Thermodynamics of rotating solutions in (n+1)-dime

arXiv:hep-th/0606237v21Jul2006Thermodynamicsofrotatingsolutionsin(n+1)-dimensionalEinstein-Maxwell-dilatongravityA.Sheykhi1,M.H.Dehghani1,2∗,N.Riazi1†andJ.Pakravan11.PhysicsDepartmentandBiruniObservatory,ShirazUniversity,Shiraz71454,Iran2.ResearchInstituteforAstrophysicsandAstronomyofMaragha(RIAAM),Maragha,IranWeconstructaclassofcharged,rotatingsolutionsof(n+1)-dimensionalEinstein-Maxwell-dilatongravitywithLiouville-typepotentialsandinvestigatetheirproperties.Thesesolu-tionsareneitherasymptoticallyflatnor(anti)-deSitter.Wefindthatthesesolutionscanrepresentblackbrane,withtwoinnerandoutereventhorizons,anextremeblackbraneoranakedsingularityprovidedtheparametersofthesolutionsarechosensuitably.Wealsocomputetemperature,entropy,charge,electricpotential,massandangularmomentumoftheblackbranesolutions,andfindthatthesequantitiessatisfythefirstlawofthermody-namics.WefindaSmarr-typeformulaandperformastabilityanalysisbycomputingtheheatcapacityinthecanonicalensemble.Wefindthatthesystemisthermallystableforα≤1,whileforα1thesystemhasanunstablephase.ThisisincommensuratewiththefactthatthereisnoHawking-Pagephasetransitionforblackobjectswithzerocurvaturehorizon.I.INTRODUCTIONItseemslikelythatgravityisnotgivenbytheEinsteinaction,atleastatsufficientlyhighenergyscales.Themostpromisingalternativeseemstobethatofferedbystringtheory,wherethegravitybecomesscalar-tensorinnature.Inthelowenergylimitofthestringtheory,onerecoversEinsteingravityalongwithascalardilatonfieldwhichisnon-minimallycoupledtogravity[1].WhenadilatoniscoupledtoEinstein-Maxwelltheory,ithasprofoundconsequencesfortheblackholesolutions.ThisfactmaybeseeninthecaseofrotatingEinstein-Maxwell-dilaton(EMD)blackholesofKaluza-Kleintheorywithcouplingconstantα=√3whichdoesnotpossessthegyromagneticratiog=2ofKerr-Newmanblackhole[2].Thus,itisworthtofindexactsolutionsofEMDgravityforanarbitrarycouplingconstant,andinvestigatehowthepropertiesofblackholes/branesaremodifiedwhenadilatonispresent.Alsoitisinterestingtoinvestigatethethermalstabilityofblackholes/branesinthepresenceofdilaton,andfindouttheeffectofdilatoninthe∗email:[e-mail]†email:[e-mail]holesolutionsofEMDgravityintheabsenceofadilatonpotentialhavebeenconstructedbymanyauthors[3,4].Thedilatonchangesthecasualstructureofthespacetimeandleadstocurvaturesingularitiesatfiniteradii.Theseblackholesareasymptoticallyflat.Inrecentyears,non-asymptoticallyflatblackholespacetimesareattractingmuchinterestinconnectionwiththesocalledAdS/CFTcorrespondence.Blackholespacetimeswhichareneitherasymptoticallyflatnor(anti)-deSitter[(A)dS]havebeenfoundandinvestigatedbymanyauthors.Theunchargedsolutionshavebeenfoundin[5],whilethechargedsolutionshavebeenconsideredin[6].InthepresenceofLiouville-typepotential,staticchargedblackholesolutionshavebeendiscoveredwithpositive,zeroornegativeconstantcurvaturehorizons[7,8].Recently,thepropertiesoftheseblackholesolutionswhicharenotasymptoticallyAdSordS,havebeenstudied[9].Theseexactsolutionsareallstatic.Chargedrotatingdilatonblackholeswithcurvedhorizonhavenotbeenconstructedinfourorhigherdimensionsforanarbitrarycouplingconstantandarbitraryrotationparameter.Indeed,exactmagneticrotatingsolutionshavebeenconsideredinthreedimensions[10],whileexactrotatingblackholesolutionsinfourdimensionshavebeenobtainedonlyforsomelimitedvaluesofthecouplingconstant[11].Forgeneraldilatoncoupling,thepropertiesofrotatingchargeddilatonblackholesonlywithinfinitesimallysmallcharge[12]orsmallangularmomentuminfour[13]andfive[14]dimensionshavebeeninvestigated.Forarbitraryvaluesofangularmomentumandchargeonlyanumericalinvestigationhasbeendone[15].Recently,twoclassesofmagneticrotatingsolutionsinfour-dimensionalEinstein-Maxwell-dilatongravitywithLiouville-typepotentialhavebeenconstructedbyoneofus[16].Althoughthesesolutionsarenotblackholesandrepresentspacetimeswithconicsingularities,chargedblackstringsolutionsinfour-dimensionalEMDgravityhavealsobeenconstructed[17].Tillnow,exactrotatingchargeddilatonblackhole/branesolutionsforanarbitrarycouplingconstantinmorethanfourdimensionshavenotbeenconstructed.Ouraiminthispaperistoconstructexact,rotatingchargeddilatonblackbranein(n+1)dimensionsforanarbitraryvalueofcouplingconstantandinvestigatetheirproperties.Specially,wewanttoperformastabilityanalysisandinvestigatetheeffectofdilatononthestabilityofthesolutions.Theoutlineofthepaperisasfollows.SectionIIisdevotedtoabriefreviewofthefieldequationsandgeneralformalismofcalculatingtheconservedquantities.InSec.III,weconstructthe(n+1)-dimensionalchargedrotatingdilatonblackbraneswithacompletesetofrotationparametersandinvestigatetheirproperties.InSec.IV,weobtaintheconservedandthermodynamicalquantitiesofthe(n+1)-dimensionalblackbranesolutionsandshowthatthesequantitiessatisfythefirstlaw3ofthermodynamics.InSec.V,weperformastabilityanalysisandshowthatthedilatoncreatesanunstablephaseforthesolutions.Wefinishourpaperwithsomeconcludingremarks.II.GENERALFORMALISMTheactionof(n+1)-dimensionaldilatonEinstein-Maxwellgravitywithonescalarfieldcanbewri