Higher dimensional black holes with a generalized gravitational action

JerzyMatyjasek∗andMa󰁒lgorzataTelecka

InstituteofPhysics,MariaCurie-Sk󰀫lodowskaUniversitypl.MariiCurie-Sk󰀫lodowskiej1,20-031Lublin,Poland

DariuszTryniecki

InstituteofTheoreticalPhysics,Wroc󰀫lawUniversity

pl.M.Borna9,50-204Wroc󰀫law,Poland

(Dated:February1,2008)

WeconsiderthemostgeneralhigherordercorrectionstothepuregravityactioninDdimensionsconstructedfromthebasisofthecurvaturemonomialinvariantsoforder4and6,anddegree2and3,respectively.Perturbativelysolvingtheresultingsixth-orderequationsweanalyzetheinfluenceofthecorrectionsuponastaticandsphericallysymmetricbackhole.Treatingthetotalmassofthesystemastheboundaryconditionwecalculatelocationoftheeventhorizon,modificationstoitstemperatureandtheentropy.TheentropyiscalculatedbyintegratingthelocalgeometrictermconstructedfromthederivativeoftheLagrangianwithrespecttotheRiemanntensoroveraspacelikesectionoftheeventhorizon.Itisdemonstratedthatidenticalresultcanbeobtainedbyintegrationofthefirstlawoftheblackholethermodynamicswithasuitablechoiceoftheintegrationconstant.WeshowthatreducingcoefficientstotheLovelockcombination,theapproximateexpressiondescribingentropybecomesexact.Finally,webrieflydiscusstheproblemoffieldredefinitionandanalyzeconsequencesofadifferentchoiceoftheboundaryconditionsinwhichtheintegrationconstantisrelatedtotheexactlocationoftheeventhorizonandthustothehorizondefinedmass.

PACSnumbers:04.50.+h,04.70.Dy

I.INTRODUCTION

Inrecentyearsgravitationtheorieswithhigherderivativetermshaveattractedagreatdealofattention.Indeed,accordingtoourpresentunderstandingthegeneralrelativityistobetreatedasthelowestordertermoftheeffectivetheoryconsistingofaseriesoflooporstringcorrections.Typicallysuchcorrectionsareconstructedfromhigherpowersofcurvatureandtheirderivatives,

2

and,hence,thegravitationalactionIcanbewrittenas

I=

m󰀇k=0

αkIk,

(1)

whereIkfork≥1isthecontributionofoperatorsofdimension2k,I0isrelatedtothecosmo-logicalconstantandαkarearbitraryconstants.AmongthehighercurvaturetheoriesagreatdealofactivityhasbeenfocusedontheLovelockgravity[1].Inthistheory,theLagrangianLmisconstructedfromthedimensionallyextendedEulerdensitiesofa2k-dimensionalmanifold

akbkc1d1...ckdka1b1

,R...RLk=2−kδackdkc1d11b1...akbk

(2)

wherethegeneralizedδfunctionistotallyantisymmetricinbothsetsofindices.Am−thorderLovelockaction,Im,isthesumofm+1termsgivenbyEq.(2)ofascendingcomplexity

Im=

󰀌

dx(−g)

D

1/2

Lm=

󰀌

dx(−g)

D1/2

m󰀇k=0

αkLk,

(3)

whereαkarearbitraryparameters.IftheLovelockactionincludesthetermsuptoLm,thedimensionofthespacetimeshouldsatisfyD≥2m+1.

VaryingtheactionfunctionalImwithrespecttothemetrictensoroneobtainstheequationsofmotionofpuregravity,whichcanbeschematicallywrittenas

Lab=

1

δgab

Im=0.

(4)

Originally,LabhasbeenintroducedbyLovelocktodemonstratethemostgeneralsymmetricanddivergence-freesecondranktensor,whichcanbeconstructedfromthemetricanditsfirstandsecondderivatives.SincetheresultingequationsoftheLovelockgravityinvolveatmostsecondderivativesofthemetricitavoidssomeofthetypicalproblemsofthehighercurvaturetheories[2,3].Specifically,attheclassicallevel,itavoidssingularperturbations[4,5,6]whichdonotapproachtheirEinsteiniancounterpartsastheperturbativeexpansionparameterissettozero,and,whenlinearized,theLovelockequationsarefreeofghosts.Moreover,thehigher-ordertermsappearquitenaturallyasthelow-energylimitofthestringtheory[2,3].

