arXiv:1509.07142v2[hep-th]4Oct2015
MI-TH-1533
BlackHoleEntropyandViscosityBound
inHorndeskiGravity
Xing-HuiFeng1,Hai-ShanLiu2,3,H.L¨u1andC.N.Pope3,4
1CenterforAdvancedQuantumStudies,DepartmentofPhysics,
BeijingNormalUniversity,Beijing100875,China
2InstituteforAdvancedPhysics&Mathematics,
ZhejiangUniversityofTechnology,Hangzhou310023,China
3GeorgeP.&CynthiaWoodsMitchellInstituteforFundamentalPhysicsandAstronomy,
TexasA&MUniversity,CollegeStation,TX77843,USA
4DAMTP,CentreforMathematicalSciences,CambridgeUniversity,
WilberforceRoad,CambridgeCB3OWA,UK
ABSTRACT
Horndeskigravitiesaretheoriesofgravitycoupledtoascalarfield,inwhichtheac-
tioncontainsanadditionalnon-minimalquadraticcouplingofthescalar,throughitsfirst
derivative,totheEinsteintensorortheanalogoushigher-derivativetensorscomingfrom
thevariationofGauss-BonnetorLovelockterms.Inthispaperwestudythethermody-
namicsofthestaticblackholesolutionsinndimensions,inthesimplestcaseofaHorndeski
couplingtotheEinsteintensor.WeapplytheWaldformalismtocalculatetheentropyof
theblackholes,andshowthatthereisanadditionalcontributionoverandabovethosethat
comefromthestandardWaldentropyformula.Theextracontributioncanbeattributed
tounusualfeaturesinthebehaviourofthescalarfield.Wealsoshowthataconventional
regularisationtocalculatetheEuclideanactionleadstoanexpressionfortheentropythat
disagreeswiththeWaldresults.Thisseemslikelytobeduetoambiguitiesinthesub-
tractionprocedure.WealsocalculatetheviscosityinthedualCFT,andshowthatthe
viscosity/entropyratiocanviolatetheη/S≥1/(4π)boundforappropriatechoicesofthe
parameters.
xhfengp@mail.bnu.edu.cnhsliu.zju@gmail.commrhonglu@gmail.compope@physics.tamu.edu
Contents
1Introduction2
2BlackHolesinHorndeskiGravity5
2.1Thetheory....................................5
2.2Staticblackholesolutions............................6
2.3UniquenessoftheHorndeskiblackholesolutions...............9
3BlackHoleEntropyandThermodynamics11
3.1Waldentropyformula..............................11
3.2Waldformalism..................................12
3.3FurthercommentsontheentropyfromWaldformalism...........16
3.4NoetherchargeandmassofAdSplanarblackholes..............18
3.5Euclideanaction.................................19
4Viscosity/EntropyRatio22
5Conclusion24
1Introduction
Inthedictionaryofgravity/gaugedualitymappingsintheAdS/CFTcorrespondence[1–3],
perturbationsofthemetricarerelatedtotheenergy-momentumtensorofthefieldtheory
intheboundaryoftheAdSspacetime[2–4].Inthispicture,anAdSplanarblackholeis
thegravitationaldualofacertainidealfluid.Awidelyvalidrelationbetweentheshear
viscosityandtheentropydensitywasestablished,namely[5–8]
η
S=
1
4π.(1.1)
Onewaytounderstandthisratioisthatitcanbeshownthattheviscosityisproportionalto
thecross-sectionoftheblackholeforlow-frequencymasslessscalarfields[8].Alternatively,
theshearviscosityisdeterminedbytheeffectivecouplingconstantofthetransversegraviton
onthehorizon,byemployingthemembraneparadigm[9].(Thiswasconfirmedbyusingthe
Kuboformulain[10,11].)In[12],itwasshownthattheblackholeentropyisdetermined
bytheeffectiveNewtoniancouplingatthehorizon,andthatitisthusnotsurprisingthat
theratiooftheshearviscositytotheentropydensityisuniversal,inthesensethatthe
dependenceofthequantitiesonthehorizoniscanceled.Recently,itwasestablishedthat
2
therelation(1.1)oftheboundarytheoryisdualtoageneralisedSmarrrelationobeyedby
thebulkAdSplanarblackholes,therebyprovidinganewunderstandingofitsuniversality,
anditsconnectiontotheblackholethermodynamics[13].Therehavebeenanumberof
papersinliteratureestablishingtheuniversalityoftheratio(1.1)[14–17].(See[18]fora
review.)
Theviscosity/entropyratio(1.1)can,however,beviolatedwhenthebulkgravitytheory
isextendedbytheadditionofhigher-ordercurvatureterms[19,20].1(Seealso,forfurther
examples,[25–27].)
Thisleadsustooneofthemotivationsforthispaper,whichistoinvestigatewhether
onecanviolatetheratio(1.1)withoutintroducinghigher-ordercurvaturetermsinthebulk
theory.InatypicaltheoryofEinsteingravity,matterfieldscoupletogravityminimally
throughthemetric.Ascalarfieldcanalsocoupletogravitynon-minimally,suchasin
Brans-Dicketheory[28],wheretheeffectiveNewtonconstantvariesinspacetime.However,
itwasestablishedin[13]thattheratio(1.1)holdsingeneralinsuchatheory.Scalar
fieldscan,however,alsocouplenon-minimallytogravityinotherways.Inparticular,
theirderivativescancoupletothecurvaturetensor.Horndeskiconsideredawideclass
ofsuchgravity/scalartheoriesintheearlyseventies[29],focusinghisattentiononcases
wherethefieldequations,bothforgravityandthescalarfield,involvenohigherthan
secondderivatives.TheHorndeskitheorieswererediscoveredrecentlyinstudiesofthe
covariantisationofGalileontheories[30].
TheHorndeskitermstaketheform
H(k)=E(k)μν?μχ?νχ,(1.2)
wheretheE(k)tensorsare“energy-momentumtensors”associatedwiththeEulerintegrands
ofvariousorder,namely
E(k)νμ≡δνρ1···ρ2kμσ1···σ2kRσ1σ2ρ1ρ2···Rσ2k?1σ2kρ2k?1ρ2k.(1.3)
TheH(k)termsareanalogoustoEulerintegrands,inthattheyhavethepropertythat
eachfieldcarriesnomorethanasinglederivativeandhencethelinearizedequationsof
motioninvolveatmostsecondderivatives.Thusalthoughthetheoryinvolveshigher-order
derivatives,itcontainsnolinearghostexcitations.Inthispaper,weshallconsiderEinstein
gravitywithacosmologicalconstant,togetherwithjustthetwolowest-orderHorndeski
1Weshallnotbeconcernedinthispaperwithothertypesofviolation,duetothebreakingoflocal
rotationalsymmetry;see,forexample,[21–24].
3
terms,namely
H(0)=gμν?μχ?μχ,H(1)=?4Gμν?μχ?νχ,(1.4)
whereGμνistheEinsteintensor.Wefindthatalthoughthetheorycontainsthecurvature
tensoronlylinearly,theviscosity/entropyratio(1.1)nolongerholds.
Itisworthcommentingthattheviscositycanbecomputedbystandardprocedures
usingtheAdS/CFTcorrespondence,involvingthestraightforwardtechniqueofstudying
linearisedperturbationsaroundthebackgroundbulksolution.Thecalculationofthevis-
cosity/entropyratiothenhingesupontheproperdefinitionoftheentropyoftheblack
hole.SinceHawkingestablishedthethermalradiationofablackhole[31,32],therehas
beennoambiguityinestablishingtheblackholeentropyinagenerally-covarianttheory.
Inparticular,inEinsteingravityminimallycoupledtomatter,theentropyisgivenbyone
quarteroftheareaofthehorizon.ThisarealawhasbeengeneralizedtotheWaldentropy
formulawhenmorecomplicatedcouplingsorhigher-ordercurvaturetermsareinvolved,
namely[33,34]
SW=?18
integraldisplay
+
dn?2x
√
h?L?Rabcd?ab?cd.(1.5)
whereLisdefinedbytheactionI=integraltextdnx√?gL.Applyingthisformulatostaticblackholes
withspherical,toricorhyperbolicisometries,theHorndeskiterms(1.4)donotcontribute
totheWaldentropySW,andhenceonemightexpectthattheentropywouldstillbejust
onequarterofthehorizonarea.However,wefindthatthisisinfactnotthecase.By
examiningtheWaldprocedure[33,34]indetail,wefindthatinatheorysuchasHorndeski
gravitythereisanadditionalcontributiontotheentropythatisnotencompassedbythe
usualWaldformula(1.5).Itarisesbecausethederivativeofthescalarfielddivergeson
thehorizonintheblack-holesolutions(althoughthereisnophysicaldivergence,sinceall
invariants,suchasgμν?μχ?νχ,remainfinite).
