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3/14/19
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EntropyoftheVacuum
TedJacobsonUniversityofMaryland
Basedonarxiv:1505.04753(TJ)andarxiv:1812.01596 (TJ&ManusVisser)
Beyond Center Workshop, 17 February 2019
Quantum Gravity: Back to Basics
Blackholeentropy General relativity and quantum field theory ensure that Bekenstein’s generalized entropy locally satisfies the second law, despite the fact that entropy can be tossed into a black hole.
Sgen = AH/4~G+ Sout
Sgen = AH/4~G⇤ + Sout,
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A~BB~
A
`c
N.B.CutoffonproperseparaMonofpairs,whichisLorentzinvariant.
At `c scale, energy uncertainty is �E ⇠ ~/`c.
Causalstructurefluctuates,blurssubsystem,cuSngoffentanglemententropyatthePlanckscale.
WHY IS VACUUM ENTANGLEMENT ENTROPY FINITE?
Gravity is strong at this scale when `c . rg ⇠ G�E ⇠ ~G/`c
i.e. when `c . `P
AdS/CFT appears to provide a realization of these dreams:
The Ryu-Takayanagi formula (& its time-dependent generalization)
relates CFT entanglement entropy to bulk acceleration horizon entropy,
with a nonzero Newton constant, 1/G ~ # fields of CFT < ∞.
The bulk Einstein equation can be derived from RT formula together with
CFT entropy properties.
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§ Is the AdS boundary essential?
§ How locally can notions of black hole thermodynamics be applied?
§ Thermodynamics of dS static patch?
§ A small causal diamond in any spacetime is a small deformation of a maximally symmetric causal diamond, and the Einstein equation is equivalent to the first law for such diamonds. Is this because entanglement entropy is maximized in vacuum?
§ Can this shed light on the cosmological constant problem?
Origin of the first law of black hole mechanics
1972,Bekenstein:ThefirstlawfromvaryingparametersintheKerr-NewmansoluMon.Didn’tknowthat,liketheangularvelocityofthehorizonΩandtheelectostaMcPotenMalΦ,thequanMtyκisanintensivevariable,northatithadtheinterpretaMonofsurfacegravity.1972,Bardeen,Carter&Hawking:DerivedthefirstlawbyvariaMonoftheSmarrFormulawhichtheyobtainedfromanidenMtywiththeKillingvector,andtheEinsteinequaMon.Onestepwastoprove,assumingthedominantenergycondiMon,thatκisconstantonthehorizon.(AproofassumingonlyacertainsymmetrybutnoenergycondiMonexists(Carter,Racz&Wald),andtheproofistrivialifoneassumesthefuturehorizonterminatesatabifurcaMonsurface,wheretheKillingvectorvanishes.)
dM =
8⇡GdA+ ⌦H dJ + � dQ
M = 2⌦HJ +1
4⇡GA+matter terms
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1992,Sudarsky&Wald;1993,Wald;1994,Iyer&Wald:Thefirstlawforanydiffeomorphisminvarianttheory,withtheroleofentropyplayedbythehorizonNoetherchargeassociatedwiththehorizongeneraMng
Killingvector,andvalidforallperturbaMons(notjuststaMonaryones).
DiffeomorphisminvarianceisresponsiblefortyingtogethervariaEonsofsurfaceintegralsatinfinityandatthehorizon.BlackholethermodynamicsisinEmatelyconnectedtodiffinvariance.
ThefirstlawindeSigerspaceMmeGibbons&Hawking,1977
TheyobtainedthisbyfirstderivingaSmarrformula,thenvarying.
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ThefirstlawindeSigerspaceMmeGibbons&Hawking,1977
NNNN
ThefirstlawindeSigerspaceMmeGibbons&Hawking,1977
A
NNNN
NegaMvetemperature!(suggestedbyKlemm&Vanzo,2004);pickedupbynobody…
NegaMvetemperaturerequiresafinitedimensionalHilbertspace……andthereareindependentreasonstothinkthedSHilbertspaceisfinitedimensional:finiteentropy(Banks&Fischler,…)
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FailedagemptstoincreasetheentropyofthedSstaMcpatch
1. Putablackholeinside.Fails:AC+AHdecreases!
2. Putmagerwithentropyinside.Fails:magerhaslessentropythanablackholeforthesamemass.
SuggeststhatentropyofthedSvacuumismaximal.
Thisiscloselyrelatedtothemaximalvacuumentanglementhypothesis,thatthegeneralizedentropyofsmallgeodesicballsismaximalatfixedvolumeinMinkowskispaceMme,wrtvariaMonsofthestateawayfromtheMinkowskivacuum(TJ,2015).
Butisn’ttheGibbons-HawkingtemperatureofdSposiEve??Yes,indeed.ThedSvacuumisathermalstatewithrespecttotheHamiltoniangeneraMngMmetranslaMononthestaMcpatch.ThiswasfoundintheoriginalGibbons-Hawkingpaper,anditisadSanalogoftheDavies-UnruheffectintheRindlerwedgeofMinkowskispaceMme.
TGH = ~c/2⇡, c = H =p
⇤/3
Sodoesn’tthiscontradictthefirstlawofdS?No!ThemagerentropyaddstothedShorizonentropy,formingthestatementthatBekenstein’sgeneralizedentropyisstaEonary:
TGHdSm = �TGHdSBH =) dSgen = 0
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Exceptincertainlimits,theyadmitonlyaconformalKillingvector.ThemetriccanbepresentedasaconformalfactorMmes(hyperbolicspace)x(Mme):
Maximallysymmetriccausaldiamonds(TJ,2015;TJ&ManusVisser,2018)
Remarkably,theslicesofconstantsformaCMCfoliaMon:
⇣ = @sisaconformalKillingvector,withunitsurfacegravity,andan“instantanous”trueKillingvectorats=0.