AteachorderLkisalinearcombinationofthebasiscurvatureinvariantswiththeparticularsetofcoefficientscalculatedfromEq.(2).Forexample,inthefirsttwo(simplest)cases,onehasacosmologicalconstantandthecurvaturescalar,fork=0andk=1respectively.Atk=2therearethreeinvariantswhicharecombinedintotheGauss-Bonnettermwhereasatk=3thebasishaseightmembers.

3

Ontheotherhand,onemayconsiderthemoregeneralcurvatureterms,witharbitrarycoeffi-cientsratherthanthoseinspiredbytheparticulartheory.(Seeforexample[7,8]andreferencescitedtherein).Inthiscase,therelationbetweendimensionofthespacetimeandtheorderofhighercurvaturetermsretainedintheactionfunctionalislostandthedynamicalequationsinevitablyinvolvehigherderivativesofthemetric.Thereisnothingwronginusingsuchequations,pro-videdonlythephysicalsolutionsareselected.However,theexpectedcomplexityoftheresultingequationsmaybeaseriousobstacleinthisregard.

Theliteraturedevotedtovariousaspectsofthehigherderivativegravityisvastindeed.Astheexamplesofsuchtheoriesonemayconsiderthequadraticorhigherordergravity(see[9,10,11,12,13,14,15,16]andthereferencescitedtherein),and,whenanegativecosmologicalconstantispresent,theEinstein-HilbertactioninD=5,supplementedwithRiem2=RabcdRabcd,whichcorrespondstoaneffectiveAdS5bulkaction[17].

Theforegoingdiscussionindicatesthattheanalysescanbecarriedoutintwodirections.First,onecanconstructandinvestigatethepossiblecandidatetermsthatmayappearinIk,whereastheseconddirectionofcalculationsistoapplythethusconstructedequationsintheparticularcontextofblackholephysicsorcosmologywithorwithoutadditionalmatterfields.ThelatterapproachhasbeensuccessfullyappliedinvariouscontextsinRefs.[17,18,19,20,21,22,23,24,25,26]andinthereferencescitedtherein.

Afewwordsofcommentareinorderhere.First,itshouldbeobservedthatwehavenoinformationonm,i.e.,thenumberoftermsthatshouldberetainedinEq.(1).Second,andthatiscloselyrelatedtotheaboveobservation,itisreallydesirableandperhapsunavoidabletoconstructthesolutionsofthedynamicalequationswhichreflectthenatureoftheirderivation.Finally,itshouldbeobservedthatbecauseofcomplexityoftheproblemthefullsystemofequationsisprobablyintractableanalyticallyandonehastoconstructeitherapproximatesolutionsorrefertothenumericalmethods.

InthisnoteweshallexplorethesecondpossibilityandperturbativelysolvetheequationsresultingfromthevariationoftheD−dimensionalaction(1)withoutacosmologicalconstantabouttheTangherliniblackhole[27,28].Indoingsotheclassofsolutionsarerestrictedtotheadmissibleones.WeshallassumethatthetotalactionfunctionalIisthesumofthefirstthreenonvanishingterms,Ik,constructedfromthebasiscurvatureinvariants.Thatis,weassumearbitrarycoefficientsratherthanthoseinspiredbytheparticulartheory.TheresultsofthispapergeneralizeresultsofLuandWise[20]andmaybethoughofasapartialgeneralizationoftheanalogousresultsobtainedwithintheframeworkoftheLovelockgravity[4,29,30,31,32].

4

II.

