixedstate (finitetemperature).

Generalprediction,onlyspecialrelativity+quantummechanics+localityVacuumishighlyentangled!

Blackholecase

r=0 horizon

interior

star

Blackholefromcollapse

singularity

Equivalenceprinciple:regionnearthehorizonissimilartoflatspace.

Ifwestayoutsideà acceleratedobserverà temperature.

Acceleratedobserver!

Blackholeshaveatemperature

Dotheyobeythelawsofthermodynamics?

Blackholeentropy

rH $ M

dM = TdS

T ⇠ ~rH

Specialrelativityneartheblackholehorizon

Einsteinequations

1st Lawofthermodynamics

Blackholeentropy

2nd Lawà areaincreasefromEinsteinequationsandpositivenullenergycondition.

Bekenstein,HawkingS =

(Area)

4~GN

Hawking

Includingthequantumeffects

SBH =(Area)

4GN+ Sentanglement

Entanglemententropyofquantumfieldsacrosstheblackholehorizon

Hasbeenunderstoodbetterinquantumfieldtheory2nd Lawextendedtoincludethisterm.

Bekenstein boundà automaticinrelativisticquantumfieldtheory.

Focusingtheoremsandbetterunderstandingofthepositivityofenergy,andnew``area’’increasestatements.

Wall2011

Casini 2008

Bousso’s talkBousso,Englehardt,Wall,Faulkner,…

Bekenstein – Casini bound

S 2⇡ER Bekenstein

Whenisthistrue?Isittrue?DoesitimposeaconstraintonQFT?

Casini 2008

ItisalwaystrueinanyrelativisticQFT.

�S 2⇡�ERindler

2nd Lawalwayssatisfied.

Generalrelativityandthermodynamics

~ kB

c

GN

Generalrelativityandthermodynamics

• Blackholeseenfromtheoutside=thermalsystem withfiniteentropy.

• Isthereanexactdescriptionwhereinformationispreserved?

• Yes…

Quantum dynamical Space-time

(General relativity)string theory

Theories of quantuminteracting particles

(very strongly interacting)

Gauge/Gravity Duality (or gauge/string duality, AdS/CFT, holography)

JM97Witten,Gubser,Polyakov,Klebanov….

GravityinasymptoticallyAntideSitterSpace

Duality

QuantuminteractingparticlesquantumfieldtheoryGravity,

Strings

Blackholesinagravitybox

HotfluidmadeoutofverystronglyInteractingparticles.Gravity,

StringsBlackhole

Lessonsforblackholes

• Blackholesasseenfromoutside(frominfinity)arelikeanordinaryquantumsystem.

• Blackholeentropy=ordinaryentropyofthequantumsystem.

• Absorptionintotheblackhole=thermalization• Chaosà nearhorizongravity

• Interior?

Shenker StanfordKitaev ,2013-2015

Mathur,Almheiri,Marolf,Polchinski,Sully

Inthemeantime

BlackholesasasourceofinformationBlackholesastoymodels

HotfluidmadeoutofstronglyInteractingparticles.

Gravity,Strings

Blackhole

Usedtomodelstronglyinteractingsystemsinhighenergyphysicsorcondensedmatterphysics.

Keyinsightsintothetheoryofhydrodynamicswithanomalies.

Damour,Herzog,Son,Kovtun,Starinets,Bhattacharyya,Hubeny,Loganayagam,Mandal,Minwalla,Morita,Rangamani,Reall,Bredberg,Keeler,Lysov,Strominger…

Letusgobacktochaos

Chaosà divergenceofnearbytrajectories

Thermalsystemà averageoveralltrajectories

GrowthàWhereyouareaftertheperturbationvs.whereyouwouldhavebeen.

�q(t) / e�t

[Q(t), P (0)]2

Classical

Quantum

�q(t)

�q(0)= {q(t), p(0)} , {q(t), p(0)}2

�q(0)

�q(t)

QuantumGeneral:

W,Varetwo``simple’’(initiallycommuting)observables.ImaginewehavealargeNsystem.ThisisthedefinitionofthequantumLiapunov exponent

h[W (t), V (0)]2i� / 1

Ne�t

V(0)

W(t) Commutatorà involvesthescatteringamplitudebetweenthesetwoexcitations.

Leadingorderà gravitonexchange

Largetà largeboostbetweenthetwoparticles.

Gravitationalinteractionhasspin2,Shapirotimedelayproportionaltoenergy.

Energygoesaset

horizon

Forquantumsystemsthathaveagravitydual

t

h[W (t), V (0)]2i� / 1

Ne

2⇡� t � =

2⇡

�= 2⇡T

Canitbedifferent?

Gravitonà phaseshift: �(s) ⇠ GNs �! � =2⇡

�

Stringà phaseshift

�(s) ⇠ GNs1+↵0t �! � =2⇡

�(1� l2s

R2)

s,t=MandelstaminvariantsRadiusofcurvatureofblackhole

Typicalsizeofstring(ofgravitoninstringtheory)

Itcanbeless…

More?

Inflatspaceaphaseshifthastoscalewithapowerofslessthanoneinordertohaveacausaltheory

Maybethereisabound…

Blackholesasthemostchaoticsystems

� 2⇡

�= 2⇡T

Forany largeN(smallhbar)quantumsystem.