Thepaperisorganisedasfollows.Insection2weintroducetheHorndeskitheorythat
weshallbeconsidering,andwereviewthestaticblackholesolutions.Theseareknown
forallthecasesofspherical,toroidalandhyperbolichorizongeometries.Ourfocuswill
beonthesphericalandthetoroidalhorizons.Wealsoincludeademonstrationofthe
uniquenessoftheknownstaticsolutions.Insection3weaddresstheproblemofcalculating
theentropy,andalsothemass,ofthestaticblackholes.Webeginbycalculatingthe
entropyusingthestandardWaldformula(1.5),andthenweconsidertheapplicationofthe
Waldformalisminmoredetail,showingthatthereisanothercontributiontotheentropy
thatisnotcapturedby(1.5).Weshowthatinthecaseoftheplanarblackholes(with
toroidalhorizons),theentropyexpressionweobtainisconsistentwiththecomputation
4
oftheNoetherchargeassociatedwithascalingsymmetryoftheblackholes.Wealso
considerthecalculationoftheEuclideanaction,showingthat,atleastwhenfollowinga
naiveregularisationprocedure,thisyieldsyetanotherresultfortheentropy,andthemass,
thatdisagreeswiththosefromtheWaldformalism.Insection4wecalculatetheshear
viscosityinthedualboundarytheoryusingtheAdS/CFTcorrespondence,andhencewe
obtainanexpressionfortheviscosity/entropyratio.Thisisdifferentfrom1/(4π)onaccount
oftheHorndeskiterm,andweshowthatforanappropriatechoiceoftheparametersitcan
violatetheη/S≥1/(4π)bound.Thepaperendswithconclusionsinsection5.
2BlackHolesinHorndeskiGravity
2.1Thetheory
Aswehavediscussedintheintroduction,Horndeskigravityrepresentsaclassofhigher-
derivativetheoriesinvolvinggravitywithanon-minimallycoupledscalar.Thecouplings
differfromthoseintheBrans-Dicketheory,sinceintheHorndeskitheoriesthescalar
couplesthroughitsderivativetothecurvaturetensors.WeshallfocusontheHorndeski
theorywhoseLagrangianinvolvesatmostonlylinearcurvatureterms.Asweshallshow,
theviscosity/entropyratio(1.1)canbeviolatedeveninsuchatheory.Theactionisgiven
by
I=116π
integraldisplay
dnx√?gL,L=κ(R?2Λ)?12(αgμν?γGμν)?μχ?νχ,(2.1)
whereκ,αandγarecouplingconstants,andGμν≡Rμν?12RgμνistheEinsteintensor.
Notethatthetheoryisinvariantunderaconstantshiftofχ.Inatypicalgravitytheory
withascalarfield,suchasBrans-Dicketheory,onecandefinedifferentmetricframesby
meansofconformalscalingsusingthescalarfield.However,fortheHorndeskitheory(2.1),
thiswouldleadtothebreakingofthemanifestconstantshiftsymmetryofthescalar,and
henceitwouldnotbeanaturalfieldredefinitiontomakehere.
Thevariationoftheaction(2.1)givesriseto
δI=116π
integraldisplay
dnx√?g(Eμνδgμν+Eδχ+?μJμ).(2.2)
where
Eμν=κ(Gμν+Λgμν)?12α
parenleftBig
?μχ?νχ?12gμν(?χ)2
parenrightBig
?12γ
parenleftBig
1
2?μχ?νχR?2?ρχ?(μχRν)
ρ
??ρχ?σχRμρνσ?(?μ?ρχ)(?ν?ρχ)+(?μ?νχ)squareχ+12Gμν(?χ)2
?gμνbracketleftbig?12(?ρ?σχ)(?ρ?σχ)+12(squareχ)2??ρχ?σχRρσbracketrightbig
parenrightBig
,
5
E=?μparenleftbig(αgμν?γGμν)?νχparenrightbig.(2.3)
Thetotalderivativetermin(2.2)playsnoroleintheequationsofmotion
Eμν=0,E=0.(2.4)
However,itdoesplayanimportantroleintheWaldformalism,whichweshallpresentin
section3.2.
2.2Staticblackholesolutions
Wenowconsiderstaticblackholes,withtheansatz
ds2n=?h(r)dt2+dr
2
f(r)+r
2d?2
n?2,?,χ=χ(r),(2.5)
whered?2n?2,?with?=1,0,?1isthemetricfortheunitSn?2,then-torusortheunit
hyperbolicn-space.Itisconvenienttotaked?2n?2,?=ˉgijdyidyjforgeneralvaluesof?to
bethemetricofconstantcurvaturesuchthatitsRiccitensorisgivenbyˉRij=(n?3)?ˉgij.
Wemay,forexample,taked?2n?2,?tobegivenby
d?2n?2,?=du
2
1??u2+u
2d?2
n?3,(2.6)
whered?2n?3isthemetricoftheunit(n?3)-sphere.
Itisclearfromtheequationsofmotionthatχ=χ0(constant)isasolution,inwhich
case,theHorndeskigravityreducestoEinsteingravitywithacosmologicalconstantΛ0.It
followsthattheSchwarzschild-AdSblackholeisasolutionofthetheory.Weshallregard
thissolutionasbeing“trivial,”inthesenseofnotyieldinganythingnew.Inaddition,a
one-parameterfamilyofblackholesolutionsforwhichthescalarfieldisnotaconstantwas
constructedin[35].(Seealso,[36,37].)Inthissection,wewouldliketoprovethatthese
aretheonlyblackholesolutionsfromtheansatz(2.5)inwhichthescalarisr-dependent.
First,wereviewtheconstructionin[35].
ThescalarequationofmotionE=0yields
parenleftBig
rn?4
radicalbigg
f
h
parenleftBig
γparenleftbig(n?2)rfh′+(n?2)(n?3)(f??)hparenrightbig?2αr2h
parenrightBig
χ′
parenrightBig′
=0.(2.7)
TherearetwomoreequationsthatfollowfromEμν=0:
4κ
parenleftBig
(n?2)rf′+(n?2)(n?3)(f??)+2Λ0r2
parenrightBig
+2αr2fχ′2
+γ(n?2)
parenleftBig
4rfχ′′+parenleftbig3rf′+(n?3)(f+?)parenrightbigχ′
parenrightBig
fχ′=0,
6
4κ
parenleftBig
(n?2)rfh′+(n?2)(n?3)h(f??)+2Λ0r2h
parenrightBig
?2αr2fhχ′2
+γ(n?2)
parenleftBig
3rfh′+(n?3)(3f??)h
parenrightBig
fχ′2=0.(2.8)
In[35],aclassofblackholesolutionwasobtainedbysolving(2.7)bytaking
γparenleftbig(n?2)rfh′+(n?2)(n?3)(f??)hparenrightbig?2αr2h=0.(2.9)
(Inotherwords,theintegrationconstantinthefirstintegralof(2.7)wastakentobezero,
andχ′wasallowedtobenon-zero,thusimplyingthatitsco-factor,givenin(2.9),mustbe
equaltozero.)Thisleadstothesolution
h=?μrn?3+8κ[g
2r2(2κ+βγ)+2?κ]
(4κ+βγ)2
+(n?1)
2β2γ2g4r4
?(n+1)(n?3)(4κ+βγ)22F1
bracketleftBig
1,12(n+1);12(n+3);?n?1(n?3)?g2r2
bracketrightBig
,
f=(4κ+βγ)
2bracketleftbig(n?1)g2r2+(n?3)?bracketrightbig2
bracketleftbig(n?1)(4κ+βγ)g2r2+4(n?3)?κbracketrightbig2h,χ′2=βf
bracketleftBig
1+(n?3)?(n?1)g2r2
bracketrightBig?1
,(2.10)
whichisvalidforallvaluesof?.Inpresentingthesolution,wehaveintroducedtwoparam-
eters(g,β)inplaceoftheoriginalparameters(α,Λ0)intheLagrangian,with
α=12(n?1)(n?2)g2γ,Λ0=?12(n?1)(n?2)g2
parenleftBig
1+βγ2κ
parenrightBig
.(2.11)
Notethatthesolutioncontainsonlyoneintegrationconstant,μ.Allotherparameters
arethoseofthetheoryitself.Notealsothatsincethedimensionnisaninteger,the
hypergeometricfunctionreducestopolynomialswithanarctanfunctioninevendimensions,
andwithalogfunctioninodddimensions.Tobeexplicit,wehave
n=even:(2.12)
2F1[1,12(n+1);12(n+3);?x]=
(?1)n/2(n+1)
xn/2
braceleftBigarctan√x
√x?
bracketleftBigarctan√x
√x
bracketrightBig
n
2?1
bracerightBig
,
n=odd:(2.13)
2F1[1,12(n+1);12(n+3);?x]=
(?1)n?12(n+1)
2xn?12
braceleftBiglog(1+x)
x?
bracketleftBiglog(1+x)
x
bracketrightBig
n?3
2
bracerightBig
,
whereweusethenotation[F(x)]mtodenotethetruncatedpowerseriesexpansionofF(x)
aroundx=0,inwhichonlythetermsuptoandincludingxmareretained.Thus
bracketleftBigarctan√x
√x
bracketrightBig
n
2?1
=
n/2?1summationdisplay
p=0
(?x)p
2p+1,
bracketleftBiglog(1+x)
x
bracketrightBig
n?3
2
=
n?3
2summationdisplay
p=0
(?x)p
p+1,(2.14)
fornevenandnodd,respectively.