EQUATIONS

Weshallconsidertheactionfunctionalbeingasumoftheterms(conventionsareRab=Rcacb∼∂cΓcab,signature−,+,+,+)

I1=a

󰀌

dDx(−g)1/2R,

(5)

I2=

󰀌

dDx(−g)

1/2

󰀎

b1R2+b2RabR

ab

+b3RabcdR

abcd

󰀐

(6)

and

I3=󰀌

dDx(−g)

1/2

󰀎

c1R3+c2RRbaRa

b+c3RRabcdRcdab+c4Rba

RdcRbdac+c5RbaRcbRa

c

+

c6Rb

aRbcdeRdeac

+

c7RabcdRefcdRefab

+

c8RceabRafcdRef

bd󰀐

,(7)

wherea=(16πGD)−1andGDisNewton’sconstant.Thatis,wewillrestrictourselvestoscalar

termsoforder2,4and6belongingtoclassesR02,1,R04,2andR06,3,respectively[8].Inthecourseof

thecalculationsweshallassumethatthecoefficientsbiandcksatisfy|bi|/a<<1and|ck/bi|<<1fori=1,..,3andk=1,...,8,respectively.Thecasebi∼cicanbeeasilyobtainedfromtheformeronesimplybyrelegatingthetermsproportionaltobibjfromtheresultingexpressions.

ItshouldbenotedthatdependingonthedimensionDthecurvaturetermsmaybesubjectedtoadditionalrelations[8].Moreover,forastaticandsphericallysymmetriclineelementwehaveaddi-tionalvanishingcombinationoftheelementsofthecurvaturebasiswiththecoefficientsdependingonD[7].

Althoughitispossibletoadopt(withsmallmodifications)theresultspresentedinRefs.[33,34,35],hereweproceeddifferentlyandusetheWeylmethod[36,37,38].Thelineelementdescribingthestatic,sphericallysymmetricD=d+2-dimensionalgeometrymaybycastintotheform

ds2=−f2(r)dt2+h−2(r)dr2+

δij+xixj

5

Rtrtr=−f−1h(f′h)′

and

j

Rtitj=−r−1f1h2f′δi,

(11)

(12)

wheretheprimedenotesdifferentiationwithrespecttothecoordinater.UponinsertingthelineelementintoIandsubsequentlyvaryingthethusobtainedreducedactionwithrespecttothefunctionsfandh,oneobtainsrathercomplicatedsystemofequations(notdisplayedhere),whichmaybefurthersimplifiedwiththesubstitution

f2(r)=e2ψ(r)1−

󰀑

2M(r)

rd−1

.(14)

ExceptforcertaincombinationsofthenumericalcoefficientsbiandckleadingtothedimensionallyextendedEulerdensities,theresultingequationsofmotionarestilltoocomplicatedtobesolvedexactly.Fortunately,onecaneasilydevisetheperturbativeapproachtotheproblem,treatingthehigherderivativetermsassmallperturbations.

Now,inordertosimplifycalculationsandtokeepcontroloftheorderoftermsincomplicatedseriesexpansions,weshallintroduceanother(dimensionless)parameterε,substitutingbi→εbiandci→ε2ci.Weshallputε=1inthefinalstageofcalculations.FortheunknownfunctionsM(r)andψ(r)weassumethattheycanbeexpandedas

M(r)=

and

ψ(r)=

m󰀇i=1m󰀇i=0

εiMi(r)+O(εm+1)

(15)

εiψi(r)+O(εm+1).

(16)

ThesystemofdifferentialequationsMi(r)andψi(r)istobesupplementedwiththeappropriate,physicallymotivatedboundaryconditions.First,itseemsnaturaltodemand

M(r+)=

d−1

r+

6

d−1

/2andMi(r+)=0fori≥1,wherer+denotestheexactlocationor,equivalently,M0(r+)=r+

oftheeventhorizon.Suchachoiceleadsnaturallytothehorizondefinedmass,

MH=

dωd

dωd

where

ωd=

M,(19)

2π(d+1)/2

ard+1

(d−2)(d−1)(21)

andψ1=0.Integrationofthesecond-orderequations,althoughstraightforward,yieldsmuchmorecomplicatedresults:M2(r)=