Firstlawformaximallysymmetriccausaldiamonds(TJ,2015;TJ&ManusVisser,2018)
ASmarrformulaandaFirstLawcanbederivedusingthediff.NoethercurrentàlaWald.FirstLawhasanaddiMonalterm,sinceckvnotakv:
Vistheball’svolume,kistheoutwardextrinsiccurvatureofitsedge.IndS,k=0.
Thevolumetermis;ithasthisgeometricformthankstoaminormiracle:d(divς)hasconstantnorm~κkonΣ(i.e.ats=0).ThatitisproporMonaltothevolumevariaMonispresumablyrelatedtotheYorkMme(-K)Hamiltonianbeingthevolume.
��Hgrav⇣
ThelasttermisthethermodynamicvolumeMmesthepressurevariaMon.ItappearedintheGHSmarrformula,andwasintroducedandinterpretedbyKastor,RayandTraschen(2009)intheAdSblackholeseSng.
�Hmatter⇣ =1
8⇡G(� �A+ k �V � V⇣ �⇤)
V⇣ =
Z
⌃⇣ · ✏
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Comments on the First Law
• ThediamondhasnegaMvetemperatureasforthedSstaMcpatch.
• Thevolumeoftheballcanbedefinedinagaugeinvariantwayasthevolumeofthemaximalslicewithfixedboundary.ThismightbeimportantwhenextendingthisrelaMontoasecondordervariaMon.
• ThevariaMonofareaatfixedvolumehasadeficit,whilethevariaMonofvolumeatfixedareahasanexcess.
• A“small”diamondinanarbitraryspaceMmecanbeviewedasavariaMonofamaximallysymmetricspace,andthisvariaMonmustsaMsfythefirstlawifthespaceMmeisasoluMontoEinstein’seqn.Conversely,allthesefirstlawsmustimplytheEinsteinequaMon.
T = �TH = �~/2⇡
CommentsontheFirstLaw,contd.
• ThemagerHamiltonianvariaMoncanbetradedforanentanglemententropyvariaMon,combiningwiththeareatomakethegeneralizedentropyvariaMon.
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�hKi = 2⇡~
Z�hTabi⇣ad⌃b
CFTEntanglemententropyinaMinkowskiball
⇢vac / e�KConsiderthegroundstateofaQFT,restrictedtothediamond:
�S = ��Tr(⇢ ln ⇢) = Tr(�⇢K) = �hKi
(Hislop&Longo’82,Casini-Huerta-Myers’11)
1/Unruhtemperature Conformalboostenergy
UsedalsobyLashkari,McDermog&vanRaamsdonk’13inholographicderivaMonoflinearizedEinsteinonAdS.
Kisthe“modularHamiltonian”.UnderastatevariaMontheentropyvariaMonis:
ForaCFT,Kislocal,=conformalboostenergy/Unruhtemperature:
=1
TH�hHmatter⇣ i�Smatter =
SemiclassicalFirstlawformaxsymmcausaldiamonds(TJ,2015;TJ&ManusVisser,2018)
�Hmatter⇣ =1
8⇡G(� �A+ k �V � V⇣ �⇤)
0 = T �Sgen +1
8⇡G(k �V � V⇣ �⇤)
ForaCFTcanbeexpressedasstaMonaryentropyatfixedVandΛ:
HoldsalsoforaQFTforsmalldiamond,withaquantummagercontribuMontoδΛ.HoldingVandnetΛfixedsMllcorrespondstostaMonaryentropy.
�hKi = 1TH
�hHmati+ V⇣ �X
Lorentzscalar,contributes“cosmological”term.
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Entangledqbits
whichbytheEinsteinequaMonimplies
hence
Isvacuumentanglementmaximal?
Bychangingthestateofmagerandgeometry,canSbeincreasedwhileholdingthevolumefixed?
Assume S = SUV + SIR, with
�E & ~/`
SUV = SBH = A/4~G
�A|V . �~G
�Stot
. 0
“Highlyentropic”systemsalsocontributetothevacuumentropy,suppressingtheirentropychange.Infact,themaximumentropyforagivenenergyisdE/Tinathermalstate.Marolf&Sorkin‘03,Marolf,Minic&Ross‘03,Marolf‘04
(N.B.massmakesitevenhardertoincreasetheentropy.)
Underwehave
VariaMonofentanglemententropy
(�gab, �| i)
�S = �SUV + �SIR
=�A
4~G + �hKi
=0atconstantV,foraCFT.
HowaboutforfinitevariaMons?PosiEvityofrelaEveentropymeanshereposiEvityoftheconformalboostfreeenergyvariaEon:
�hKi � �SIR � 0
<
(onafixedalgebra)
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UnderfinitevariaMons
VariaMonofentanglemententropy
(�gab, �| i)posiEvityofrelaEveentropyplustheEinsteinequaMonimplies
�Stot
|V 0
Maximal Vacuum Entanglement Entropy
N.B.OnlytheEinsteinHamiltonianconstraintequaMonisinvoked.Thatis,onlyrestricMontothephysicalphasespace.ThisisconsistentwiththenoMonthatthevacuumisanequilibriumstate,maximizingentropyoverALLstatesinthephasespace/Hilbertspace.
Conversely, one can “almost” derive the Einstein equation from maximal
vacuum entanglement.
Not yet clear if/how the argument can be extended to apply to
coherent states of matter (which appear to carry energy
without modifying entanglement entropy of matter).
BEYOND, two questions seem pressing:
Is there a well-defined partition function for a diamond?
(First, does the G-H dS partition function make sense?)
Is there a well-defined regional quantum gravity,
enclosed in a finite boundary?
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