7
Forstaticsolutionsofthiskind,itisinfactalwayssufficienttoconstructthesolution
with?=1.Thesolutionsforallothervaluesof?,whichwepresentedabove,canthenbe
obtainedfromthe?=1solutionbymeansoftherescalings
r?→r√?,t?→√?t,d?2n?2?→?d?2n?2,?,μ?→??(n?1)/2μ(2.15)
Fromnowon,weshallpresentresultsforthetwospecificcases?=0and?=1.
?=0solution:
When?=0,thesolutionreducestotheverysimpleform
h=f=g2r2?μrn?3,χ′2=βf.(2.16)
Notethatinthis?=0case,χcanbesolvedforexplicitly,giving
χ=2
√β
(n?1)glog
parenleftbigradicalbig(gr)n?1+radicalbig(gr)n?1?μgn?3parenrightbig+χ
0.(2.17)
Thusthe?=0solutiondescribesanAdSplanarblackhole,withtherequirementsthat
μ>0andβ≥0.Thehorizonradiusr=r0isgivenbyμ=g2rn?10.TheHawking
temperatureisgivenby
T=(n?1)g
2
4πr0.(2.18)
?=1solution:
For?=1,thesolutiondescribesaspherically-symmetricandstaticblackhole.Ina
large-rexpansion,ifniseventhefunctionshandfhavetheasymptoticforms
h=g2r2?μrn?3+
summationdisplay
k=0
ck
r2k=g
2r2+4κ?βγ
4κ+βγ?+···,
f=g2r2?μrn?3+
summationdisplay
k=0
dk
r2k=g
2r2+4(n?1)κ+(n?5)βγ
(n?1)(4κ+βγ)?+···,(2.19)
where(ck,dk)areconstants,whicharefunctionsoftheparameters(κ,g,β)butindependent
ofμ.Ifnisodd,thenfork=(n?3)/2,thequantityckhasanadditionaltermproportional
tologr.Thisamountstoalogarithmicallydivergingadditiontothemasscoefficientμat
order1/rn?3.Thisinturnimpliesthatdkhasadditionallogrtermsforallk≥(n?3)/2.
Notethatallthe(ck,dk)vanishfor?=0.
ThemetricisasymptoticlocallytoAdSspacetime,anditcannotbecomepureAdS
spacetime,regardlessofthechoiceoftheparameterμ.Toseethatthesolutiondescribesa
blackhole,wenotethathispositiveasrgoestoinfinity,butbecomesoforder?μ/rn?3
8
asr→0,wherethereisaspacetimecurvaturesingularity.Thuswhenμ>0,theremust
existsomeintermediatevalueofr,beaneventhorizonr=r0,forwhich
h(r0)=0=f(r0).(2.20)
Thisimpliesthattheparameterμcanbeexpressedintermsofthehorizonradiusr0inthis
?=1caseas
μ=8κr
n?3
0
(4κ+βγ)2
parenleftBig
2κ+(2κ+βγ)g2r20
+(n?1)
2β2γ2g4r40
8κ(n?1)(n?3)2F1[1,
1
2(n+1);
1
2(n+3);?
n?1
(n?3)g
2r2
0]
parenrightBig
.(2.21)
Notethatthisrelationbetweenμandr0isfarmorecomplicatedthanthesimpleexpression
μ=g2rn?10thatholdsinthe?=0case.Thetemperatureofthe?=1blackholeisgiven
by
T=
radicalbigh′(r
0)f′(r0)
4π=
(n?1)g2
4πr0+
(n?3)κ
π(4κ+βγ)r0.(2.22)
Notethatifwesetμ=0,thenthesolutionhasnoeventhorizon,andnearr=0the
functionsh,fandχhavetheforms
h=16κ
2
(4κ+βγ)2
parenleftBig
1+(2κ+βγ)g
2r2
2κ+···
parenrightBig
,
f=1+((n?3)κ?βγ)g
2r2
(n?3)κ+···,
χ=χ0+(n?1)β2(n?3)gr2+···.(2.23)
Thustheμ=0solutionisasmoothspherically-symmetricsoliton,withoutanyfreeparam-
eters,thatisasymptoticlocallytoAdSspacetime.Therealsoexistsasolutionfor?=1in
thelimitof4κ+βγ=0,butitdoesnotdescribeablackhole.
2.3UniquenessoftheHorndeskiblackholesolutions
Weshallleavethediscussionofthemassandentropyoftheblackholestothenextsection.
Toclosethissection,weshallshowthatthesolutionsdiscussedaboveareinfacttheonly
blackholeswithnon-constantχthatarecontainedwithintheansatz(2.5)inthetheory.
Toshowthis,wereturntotheequationofmotion(2.7)forthescalarfield.Onecan
immediatelywritedownthefirstintegral
χ′=qr
4?nradicalbigh/f
γparenleftbig(n?2)rfh′+(n?2)(n?3)(f??)hparenrightbig?2αr2h,(2.24)
whereqisanintegrationconstant.Thesolutionswediscussedabovewereobtainedby
takingq=0.Itwaspossibletofindsuchsolutionswithχ′negationslash=0byimposingtherelation
9
(2.9),whichinfactrenderedthescalarequationofmotion(2.7)trivial.Ifinsteadwetake
theintegrationconstantqtobenon-zero,thenχ′isnowdeterminedby(2.24).
Ifasolutionwithqnegationslash=0istodescribedescribeablackhole,theremustbeanevent
horizonatsomeradiusr=r0.ThefunctionshandfnearthehorizonwillhaveTaylor
expansionsoftheform
f=f1(r?r0)+f2(r?r0)2+···,h=h1(r?r0)+h2(r?r0)2+···.(2.25)
Itfollowsfrom(2.24)thatχ′nearthehorizonhastheexpansion
χ′=?χ?1r?r
0
+?χ0+?χ1(r?r0)+···.(2.26)
Substitutingtheseexpansionsintotheotherequationsofmotion,wefindthatnosuch
solutionscanexist.Inotherwords,theassumptionthatthereexistsahorizon,nearwhich
theexpansions(2.25)wouldhold,isinconsistentwiththeequationsofmotionwhenqnegationslash=0..
Inordertohaveasolutionwithahorizon,wemustthereforesetq=0,whichthenreduces
tothepreviouscasediscussedabove.However,asmentionedalready,inorderforthis
solutionnottobetrivial,i.e.forχ′tobenon-vanishing,wemustthenalsoimposethe
condition(2.24).Thisleadsthetotheblackholesolution(2.10).
Inthenear-horizonregion,thefunctionχintheblack-holesolutions(2.10)hasan
expansionoftheform
χ=?χ0+?χ1(r?r0)12+?χ2(r?r0)32+···.(2.27)
Substitutingbackintotheequationsofmotion,wefindthatallthecoefficientsinthe
expansionscanbeexpressedintermsoftwoparameters,h1andr0.Forexample,
f1=(n?2)(n?3)γ?+2αr
20
(n?2)γr0,χ1=
2radicalbig(n?1)βgr
3
20
(n?1)g2r20+(n?3)?,···.(2.28)
Thusthesolutionhasthreeintegrationconstants(?χ0,h1,r0).However,theparameters
(?χ0,h1)aretrivial.Itfollowsthattheonlynon-trivialparameterisr0,whichisdetermined
byμinthefinalsolution.
Finallywewouldliketoemphasizeagainthatβisnotanintegrationconstant,buta
parameterofthetheory.Forβnegationslash=0,therearetwoblackholes,buteachassociatedwitha
differentvacuum.Whenβ=0,thereisonlytheSchwarzschild-AdSblackholesolutionin
thetheory.
10
3BlackHoleEntropyandThermodynamics
Intheprevioussection,wereviewedtheHorndeskigravitytheory,anditsstaticblackhole
solutions.Weidentifiedthehorizonandcomputedthetemperatureoftheseblackholes.
Inthissection,weconsidervariouspossiblemethodsforcalculatingtheirentropy.Itturns
outthatdifferentwell-establishedmethodsyielddifferentanswers.Acorrectanswerofthe
entropyisimportantforstudyingtheblackholethermodynamics,anditisparamountfor
determiningtheη/Sratio,aswediscussedintheintroduction.