C3(d−1)

c8(2+25d−24d2−7d3)−

8b1b3

a

ard+3

16b1b3

a

(d−2)−

(d−2)(d+1)(d+2)

−

2

8b23

[8c3d−c6(2−3d)+12c7(d−1)32b23

−3c8(d−1)−

7

andψ2(r)=−

C2(d2−1)

c8(2−3d−d2

2

)−

8b1b3

G3dωdrd−1

+4096

π3DM

2+α365536

π4G4DM

3a

(d−2)(d+2)−

8b23

dωdr

d+1

(d−2)(d−1)󰀃

(d−2)(d−1)8

realrootsofthem−thorderpolynomialequation

m󰀇k=0

cˆkFk=

2C

1a

2k󰀊

a

(d+2−n)fork>1.(27)

n=3

Notethatifthecosmologicalconstantistakentobezerothenα0=0.Finally,assumingα2=b∼εandα3=c∼ε2andexpandingEq.(25)inpowersofε,oneeasilyreproducesEq.(24).

IV.TEMPERATUREANDENTROPY

Theapproximatelocationoftheeventhorizonoftheblackholesolutionderivedintheprevioussectionisgivenbyr+=(2C)

2

1

1/(d−1)

−

d−2

1

a

(2C)

−3/(d−1)

c7[2−(d−1)(5d−4)d]+

󰀔1c3(d2−1)d+

a

(b1b3+b2b3)(d2−1)(d−2)

+

G3M4

󰀑

6c3+5c7+

3

dr

gtt

󰀓−1

.(30)

Itsreciprocalisidentifiedwiththeblackholetemperature,which,inthecaseonhand,reads

T=++d−1d−1

2

c8

2a4πa

b3(2C)−3/(d−1)

(d−4)[2+(d−2)(d−1)d]

(d−2)2(4+7d−4d2)}.

(31)

9

ItshouldbenotedthattheHawkingtemperaturedoesnotdependonb1,b2,c3andc6.Thisbehaviorcanbetracedbacktothepossibilityofremovingcurvaturetermsproportionaltothisverycoefficientsbymeansoftheappropriatefieldredefinition.Sucharedefinitioncertainlymodifiesequationsofmotionofthetestparticlesbutshouldnotmodifythetemperature,which,inturn,istobemeasuredatinfinity.Ontheotherhand,thehorizondefinedmassleadstotheexpressionfortemperaturewhichdependsonthefullsetofparameters.Thereasonisthatthehorizondefinedmassisnotthemassmeasuredatinfinity.Weshallreturntothisproblemlater.

FromEq.(31)oneseesthattheheatcapacityCBH=∂M/∂TcalculatedfortheTangherliniblackholeisgivenby

TCBH=−

dωd

4πT

󰀓d

(32)

andisalwaysnegative.Thatmeansthatford>1suchablackholeisthermodynamicallyunstable,i.e.,itincreasesitstemperaturewhenradiating.Ontheotherhand,forT>>1thehigherderivativecorrectionscanmodifythisbehaviour,and,dependingonthesignsandvaluesofthecouplingconstantstheycangiveanonnegativecontributiontothetotalheatcapacityascanbeeasilyinferredfrom

TCBH=CBH+∆CBH,

(33)

where∆CBH=

×ωd󰀋

4πT

1

4

󰀓d−2

b3+

dωd

4πT

1

󰀓d−4

[8+4(d−a)(d−1)d]c7−

∂M

|<<1.(35)

10

Now,makinguseof(31)onehas

∂T

dωda2dω+

5

(2C)−d/d−1+

12GDb3

d

(d−2)2(4d2−7d−4)(2C)−(d+4)/(d−1)

4GD

AH+4π

󰀌

∂L

(d)gddx,

(37)

H

whereg˜abdenotesthemetricinthesubspaceorthogonaltotheeventhorizon,and,inthecaseonhand,ListhesumoftheLagrangiansgivenbyEqs.(6)and(7).Itshouldbenotedthattocalculatetheentropytotherequiredorderitsufficestoretaininthelineelementthetermswhicharelinearinε.