JM,Shenker,Stanford

Sekino Susskind

(Stringsconnectweaklycoupledtostronglycoupledsystems)

Howdowegetorderfromchaos?

Howdowegetthevacuumfromchaos,orfromachaoticquantumsystem?

horizon?

Example:hydrodynamics,wegetsomethingsimpleforsomeinteractions,butitismorecomplicatedwithverysmallinteractions(Boltzman equation).

ThefullSchwarzschildwormhole

horizon

Rightexterior

leftexterior

Noneedtopostulateanyexoticmatter

Nomatteratall!

Viewitasanentangledstate

horizon

Rightexterior

leftexterior

|TFDi =X

n

e��En/2|EniL|EniRIsrael70’sJM00’s

Entangled

Symmetry

horizon

Rightexterior

leftexterior

Whatisthisfunny“timetranslationsymmetry”?

|TFDi =X

n

e��En/2|EniL|EniR

U = eit(HR�HL)

ExactsymmetryExactboost symmetry!

Truecausalseparation

|TFDi =X

n

e��En/2|EniL|EniR

IfBobsendsasignal,thenAlicecannotreceiveit.

Thesewormholesarenottraversable

Notgoodforsciencefiction.

Goodforscience!

RelayonIntegratednullenergycondition(Nowatheorem,provenusingentanglementmethods) Balakrishnan,Faulkner,

Khandker,Wang

Interioriscommon

|TFDi =X

n

e��En/2|EniL|EniR

Iftheyjumpin,theycanmeetintheinterior!

Buttheycannottellanyone.

Spacetime connectivityfromentanglement

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

ER=EPRVanRaamsdonkVerlinde2Papadodimas Raju

JMSusskind

Entanglementandgeometry

Localboundaryquantumbitsarehighlyinteractingandveryentangled

S(R) =Amin

4GN

Ryu,Takayanagi,Hubeny,Rangamani

GeneralizationoftheBlackholeentropyformula.

Interestingconnectionstoquantuminformationtheory

Quantumerrorcorrection

Complexitytheory Harlow,Hayden,Brown,Susskind

Almheiri,Dong,Harlow,Preskill,Yoshida,Pastawski

Conclusions

• Blackholesareextremeobjects:mostcompact,mostefficientenergyconversion,mostentropic,mostchaotic,…

• Mostconfusing...• Theprocessofunravellingtheseconfusionshasleadtobetterunderstandofgravity,quantumsystems,stringtheoryandtheirinterconnections.

• Blackholesarenotonlyinthecosmos,butcanalsobepresentinthelab.

• Andtherearestillveryimportantconfusionsandopenproblems:Interiorandsingularity?

Thankyou!

Extraslides

FullSchwarzschildsolution

SimplestsphericallysymmetricsolutionofpureEinsteingravity(withnomatter)

Eddington,Lemaitre,Einstein,Rosen,FinkelsteinKruskal

ER

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

Wormholeinterpretation.

Wormholeinterpretation.

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

Note:Ifyoufindtwoblackholesinnature,producedbygravitationalcollapse,theywillnotbe describedbythisgeometry

Nofasterthanlighttravel

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

Non travesable

No signals

No causality violation

Fuller, Wheeler, Friedman, Schleich, Witt, Galloway, WooglarFigure 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

Intheexacttheory,eachblackholeisdescribedbyasetofmicrostatesfromtheoutside

Wormholeisanentangledstate

| i =X

n

e��En/2|EniL ⇥ |EniR EPR

Geometric connectionfrom entanglement. ER = EPR

IsraelJM

SusskindJMStanford,Shenker,Roberts,Susskind

EPR

ER

Aforbiddenmeeting

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

Romeo Juliet

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

Stringtheory

• Stringtheorystartedoutdefinedasaperturbative expansion.

• Stringtheorycontainsinterestingsolitons:D-branes.

• UsingD-branes onecan``count’’thenumberofstatesofextremal chargedblackholesincertainsuperstringtheories.

• D-branes inspiredsomenon-perturbativedefinitionsofthetheoryinsomecases.

Matrixtheory:Banks,Fischler,Shenker,SusskindGauge/gravityduality:JM,Gubser,Klebanov,Polyakov,Witten

Polchinski

Strominger Vafa

Entanglementandgeometry• Theentanglementpatternpresentinthestateoftheboundarytheorycantranslateintogeometricalfeaturesoftheinterior.

• Spacetime iscloselyconnectedtotheentanglementpropertiesofthefundamentaldegreesoffreedom.

• Slogan:Entanglementisthegluethatholdsspacetimetogether…

• Spacetime isthehydrodynamicsofentanglement.

VanRaamsdonk

Questions

• Blackholeslooklikeordinarythermalsystemsifwelookatthemfromtheoutside.Weevenhavesomeconjecturedexactdescriptions.

• Howdowedescribetheinteriorwithinthesameframeworkthatwedescribetheexterior?

• Oncewefigureitout:whatisthesingularity?• Whatlessonsdowelearnforcosmology?

Modernversionoftheinformationparadox;Mathur,Almheiri,Marolf,Polchinski,Stanford,Sully,..