3.1Waldentropyformula
Firstletusconsiderthewell-knownWaldentropyformula(1.5).Itisstraightforwardto
seethatfortheHorndeskiLagrangianLgivenin(2.1),onehas
Tμνρσ≡?L?R
μνρσ
=12κ(gμρgνσ?gνρgμσ)(3.1)
+18γ[gμρχνχσ?gνρχμχσ+gνσχμχρ?gμσχνχρ?(gμρgνσ?gνρgμσ)χλχλ],
wherewehavedefinedχμ=?μχ.ForthestaticblackholesintheHorndeskitheory,
describedinsection2,wefindfrom(3.1)thattheWaldentropyformula(1.5)forthe
entropygivesthesameresultasinstandardEinsteingravity,namelyonequarterofthe
areaoftheeventhorizon,
SW=14κrn?20ωn?2,(3.2)
whereωn?2isthevolumeofaunitSn?2inthe?=1case.For?=0,correspondingto
atoroidalhorizon,theperiodsofthecirclesformingthetoruscanbechosenarbitrarily,
andweshall,forconvenience,thentakeωn?2=1inthispaper,andsocorrespondinglyS
shouldthenbeviewedastheentropydensity.
Sincethestaticblackholesolutionsarecharacterisedbyonlyoneparameter(i.e.one
integrationconstant),itisguaranteedthatonecanobtainanexpressionfora“thermody-
namicmass”byintegratingthefirstlawofblackholethermodynamics2
dM=TdS.(3.3)
2Inamoregeneralsituationwheretherearefurtherintensive/extensivepairsofthermodynamicvariables
contributingontheright-handsideofthefirstlawformulti-parametersolutions,theintegrabilityofthe
right-handsidecanprovideanon-trivialcheckonthecorrectnessofthethermodynamicquantities.Nosuch
consistencycheckarisesinthecaseofaone-parameterfamilyofsolutions,sinceall1-formsareexactinone
dimension.
11
Ifweusetheexpression(3.2)fortheentropy,thenfromtheresultfortheHawkingtem-
peratureobtainedintheprevioussectionwethereforefind
?=0:M=κ(n?2)16πμ,(3.4)
?=1:M=
parenleftBigκ(n?2)
16πg
2rn?1
0+
κ2(n?2)
4π(4κ+βγ)r
n?3
0
parenrightBig
ωn?2.(3.5)
Notethatinthe?=0caseitwasstraightforwardtoexpressthemassintermsofthe“mass
parameter”μ,becauseofthesimplerelationμ=g2rn?10fortheseplanarblackholes.On
theotherhand,therelationbetweenμandr0ismuchmorecomplicatedinthe?=1
case,andisgivenin(2.21).Thuswhen?=1theexpression(3.5)forMwouldbecomea
complicatedtranscendentalfunctionofthemassparameterμ.
Onthefaceofit,themassformula(3.4)forthe?=0caselooksnotunreasonable.In
factthethermodynamicalquantitiessatisfyalsotheexpectedgeneralisedSmarrrelation
M=n?2n?1TSW.(3.6)
However,forthe?=1case,themassformula(3.5)lookslessreasonable.Asmentioned
above,itwouldbeacomplicatedtranscendentalfunctionofthe“massparameter”μ.Whilst
thisfact,ofitself,doesnotconclusivelyshowthatitmustbeincorrect,itdoesperhapsraise
doubtsaboutitslikelyvalidity,sinceitwouldbeaveryunusualkindofrelationthatisnot
normallyseeninotherblackholesolutions.Furthermore,ifthe?=1massformulaiscalled
intoquestionthenthisalsoraisesquestionsaboutthevalidityofthe?=0massformula.
Inordertoexploretheseissuesingreaterdepth,weshallmakeamoredetailedinves-
tigationoftheWaldprocedure,inordertoseewhethertherearenewsubtletiesthatcan
ariseinatheorysuchasthatofHorndeski.
3.2Waldformalism
Waldhasdevelopedaprocedureforderivingthefirstlawofthermodynamicsbycalculating
thevariationofaHamiltonianderivedfromaconservedNoethercurrent.Thegeneral
procedurewaspresentedin[33,34].TheWaldentropyformula(1.5)isaconsequenceof
applyingthisprocedureinrathergenerichigher-derivativetheories.TheWaldformalism
hasbeenusedtostudythefirstlawofthermodynamicsforasymptotically-AdSblackholes
invarietyoftheories,includingEinstein-scalar[39,40],Einstein-Proca[41],Einstein-Yang-
Mills[42],ingravitiesextendedwithquadratic-curvatureinvariants[43],andalsoforLifshitz
blackholes[44].However,theratherunusual-lookingresultsthatitledtoforthemassof
the?=1blackholesinsection3.1raisedthepossibilitythattheformula(1.5)mightnot
12
bevalidforHorndeskigravity.Forthisreason,weshallnowstudyindetailtheapplication
oftheWaldformalismfortheaction(2.1).
Ageneralvariationofthefieldsintheaction(2.1)wasgivenin(2.2).Thesurfaceterm
Jμisgivenby
Jμ=2?L?R
ρσμν
?σδgρν?2?ν?L?R
ρμνσ
δgρσ+?L?(?
μχ)
δχ
=
parenleftBig
κJμg+αJμχ+γ(Jμgc+Jμχc)
parenrightBig
,(3.7)
with
Jμg=gμρgνσ(?σδgνρ??ρδgνσ),Jμχ=?gμν?νχδχ,Jμχc=Gμν?νχδχ,
Jμgc=?14(?χ)2Jμg+14gμρgνσ[?σ(?χ)2δgνρ??ρ(?χ)2δgνσ]
+12gμλ?ρχ?σχ?ρδgσλ?12?ρ(?μχ?σχ)gρλδgσλ
?14gμλ?ρχ?σχ?λδgρσ+14?λ(?ρχ?σχ)gλμδgρσ
?14gρλ?μχ?σχ?σδgρλ+14?σ(?σχ?μχ)gρλδgρλ.(3.8)
FollowingtheWaldprocedure,wecannowdefinea1-formJ(1)=JμdxμanditsHodgedual
Θ(n?1)=(?1)n+1?J(1).(3.9)
Wenowspecialisetoavariationthatisinducedbyaninfinitesimaldiffeomorphism
δxμ=ξμ.Onecanshowthat
J(n?1)≡Θ(n?1)?iξ?L0=?d?J(2),(3.10)
aftermakinguseoftheequationsofmotion.Hereiξdenotesacontractionofξμonthe
firstindexofthen-form?L0.Onecanthusdefinean(n?2)-formQ(n?2)≡?J(2),such
thatJ(n?1)=dQ(n?2).Notethatweusethesubscriptnotation“(p)”todenoteap-form.
Tomakecontactwiththefirstlawofblackholethermodynamics,wetakeξμtobethe
time-likeKillingvectorthatisnullonthehorizon.Waldshowsthatthevariationofthe
Hamiltonianwithrespecttotheintegrationconstantsofaspecificsolutionisgivenby
δH=116πδ
integraldisplay
c
J(n?1)?116π
integraldisplay
c
d(iξΘ(n?1))=116π
integraldisplay
Σ(n?2)
parenleftBig
δQ(n?2)?iξΘ(n?1)
parenrightBig
,(3.11)
wherecdenotesaCauchysurfaceandΣ(n?2)isitsboundary,whichhastwocomponents,
oneatinfinityandoneonthehorizon.ThusaccordingtotheWaldformalism,thefirstlaw
ofblackholethermodynamicsisaconsequenceof
δH∞=δH+.(3.12)
13
FortheHorndeskigravityconsideredinthispaper,wefind
Jα1···αn?1=E.O.M+2?α1···αn?1μ?ν
braceleftBig
κ?[νξμ]?14γ(?χ)2?[νξμ]+12γ?[ν(?χ)2ξμ]
+12γ?σχ?[νχ?σξμ]?12γ?σ(?σχ?[νχ)ξμ]?12γ?[ν(?μ]χ?σχ)ξσ
bracerightBig
,
Qα1···αn?2=?α1···αn?2μν
braceleftBig?L
?Rμνρσ?ρξσ?2ξ[σ?ρ]
parenleftBig?L
?Rμνρσ
parenrightBigbracerightBig
=?i1···in?2μνbraceleftbigκ?μξν?14γ(?χ)2?μξν+12γ?σχ?μχ?σξν
+12γparenleftbig?μ(?χ)2parenrightbigξν?12γ?σ(?σχ?μχ)ξν?12γ?μ(?νχ?σχ)ξσbracerightbig,
(iξΘ)α1···αn?2=?α1···αn?2μλ
parenleftBig
2?L?R
ρσμν
?σδgρν?2?ν?L?R
ρμνσ
δgρσ+?L?(?