ThecalculationofSconsistsofthreesteps.First,itisnecessarytoexpressthelineelementintermsofr+,whichcanbeeasilyachievedbyinvertingrelation(28).Simplecalculationsyield

C=

1

d−3

−(d−2)(d−1)r+

2a

󰀒c62

(d−1)(d+1)c3d+

1

d−5

(d−1)c7[2+d(d−1)(4−5d)]r+

+

2a1

4a

∂Rabcd

(39)

11

rememberingthatJabcdsharesallsymmetriesoftheRiemanntensor.Finally,performingsimple

d),afterintegration(whichactuallyreducestothemultiplicationoftheresultbythefactorωdr+

somealgebra,onegetsthedesiredresult:

S=

AH

+

4GD

d(d−1)

2

+

1

2a2GD

󰀄󰀅

(d−2)(d−1)d(d+1)b1b3+b2b3+3b23

3

(d2−1)c6+3d(d−1)c7−

∂r+

󰀓

dr++S0,

Qi

(43)

wheretheintegrationconstantS0doesnotdependonr+,butpossiblydependsonthecouplingconstantsandthespacetimedimension.Itshouldbenotedthatinthepresentapproachitisnecessarytoretaininthelineelementallthetermsproportionaltoε2also.Aftersomealgebraoneobtainstheexpressiondescribingtheentropy,whichforS0=0coincideswiththeonegivenbyEq.(41).

Theintegrationconstantcanbedeterminedfromthephysicalrequirementthattheentropyvanisheswhenthehorizonradiusshrinkstozero[32,46].FortheLovelocktheoryithasbeen

12

shownthatthisconditionleadstotheresultswhichareidenticalwiththoseobtainedwithintheframeworkoftheEuclideanapproach.Ford>4thisproceduregivesS0=0.

TheentropyasgivenbyEq.(41)isexpressedintermsoftheexactlocationoftheeventhori-zon,r+,andthereforeitdependsonthefullsetofthecouplingconstants.However,accordingtoourpreviousdiscussion,onecaneasilyreducetheirnumberbyasuitablechoiceofrepresentation.Indeed,expressingtheentropyintermsofthetotalmassasseenbyadistantobserver(or,equiv-alently,C)reducesthenumberofremainingcouplingconstantstothree.Todemonstratethis,letussubstituter+givenbyEq.(28)intoEq.(41)andretainthetermsuptosecondorderinε.Aftersomerearrangement,oneobtainsS=

ωd

(d−2)/(d−1)24aG(2C)d2

b3+

εωd

4

c8󰀄D

4+5d−6d2+d3

󰀅−

1G2M2

(4c4+c8)ε2.

(45)

Inspectionof(45)showsthatitcontainsthetermproportionaltob3,whichisabsentintheLu-Wisepaper.ThiscanbeeasilyunderstoodasLuandWiseignoredtheGauss-Bonnetterm,which,infourdimensions,doesnotaffectblackholesolutionofthefieldequations.Itaffects,ofcourse,theactionitselfandconsequentlytheentropy,leadingtoappearanceofaconstantterminSthatisindependentofM.

Finally,letusrestrictvaluesofthecoefficientstoitsLovelockcombinations.Aftersomema-nipulationsitcouldbeshown,that(41)reducestoasimpleexpression

rdS=

+ωd

3α3

r2d(d−1)+

+

4GD

n󰀇

mnd

=1

13

V.

FINALREMARKS

Sofarwehaveconsideredtheboundaryconditionsofthesecondtypeonly.Now,weshallbrieflyexamineappropriatesolutionconstructedwiththeaidoftheconditions(17)andψ(∞)=0.Sincebothsolutionsarenotindependentonecantreatthesolutionoftheonetypeastheusefulcheckoftheother.BeforeweproceedfurtherletusanalysesomegeneralfeaturesofthefunctionM(r).Natureoftheproblemandourpreviousanalysissuggeststhatthesolutionhastheform

˜(r)+C1,M(r)=M

(48)

