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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?