μχ)
δχ
parenrightBig
ξλ.(3.13)
Tospecialisetoourstaticblackholeansatz(2.5),theresultfortheLagrangianwithγ=0
iswellestablished(see,forexample,[39,40]),andisgivenby
Qκ,α=rn?2κ
radicalbigg
f
hh
′?(n?2),
iξΘκ,α=?rn?2
radicalBigg
h
f
parenleftBig
κ(?fhδh′+fh
′
2h2δh?
h′
2hδf?
n?2
rδf)?αfχ
′δχ
parenrightBig
?(n?2),
(δQ?iξΘ)κ,α=?rn?2
radicalBigg
h
f
parenleftBig
κn?2rδf+αfχ′δχ
parenrightBig
?(n?2),(3.14)
Wefindthatthecontributionsassociatedwiththeγtermintheactionaregivenby
Qγ=?12(n?2)γrn?3
radicalBigg
h
ff
2χ′2?(n?2),
iξΘγ=?12(n?2)γrn?3
radicalBigg
h
ff
2parenleftbigχ′2δh
2h+(
n?3
r(1?
?
f)+
h′
h)χ
′δχparenrightbig?(n?2),
(δQ?iξΘ)γ=12(n?2)γrn?3
radicalBigg
h
ff
2
parenleftBig
?32χ′2δff?δ(χ′2)
+(n?3r(1??f)+h
′
h)χ
′δχ
parenrightBig
?(n?2),(3.15)
WenowapplytheWaldformalismtotheblackholesolutions.First,wenotethatasa
consequenceofequation(2.9),whenweaddthecontributionsin(3.14)and(3.15)theχ′δχ
inthetotalexpressioncancel,givingtheresult
δQ?iξΘ=?(n?2)rn?3
radicalBigg
h
f
bracketleftBig
(κ?34γfχ′2)δf+γf2δ(χ′2)
bracketrightBig
.(3.16)
Infact,ascanbeseenfromtheexpressionforχ′2infortheblackholesolutionsin(2.10),
wehaveδ(fχ′2)=0,andso(3.16)canbefurthersimplified,togive
δQ?iξΘ=?(n?2)rn?3
radicalBigg
h
f
parenleftBig
κ+14γfχ′2
parenrightBig
δf.(3.17)
14
Wefirstconsiderthesimplercaseofthe?=0AdSplanarblackholes,forwhich
fχ′2=β.Wefind
δH∞=(n?2)κ16π
parenleftBig
1+βγ4κ
parenrightBig
δμ,
δH+=(n?1)(n?2)g
2κ
16π
parenleftBig
1+βγ4κ
parenrightBig
rn?20δr0.(3.18)
ThusweseeindeedthatδH∞=δH+,sinceμ=g2rn?10.Thisimpliesthatthatwecan
definethemassandentropyas
M=(n?2)κ16π
parenleftBig
1+βγ4κ
parenrightBig
μ,S=14κ
parenleftBig
1+βγ4κ
parenrightBig
rn?20,(3.19)
suchthat
δH∞=δM,δH+=TδS.(3.20)
Thefirstlawofblackholethermodynamics(3.3)thenfollowsstraightforwardlyfromthe
Waldidentity(3.12).Howeverthefactor1+βγ/(4κ)inboththeentropyandthemass
disagreeswiththeresultsin(3.2)and(3.4)thatweobtainedinsection3.1fromadirect
applicationoftheWaldentropyformula(1.5)andtheintegrationofthefirstlawdM=TdS.
Thecaseofthespherically-symmetricblackholeswith(?=1)ismorecomplicated.We
findthatδHevaluatedonthehorizontakesthegeneralform
δH+=(n?2)ωn?2T64(16κ+γf1?χ21)rn?30δr0,(3.21)
wheref1and?χ1arecoefficientsinthenear-horizonexpansionsdefinedin(2.25)and(2.27).
Forourspecific?=1solution,wehave
f1=(n?1)g2r0+n?3r
0
,?χ1=2
radicalbig(n?1)βgr3
20
(n?1)g2r20+n?3,
h1=
parenleftBig
(n?1)(4κ+βγ)g2r20+4(n?3)κ
parenrightBig2
(4κ+βγ)parenleftbig(n?2)g2r20+n?3parenrightbigr0.(3.22)
ThusifwedefineδH+=TdS,withTgivenin(2.22),wefindthattheentropyisgivenby
S=ωn?2
bracketleftBig
1
4κr
n?2
0+
(n?1)(n?2)βγg2rn0
16n(n+2)(n?3)2
parenleftBig
(n+2)(n?3)
?n(n?1)g2r202F1[1,12(n+2);12(n+4);?n?1n?3g2r20]
parenrightBigbracketrightBig
.(3.23)
Notethatthefirstterminsidethesquarebracketsgivespreciselytheresultwesawearlier
(3.2)forWaldentropySW,derivedusingtheformula(1.5).Theremainingcontributionin
thesquarebracketsisproportionaltoγ,thecoefficientoftheHorndeskitermintheaction
(2.1).
15
Toderivethefirstlaw,weevaluatetheδHatasymptoticinfinity,andwefind
δH∞=(n?2)κωn?216π
parenleftBig
1+βγ4κ
parenrightBig
δμ,(3.24)
Thisimpliesthatthemassisgivenby
M=(n?2)κωn?216π
parenleftBig
1+βγ4κ
parenrightBig
μ.(3.25)
Thisturnsouttobetheexactlythesameformasthatinthe?=0AdSplanarblackhole.
Itisnowstraightforwardtoverifythatthefirstlaw(3.3)isindeedsatisfied.Notethatχ0,
beingaconstantshiftintegrationconstantofχ,playsnoroleinthefirstlaw.
Itisworthcommentingthatforthe?=0solutions,themassesweobtainedin(3.4)and
in(3.19)bythetwodifferentmethodsarebothproportionaltoμ.Theonlydifferenceisin
theconstantprefactorcoefficient.Thisonitsownmakesitdifficulttojudgewhichisthe
morereasonableresult.However,when?=1,thedifferencebecomesmorestriking.The
result(3.25)fromthedetailedWaldprocedurethatwepresentedinthispaperisseemingly
moreplausible,fortworeasons.Firstly,themassissimplyproportionaltotheparameterμ,
insteadofbeingaconvolutedtranscendentalfunctionofμ.Secondly,themassdependence
onμisthesameforboththe?=0and?=1solution.Insolutionswithnoadditional
scalarhair,andsincethe?=0solutioncanbeobtainedasascalinglimitofthe?=1
solution,thisconclusionwouldseemtobereasonable.
3.3FurthercommentsontheentropyfromWaldformalism
Havingderivedthefirstlawofthermodynamicsandalsotheentropyinsection3.2,using
thegeneralWaldformalism,wenowexaminethesomewhatunusualfeaturesoftheblack
holesinHorndeskigravitythatleadtothebreakdownofthestandardWaldentropyformula
(1.5).Itfollowsfrom(3.13)thatforthestaticansatz(2.5)that
Q(n?2)=2h′
radicalbigg
f
hS
?0?1?0?1
(n?2)?4hT
0101;1?(n?2),(3.26)
wherethehattedindicesaretangent-spaceindices,thesemicolondenotesacovariantderiva-
tiveand
Tμνρσ≡?L?R
μνρσ
,S?0?1?0?1(n?2)=T?0?1?0?1rn?2?(n?2).(3.27)
Notethat0isthetimedirectionand1istherdirection.TheexpressionforTμνρσfor
theHorndeskigravityisgivenby(3.1).Typically,oneevaluatesQ(n?2)onthehorizonat
r=r0,withh=h1(r?r0)+···andf=f1(r?r0)+···,andsothesecondtermonthe
16
right-handsideof(3.26)vanishesandhence,aswasobservedin[33,34],wefind
1
16π
integraldisplay
r=r0
Q(n?2)=TSW,(3.28)
whereSWisthestandardWaldentropy,givenby(1.5).
Establishingthevariationalidentity(3.12)ismoresubtle,evenforthestandardcaseof
Einsteingravity.ItrequiresthatweevaluateδQonthehorizon.Naively,onewouldsimply
obtainδTSW+TδSWfrom(3.28),andthenonewouldexpectthattheδTSWtermwould
becancelledbytheiξΘcontributionin(3.11),leadingto
δH+=TδSW.(3.29)
However,inordertoevaluatethevariationproperly,weneedtoexpand(3.28)uptoorder
(r?r0),sinceδ(r?r0)=?δr0andsoitisnon-zeroeveninthelimitwhenonesetsr=r0
onthehorizon.TheneteffectisthatallthetermsinδQ(n?2)arecancelledoutbyterms
iniξΘ,andinfacttheTδStermarisesfromtheremainingtermsiniξΘalone.