˜(∞)=0andM˜(r+)=rd−1/2−C1,and,consequently,thetotalmassofthesystemaswithM

seenbyadistantobserver,M,canbeobtainedfrom

M(∞)=C1=

8πGD

2a

(d−2)(d−1)

rd+3

2d−2r+

3d−3r+

a2

(d−2)(d−1)

22

2d−4r+

+A2(r)

a

+

󰀁

−2dc3+

1

4

(d−1)c8

󰀃

(d−2)(d−1)(d+1)2

−−

ac71

󰀒c

3

󰀄󰀅

(d+1)3−3d−2d24󰀄󰀅󰀕232+25d−24d−7d4

14

Similarcalculationscarriedoutforthefunctionψ(r)yield

ψ(r)=

2−+

(d−1)(d+1)

2

3

r2d+2

.

r2d+2

󰀄󰀅2c72−2d−d

(53)

Since,byassumption,theradiusoftheeventhorizonistreatedastheexactquantitynow,thethusderivedlineelementmaybeeasilyemployedinconstructionoftheentropy.First,observethattheHawkingtemperaturecalculatedwiththeaidoftheEq.(30)isgivenintermsofr+anddependsonallrelevantcoefficients.Ontheotherhandtherelation(31)isindependentofb1,b2,c4andc5.Thisbehaviourcanbeascribedtothepossibilitytoremovethedependenceofthelineelementonthisverycoefficientsattheexpenseofmodificationsoftheequationsofmotionoftestparticles.Indeed,itcouldbedemonstratedthatbymeansofthefieldredefinitionoftheform

gab→gab+εAab+ε2Aab,

where

Aab=q1Rgab+q2Rab

and

Aab=q1R2gab+q2RRab+q3gabRcdefRcdef+q4gabRcdRcd

c

+q5RacRb+q6RacdeRbcde,

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(1)

(1)

(1)

(1)

(2)

(54)

(55)

(56)

onecanremoveallthetermsintheactionexcepttheseproportionaltotheparametersb3,c7andc8.Thecoefficientsqi

(1)

andqk

(2)

canbedeterminedbysolving,ateachorderoftheexpansion,

theappropriatesystemsofequations.Astheresultofthefieldredefinition(54),oneobtainstwoadditionaltermsR2RandRab2Rab,whichcanalsoberemovedfromtheactionfunctional.Itshouldbenoted,however,thatsuchtermsappearnaturallyintheeffectiveactionofthequantizedmassivefieldsinalargemasslimit[48,49,50].

Thegeodesicequationafterthefieldredefinitionbecomes

󰀓j

dxkdxd2xi(1)

Ajk;m

ds2ds󰀑

dxk1(2)2im

+εgAmk;j−

ds

(1)Ajk;m2

󰀓

dxj

ds

=0,(57)

15

wheresistheaffineparameteroftheoriginalmetric.Now,onecanrepeattheargumentsofRef.[20].BothMandTcanbemeasuredatinfinityanddonotdependontheparticularformoftheequationsofmotion.Consequently,thetemperaturemassrelationisindependentoftheremovedterms.However,todeterminetheradiusoftheeventhorizononeperformslocalmeasurementsandtheequationsofmotionoftestparticlesareimportant.

Itcouldbeeasilyseenthat,asexpected,theentropyispreciselythesameforbothchoicesofboundaryconditionsandisdescribedbytheformula(41).Sincethecalculationsforbothtypesoftheboundaryconditionshavebeencarriedoutindependently,thisequalitymayberegardedastheadditionalconsistencycheck.

APPENDIX

InthisAppendixwelistsomeformulasusedinSectionIV.TheexpansionsofthecomponentsoftheRiemanntensorandsomeofitscontractionsuptothefirstorderinε,whicharenecessaryincalculationoftheentropy,aregivenby

Rtitj=Rrirj=−

(d−1)

42ar+

󰀄󰀅j

+Oε2,δi

(A.1)

Rijkl=

1

22r+

−εb3

3(d−2)(d−1)d(d+1)

42ar+

󰀄󰀅

+Oε2

(A.4)

and

R=−εb3

(d−2)(d−1)d(d+1)

16

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