black holes and quantum mechanics · einstein rosen 1935 (still call it a ``singularity’’)...
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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,..