Tobespecific,letusexamineδQ?iξΘforaspherically-symmetricblackholeinpure
Einsteingravitycoupledtoamasslessscalar,asgivenby(3.14).IfwefirstperformTaylor
expansionsofQandiξΘ,asgiveninthefirsttwoequationsin(3.14),aroundthehorizonat
r=r0,thenindeedtheabovestatementcanbeverified.Thefinalequationin(3.14)gives
analternativebutequivalentevaluationwiththevariationδQ,whichmakestheobservation
moreapparent.WemayevaluateδQfirst,andthensetr=r0.Inthiscase,thern?2factor
inQκ,αjustdependsonthecoordinater,andhenceisnotvaried.Withthisprocedure,
wefindthatallthetermsinδQκ,αarecancelledoutbytermsiniξΘκ,α,leadingtothe
thirdequationof(3.14).Thususingthisprocedure,wefindthattheδH+=TδStermfor
theusualEinsteingravityarisesfromthe(n?2)δf/rterminiξΘin(3.14).Thisterm
correspondsto
rn?2
radicalBigg
h
f
2
rfgijT
1i1jδf?(n?2).(3.30)
ItisratherintriguinghowthistermisultimatelyrelatedtoSWwhichinvolvesonlyT0101.
Indeed,weseefrom(3.1)thatinvielbeincomponents,T?0?1?0?1=?12κandT?1?i?1?j=12κδijfor
theHorndeskiblackholesolutions.Inparticular,theγtermdoesnotcontributeineither
case.
IntheblackholesofHorndeskigravitytherearefurthersubtleties.Firstly,theαterm
iniξΘκ,αin(3.14)doesnotvanishforthesesolutions,andcancontributeatermtothe
entropythatisnotcontainedinSW.Furthermore,althoughthesecondtermin(3.26)
17
vanishesonthehorizon,itsvariationdoesnot.Thisextratermcanbeseenintheformof
Qγin(3.15).Thus(δQ?iξΘ)γin(3.15)willgiveanadditionalcontributiontotheentropy
thatisoverandabovethatofthestandardWaldcontributionSW.Thuswenowhave
δH+=TδS,withSnegationslash=SW.(3.31)
However,theWaldidentity(3.12),aswehaveseen,continuestohold.Thenon-vanishing
contributionsfromboththeαandtheγtermshavethesameessentialorigin,namelythat
thescalarfieldχisnotregularonthehorizon,butrather,ithasabranchcutsingularity,
asshownin(2.27).
Onemightquestionwhetherthisiscompatiblewiththeinterpretationofthesolutionsas
blackholes.However,aswehaveremarkedinsection2.1,thescalarχinHorndeskigravityis
likeanaxion,inthesensethatitentersthetheoryonlythroughitsderivative.Inparticular,
therefore,itwouldnotbenaturaltodefinedifferentconformally-scaledmetricframes(in
themannerthatonedoeswiththedilatoninstringtheory),sincethatwouldbreakthe
manifestaxionicshiftsymmetryofχ.Furthermore,allinvariantpolynomialsconstructed
from?μχwiththemetricandtheRiemanntensorareregularonthehorizon.Forexample,
gμν?μχ?νχisfiniteandnon-zeroonthehorizon.(Thesepropertiescanbeseenfromthe
factthatthevielbeincomponentsofthegradientofχarefiniteeverywhere,includingon
thehorizon,sinceonejusthasEμ?1?μχ=√fχ′=√β[1?(n?3)?/((n?2)g2r2)]?1/2,with
allothercomponentsvanishing,whereEμ?aistheinversevielbein.)Thissupportstheidea
thatthesesolutionsadmitavalidblackholeinterpretation,butatthepricethattheWald
entropyformula(1.5)nolongerprovidesthecompleteexpressionfortheentropy.However,
theidentity(3.12),andhencethefirstlawofblackholethermodynamics,continuetohold,
withtheentropybeingderivedfromthestrictapplicationoftheWaldformalism.
3.4NoetherchargeandmassofAdSplanarblackholes
Intheprevioussubsections,wedescribedtwodifferentmethodsforcalculatingtheentropy
andmassoftheHorndeskiblackholes,onebasedontheuseoftheWaldformula(1.5)for
theentropy,andtheotherbasedonamoredetailedconsiderationoftheWaldformalism.
Inboththeseapproaches,wedidnotuseindependentprocedurestocalculatethemass
andentropy,butrather,wereliedontheuseofthefirstlawofthermodynamicstoobtain
onefromtheother.Sincetheblack-holesolutionsarecharacterisedbyonlyoneparameter,
thereisnonon-trivialintegrabilitycheck,inthesensethattheright-handsideofthefirst
lawdM=TdSwouldbeintegrableregardlessofwhethertheexpressionfortheentropy
18
wascorrectornot.Thefactthatthetwoapproachesledtodifferentresultscallsforan
independentcheckonthecalculationofthemass,ortheentropy.Eventhoughthemass
andentropyobtainedfromtheWaldformalisminsection3.2seemstobemorereasonable,
themassisdeterminedthroughanintegrationofthefirstlaw,ratherthandirectly,inthis
case.Aquestiononecanaskiswhetherthemassisindeedaconservedquantity.
FortheAdSplanarblackholes(i.e.the?=0solutions),thisquestioncanbeanswered
bymeansofasimpleNoethercalculation.For?=0,werewritetheansatzas
ds2=dρ2?a(ρ)2dt2+b(ρ)2d?2?,χ=χ(ρ).(3.32)
Theeffectiveone-dimensionalLagrangianbecomes
L=116πabn?2
parenleftBig
κ(R?2Λ0)?12αχ′2+12γG11χ′2
parenrightBig
,
R=?2a
′′
a?
2(n?2)b′′
b?
2(n?2)a′b′
ab?
(n?2)(n?3)b′2
b2+
?(n?2)(n?3)
b2,
G11=(n?2)a
′b′
ab+
(n?2)(n?3)b′2
2b2?
?(n?2)(n?3)
2b2.(3.33)
whereaprimedenotesaderivativewithrespecttoρ.TheLagrangianisinvariantunder
theglobalscaling
a→λ2?na,b→λb.(3.34)
ThisglobalsymmetryyieldsaconservedNoethercharge
QN=116π(n?2)bn?3(ba′?ab′)(4κ+γχ′2).(3.35)
Intermsofthecoordinatesoftheoriginalansatz(2.5),wehave
QN=n?232πrn?3
radicalbigg
f
h(rh
′?2h)(4κ+γfχ′2).(3.36)
SubstitutingtheAdSplanarblackholesolutionintothisNoetherchargeformula,wefind
QN=(n?1)(n?2)κ8π
parenleftBig
1+βγ4κ
parenrightBig
μ=2(n?1)M.(3.37)
ThusweseethatQNisthesameasthemassobtainedfromtheWaldformalisminsection
3.2,uptosomepurelynumericalconstants.Thissupportstheconclusionthatthemass
andentropyobtainedinsection3.2arevalid,whilsttheresultsinsection3.1arenot.
3.5Euclideanaction
Analternativemethodthathasbeenusedforcalculatingthermodynamicquantitiesfor
blackholesolutionsisbymeansofthequantumstatisticalrelation
Φthermo≡M?TS=IT,(3.38)
19
firstproposedforquantumgravityin[38].HereΦthermodenotesthethermodynamicpoten-
tial,orthefreeenergy,andIistheEuclideanaction.TheregularisedEuclideanactionwas
calculatedforthe?=1Horndeskiblackholeinfourdimensionsin[35].Wehaverepeated
thatcalculation,andobtainedthesameresult(saveforanoverallfactorof2discrepancy).
However,theresultingexpressionsforMandentropyarequitedifferentfromthosein
sections3.1or3.2,andaregivenby
M=12κ
parenleftBig
1+βγ4κ
parenrightBig
μ?3βγg
2r30parenleftbig4κ(3g2r20+1)+3βγg2r20parenrightbig
8(4κ+βγ)(1+3g2r20)parenleftbig4κ(3g2r20?1)+3βγg2r20parenrightbig,
S=κπr20+3πβγg
2r40parenleftbig4κ+3(4κ+βγ)g2r20parenrightbig
2(1+3g2r20)parenleftbig4κ?3(4κ+βγ)g2r20parenrightbig.(3.39)
Notethatwhenβ=0,forwhichtheblackholereducestothestandardSchwarzschild-AdS
one,wegetM=12μandS=κπr20,asonewouldexpect.Itisclearthatthemasssuffers
fromthesameshortcomingastheoneweobtainedfromtheWaldentropyformulain(3.5),
inthatitbecomesaconvolutedtranscendentalfunctionofμfornon-vanishingβ.(Itisa
differenttranscendentalfunctionfromtheonefollowingfrom(3.5),however.)
Thecalculationforthe?=0AdSplanarblackholes(2.16)ismucheasier,andcan
bestraightforwardlycarriedoutforageneralspacetimedimensionn.Theregularised
Euclideanactioncanbedefinedbysubtractingtheactionofthebackgroundμ=0vacuum
fromtheactionfortheblackholeitself,namely
Ireg=IE[gμν,χ]?IE[g(0)μν,χ(0)],(3.40)
whereg(0)μνandχ(0)arethebackgroundfieldobtainedbysettingμ=0intheblackhole
solution(2.16).Wefind
Ireg=?κ16(n?1)
parenleftBig
1?(n?2)βγ4κ
parenrightBig
rn?20.(3.41)
Notethatinthiscalculation,wehavesetωn?2=1,sothattheresultingextensivequantities
aredensities.Usingthequantumstatisticalrelation(3.38)andthethermodynamicfirst
law(3.3),wethenfindthatthefreeenergy,mass,temperatureandentropyforthe?=0
blackholesaregivenby
F=?κμ16π
parenleftBig
1?(n?2)βγ4κ
parenrightBig
,M=?(n?2)F,
T=g
2(n?1)r0
4π,S=
1
4κr
n?2
0?
1
16(n?2)βγr
n?2
0.(3.42)
Theseexpressionsalsodisagree,inthiscasebyconstantoverallfactors,withthe?=0
resultsobtainedinsections3.1and3.2.Takeninisolation,itwouldbehardtomake
20
anyjudgmentastowhethertheseexpressionsweretrustworthyornot.Interestinglythe
generalizedSmarrrelation(3.6)isalsosatisfied.However,the?=1results(3.39)forthe
massandtheentropycertainlyraisequestionsaboutthevalidityofthiscalculationusing
theEuclideanaction.
ThereisanothermethodthathasbeenusedinordertoobtainafiniteEuclideanaction,
byaddingasurfacetermandacounterterm.Takingn=4dimensionsasanexample,the
wholeactionisthengivenby
I=Ibulk?2IGH?Ict,(3.43)
whereIGHisthestandardGibbons-Hawkingsurfaceterm,andfor?=0,thecounterterm
isgivenby
Ict=κ
integraldisplay
dx3√γc1g,withc1=4+βγκ,(3.44)
Theγinthesquarerootisthedeterminantofinducedmetricγμν.Withthesecombinations,
thetotalactionisthesameastheresultofregularization.For?=1,thecountertermis
Ict=κ
integraldisplay
dx3√γ(c1g+c2R[γ]g),withc1=4+βγκ,c2=1?βγ4κ(3.45)
andthevalueoftheactionhasanadditionaltermlineartheimaginary-timeperiod(i.e.
inverselyproportionaltothetemperature),incomparisontothatoftheregularizedcalcu-
lationabove:
Irenorm=Ireg+
√3πβ2γ2
12g(4κ+βγ)
?
T.(3.46)
Theeffectonthethermodynamicsisthattheentropyisunchanged,butthemassacquires
anadditivecontributioninthespherically-symmetric?=1solutions,independentofthe
parameterinthesolutions.Thisisnotsurprising,sincewhen?=1,theμ=0solutionis
notvacuumAdSspacetime,butinsteadasmoothsoliton,whichhasaconstantmass.Inthe
earlierregularisationbysubtractingthebackground,thisconstantenergywassubtracted
out.
Thequestionremainsastohowonemightreconciletheresultsfortheentropyandthe
mass,ascalculatedfromtheregularisedEuclideanaction,withourprevious,anddifferent,
resultsobtainedusingtheWaldformalism.Wedonothaveadefinitiveresolutiontothis
puzzle,otherthantosuggestthatbecauseoftheratherunusualfeaturesoftheblack-hole
solutionsinHorndeskigravity,itmaybethatthenaiveapplicationofasubtractionproce-
duretoobtainaregularisedEuclideanactionmaybeinherentlyambiguous.Inasomewhat
relatedcontext,itwasfoundin[45]thatattemptstoemploytheAbbott-Desermethod[46]
tocalculatethemassofasymptotically-AdSblackholesfounderedonambiguitiesinthesub-
tractionprocedureinsomecases,forsolutionsingaugedsupergravitieswherescalarfields
21
wereinvolved.Intheabsenceofarigorousderivationofavalidsubtractionschemeforthe
calculationoftheEuclideanaction,itseemsthatonecouldengineerdifferentschemesthat
gavedifferentresults,withnoguideastowhichresultshouldberegardedasthecorrect
one.
4Viscosity/EntropyRatio
Oneofthemotivationsforthispaperwastostudytheviscosity/entropyratioinHorndeski
gravity.Havingobtainedaformulafortheentropyoftheblackholes,wearenowin
apositiontoproceed.Tocalculatetheshearviscosityoftheboundaryfieldtheory,we
consideratransverseandtracelessperturbationoftheAdSplanarblackhole,namely
ds2=?fdt2+dr
2
f+r
2parenleftbigdxidxi+2Ψ(r,t)dx1dx2parenrightbig,(4.1)
wherethebackgroundsolutionisgivenby(2.11),(2.16)and(2.17).Wefindthatthemode
Ψ(r,t)satisfiesthelinearisedequation
r(4κ+βγ)(g2rn?1?μ)2Ψ′′+(4κ+βγ)(g2rn?1?μ)(ng2rn?1?μ)Ψ′
?r2n?5(4κ?βγ)¨Ψ=0.(4.2)
Foraninfallingwavewhichispurelyingoingatthehorizon,thesolutionforawavewith
lowfrequencyωisgivenby
Ψ=e?iωtψ(r),ψ(r)=exp(?iωKlogf(r)g2r2)+O(ω2),
K=14πT
radicalBigg
4κ?βγ
4κ+βγ.(4.3)
NotethattheconstantparameterKisdeterminedbythehorizonboundarycondition.The
overallintegrationconstantisfixedsothatΨisunimodularasymptotically,asr→∞.
InordertostudytheboundaryfieldtheoryusingtheAdS/CFTcorrespondence,we
substitutetheansatzwiththelinearisedperturbationintotheaction.Thequadraticterms
intheLagrangian,afterremovingthesecond-derivativecontributionsusingtheGibbons-
Hawkingterm,canbewrittenas
L2=P1Ψ′2+P2ΨΨ′+P3Ψ2+P4˙Ψ2,(4.4)
with
P1=?18(4κ+βγ)(g2rn?1?μ)r,P2=12g2rn?1[4κ?(n?2)βγ]?μ(2κ?n?34βγ),
22
P3=n?14g2rn?2[4κ?(n?2)βγ],P4=r
2n?5(4κ?βγ)
8(g2rn?1?μ)(4.5)
NotethatP3=12P′2.WethenfindthatthetermsquadraticinΨintheLagrangianare
givenby
L2=ddr(P1ΨΨ′+12P2Ψ2)+ddt(P4Ψ˙Ψ)?Ψ
bracketleftBig
P1Ψ′′+P′1Ψ′+P4¨Ψ
bracketrightBig
.(4.6)
Thelastterm,enclosedinsquarebrackets,vanishesbyvirtueofthelinearisedperturbation
equation(4.2),andsothequadraticLagrangianisatotalderivative.Theviscosityis
determinedfromtheP1ΨΨ′term,followingtheproceduredescribedin[6,20].Usingthis,
wefindthattheviscosityisgivenby
η=κ(n?1)μ64π2T
radicalbigg
1?β
2γ2
16κ2.(4.7)
Wehave,fortheplanarblackholes,
μ=g2rn?10,T=(n?1)g
2r0
4π,(4.8)
andtheentropythatwederivedinsection3.2usingtheWaldformalismisgivenby
S=14κ
parenleftBig
1+βγ4κ
parenrightBig
rn?20.(4.9)
Wethereforefindthattheviscosity/entropyratioisgivenby
η
S=
1
4π
radicalBigg
4κ?βγ
4κ+βγ(4.10)
fortheHorndeskiblackholes.3Notethatκandβarebothpositive.Forreality,wemust
have
?4κβ<γ<4κβ.(4.11)
Whenβ=0,whichturnsoffthescalarfield,theratiogoesbacktotheuniversalvalueof
1/(4π).Whenγ>0,theratioislessthan1/(4π)andhencetheboundisviolated.For
γ<0,theratioisgreaterthan1/(4π).
Finally,wenotethatintermsoftheoriginalparametersofthetheory(2.1),theviscos-
ity/entropyratioisgivenby
η
S=
1
4π
radicalBigg
3α+γΛ0
α?γΛ0.(4.12)
Interestingly,theratioisindependentoftheparameterκ.
3Intriguingly,althoughtheratioiscalculatedfortheAdSplanarblackhole(?=0),thesameratio
(4κ?βγ)/(4κ+βγ)appearsinthesub-leadingconstantterminthelarge-rexpansionofh=?gttgivenin
(2.19),butonlyforthespherically-symmetric(?=1)solutions(itvanishesforthe?=0solutions).
23
5Conclusion
MotivatedbyapplicationsfortheAdS/CFTcorrespondence,westudiedtheblackholesin
atheoryofEinsteingravitycoupledtoascalarfield,includinganon-minimalHorndeski
termwherethegradientofthescalarcouplestotheEinsteintensor.Therearetwotypes
ofstaticblackholesinthisHorndeskigravity.OneoftheseistheusualSchwarzschild-AdS
blackhole,forwhichthescalarfieldisconstant.Ourfocusisontheothernon-trivial
one-parameterfamilyofstaticblackholes,forwhichthescalardependsnon-triviallyon
theradialcoordinate.Althoughthescalarhasabranch-cutsingularityonthehorizon,it
isaxion-likeandentersthetheoryonlythroughaderivative.Furthermore,inanorthonor-
malframe,?aχisregulareverywhere,bothonandoutsidethehorizon,andallinvariants
involvingthescalarfieldarefiniteeverywhere.Wealsodemonstratedtheuniquenessof
thesestaticblackholesolutionsinthetheory.
Westudiedthethermodynamicsoftheblackholesandfoundthreesurprises.Thefirst
isthatthestandardWaldentropyformula(1.5)doesnotgivethecompleteexpressionfor
theentropyoftheseblackholes.Thiscanbeattributedtothefactthatthederivationofthe
Waldentropy(1.5)requiresthatthescalarberegularonthehorizon.Infact,thebranch
cutsingularityofthescalaronthehorizonimpliesthatthereisanextracontributionto
theentropy.WestudiedtheWaldformalismindetail,andexhibitedthenewcontribution
explicitly.ItturnsoutthattheWaldidentity(3.12)continuestoholdfortheseblackholes,
andsodoesthefirstlawofblackholethermodynamics.Theentropy,however,isnolonger
givenby(1.5),butcanbedeterminedfromtheimplementationoftheWaldprocedure.We
furtherestablished,usingasimpleconstructionoftheNoetherchargederivablefromthe
scalingsymmetryoftheplanarblackholes,thatthemassoftheAdSplanarblackhole,as
wederivedfromtheWaldprocedure,isindeedaconservedquantity.
ThesecondsurpriseconcernstheuseofthequantumstatisticalrelationE?TS=TIto
calculatethethermodynamicparametersoftheblackholesolutions.Inordertoapplythis
method,itisnecessarytocalculatetheEuclideanactionIoftheblackholesolution.The
problemisthatadirectintegrationoftheEuclideanisedactionyieldsaresultthatdiverges
attheupperendoftheradialintegration,andsoitisnecessarytoadoptsomeregularisation
procedure.Wetriedtoapplytwodifferentsuchprocedures.Thefirstinvolvedsubtracting
thedivergingcontributionofabackgroundwherethemassissettozerofromthediverging
contributionfromtheblackholewithnon-zeromass.Theotherprocedureinvolvedaddinga
boundarycounterterm.Thetwomethodsgavethesameresultsforthemassandtheentropy,
buttheseresultsdifferedfromthosethatweobtainedbyusingtheWaldformalism.The
24
originofthismismatchisnotcleartous;itmayberelatedtointrinsicambiguitiesinthe
subtractionschemesthatweusedinordertoregularisethedivergences.Suchambiguities
arepossiblymorelikelyinatheorysuchasHorndeskigravity,withitssomewhatunusual
features,andsoregularisationschemesforcalculatingtheEuclideanactionthatusually
workinlessexactingsituationsmayneedtobescrutinisedmorecarefullyhere.
Thethirdsurpriseconcernstheresultsinsection4fortheviscosity/entropyratio.In
wideclassesofconventionaltheorieswithnohigher-derivativetermsintheLagrangian,one
findsaratheruniversalresultthatη/S=1/(4π).Counter-examplestotheuniversalityof
theratiohavebeenfound,butforisotropicsituationssuchaswehaveconsideredtheyare
alwaysassociatedwithhigher-derivativegravities,suchasGauss-Bonnetormoregeneral
Lovelockgravities.Asfarasweareaware,ourfindingsfortheblackholesintheHorndeski
theorywestudiedinthispaperprovidethefirstexampleoftheviolationoftheη/S=1/(4π)
resultinatheorywhoseLagrangianisatmostlinearincurvaturetensor.
AwordofcautionabouttheuseoftheWaldformalismtocalculatetheentropyis
perhapsappropriatehere.IfweconsiderEinstein-Maxwelltheoryasanexample,thefirst
lawdM=TdS+ΦdQforReissner-Nordstr¨omblackholescanbederivedfromtheWald
formalismbycalculatingδH∞andδH+,andusingthefactthatδH∞=δH+.TheΦdQ
contributioncaneitherenterinδH+alone,ifoneusesthegaugewherethepotentialvanishes
atinfinity,orinδH∞alone,ifoneusesthegaugewherethepotentialvanishesonthe
horizon,orelseinbothδH∞andδH+,ifoneusessomeintermediategaugewherethe
potentialvanishesneitheratinfinitynoronthehorizon.Inthefirstlaw,onlythepotential
differenceΦ≡Φ+?Φ∞contributes.Ifthegaugewherethepotentialvanishesonthe
horizonischosen,thenδH+=TδSandsoδH+/Tisanexactdifferential,whichcanbe
integratedtogivetheentropy,whileδH∞=dM+Φ∞dQ,andisnotexact.Inthegauge
wherethepotentialinsteadvanishesatinfinity,δH∞=dM,whichisanexactdifferential,
whileδH+=TdS+Φ+dQ,andsoδH+/Tisnotexact.
Morecomplicatedsituationswereencounteredrecentlywhereasymptotically-AdSdy-
onicallychargedblackholeswereconstructedinafour-dimensionalgaugedsupergravity
involvingascalarandaMaxwellfield[47,48].ItwasfoundthatδH∞wasnon-exact,
andhencenon-integrable,evenwhenagaugewheretheelectricandmagneticpotentials
vanishedatinfinitywaschosen,becauseofavaryingcontributionfromtheasymptotic
coefficientsinthelarge-distanceexpansionofthescalarfield.Thefirstlawofblackhole
(thermo)dynamics,involvingthescalarcontribution,couldneverthelessbederivedusing
thestrictWaldformalism[47].Theresultswerelatergeneralisedtoblackholesingeneral
25
Einstein-scalartheories[39,40],Einstein-Procatheories[41],andgravityextendedwith
quadraticcurvatureinvariants[43].
AnalogousissuescouldinprinciplearisewhenconsideringδH+:itiscommonlythecase
thatδH+onthehorizoncanbeexpressedasTδS.InatheorysuchasEinstein-Maxwell,
thisisagauge-dependentpropertyaswediscussedabove,andinordertohaveδH+/Tbe
anexactdifferentialinthiscaseonewouldneedtoworkinthegaugewheretheelectric
potentialvanishedonthehorizon.Inmosttheoriesthathavebeenstudied,theentropyis
simplygivenbySWdefinedbytheWaldentropyformula(1.5).Thewidespreadvalidityof
theWaldentropyformulaisrelatedtothefactthattypically,matterfieldsvanishonthe
horizonofablackhole(andMaxwellpotentialscanbesettozerobymeansofappropriate
gaugechoices).IntheHorndeskigravityconsideredinthispaper,however,theaxion-like
scalarχhasanunusualbehaviournearthehorizonandnearinfinity,andindeedwehave
alreadyseenthatδH+negationslash=TδSW.Weneverthelessassumedthatitwasstillthecasethat
δH+=TδS,i.e.thatδH+/Tcouldbeintegratedtodefineanentropyfunction.That
δH+/Tisintegrableisguaranteedintheone-parameterfamilyofsolutionsconsideredin
thispaper,sinceall1-formsinonedimensionareexact.Inamultiple-parameterblackhole
solution,however,theredoesnotappeartobeanyguarantee,apriori,thatδH+/Tmustbe
atotaldifferentialinatheorysuchasHorndeskigravity.Thenon-integrabilityofthesort
thatoccursinδH∞inthedyonicasymptotically-AdSblackholeswediscussedabovemight
also,inprinciple,occurforδH+/Tonthehorizon,ifnotallthefieldsarestrictlyvanishing
onthehorizon.Itwouldbeinterestingtostudythisfurtherinmoregeneralsolutionsin
theoriessuchasHorndeskigravities.
ThefindingsinthispaperindicatethatHorndeskigravity,anditsblackholesolutions
inparticular,deservefurtherinvestigationbothintheirownright,andalsointhecontext
oftheAdS/CFTcorrespondence.
Acknowledgements
WearegratefultoSeraCremoniniforhelpfuldiscussions.H-S.L.issupportedinpart
byNSFCgrants11305140,11375153and11475148,SFZJEDgrantY201329687andCSC
scholarshipNo.201408330017.C.N.P.issupportedinpartbyDOEgrantDE-FG02-
13ER42020.TheworkofX-H.FengandH.L.aresupportedinpartbyNSFCgrantsNO.
11175269,NO.11475024andNO.11235003.
26
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