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FortschrittederPhysik,27January2012 Asymptotically Lifshitz Black Holes in Einstein-Maxwell- Dilaton Theories JavierTarrio InstituteforTheoreticalPhysicsandSpinozaInstitute,UniversiteitUtrecht,3584CE,Utrecht,TheNether- lands. 2 E-mail:[email protected] 1 0 2 Receivedxxx,revisedxxx,acceptedxxx Publishedonlinexxx n a J Keywords Holography,Lifshitzspacetimes,holographicrenormalization. 6 We study Einstein-Maxwell-dilaton theories with a cosmological constant and U(1)N gauge symmetry, 2 consideringmetricsasymptoticallyapproachingtheLifshiftzmetric.Westudythedependenceofthephase diagramonthevalueofthedynamicalexponent.Alongtheway,weapplyholographicrenormalizationand ] h proposeacountertermvalidforarbitrarydimensionanddynamicalexponentinoursetup. t - p Copyrightlinewillbeprovidedbythepublisher e h [ 1 Context andsetup 1 v ThisnoteisconcernedwiththestudyofblackholesolutionstotheEinstein-Maxwell-Dilatonsystemwith 0 negative cosmological constant, and in particular solutions in which the time and the space coordinates 8 4 scaledifferently: t λzt,xi λxi ,wherez iscalledthedynamicalexponent. Thiskindofscalingis → → 5 observedinmanycondensedmattersystems,andhasbeeninvestigatedwithgreatdetailfromaholographic 1. perspectivesincetheseminalwork[1],inwhichoneincludesanextraholographiccoordinate,scalingas 0 r λ 1r,withaboundaryinwhichthefieldtheorylives.Thepreciserelationbetweenthefieldtheoryof − → 2 interestandthedualholographictheoryisnotyetknown,andaconstructionoftheholographicdictionary 1 forthistheoriesisundercurrentinvestigation[2,3,4]. : v In[1], theauthorsproposedthatgravitationaltheoriesrealizingtheLifshitz symmetryshouldasymp- i toticallyapproachtheso-calledLifshitzmetric X ar ds2 =ℓ2dr2 r2zdt2+ r2d~x2 , (1) r2 − ℓ2z ℓ2 d−1 whichhas1+d(d+1)/2Killing vectors,relatedto thegeneratorsof thesymmetryalgebrain thefield theory(spatialrotations, spatialtranslations, time translationsand dilatations)[5]. Such a metriccan be obtained in Einstein gravity only in the presence of matter fields. Most of the solutions constructed in the literature consider a system in which gravity is coupled to a massive vector field, described by the Procaaction.IfthemassofthevectorfieldhastheappropriatevaluethemetricisasymptoticallyLifshitz. The caveat with the Proca approach to asymptotically Lifshitz black holes is that only a few solutions are knownanalytically (andfor specific valuesof dimensions, d, and the dynamicalexponentz), and in top-downapproachesthesolutionreliesonnumericcalculations[6]. Tofindagenericsolutionford > 2 and z 1 it has proven useful to consider instead the Einstein-Maxwell-Dilaton action with negative ≥ cosmologicalconstant[7]. Followingtheholographicphilosophy,theconstructionofblackholesetupsasymptoticallyapproaching (1)areinterestingtodescribethermalfieldtheories.Furthermore,iftheeffectsofaglobalU(1)symmetry inthefieldtheoryaretobeincluded,onehastoconsideragaugefieldinthebulkofthed+1-dimensional Copyrightlinewillbeprovidedbythepublisher 2 J.Tarrio:LifshitzBlackHoles gravitationaltheory.Withthisinmindwestudytheaction[8] N 1 1 1 S = dd+1x√ g R 2Λ (∂φ)2 eλiφF2 +S , (2) −16πGd+1 ZM − " − − 2 − 4 i=1 i # GH X wherethe secondtermis the Gibbons-Hawkingterm, neededto have a welldefinedvariationalproblem forthegraviton.Belowwewillseethatadditionalboundarytermsmustbeaddedtoobtainafiniteon-shell action. ThesetermsandtheGHtermdonotaffecttheequationsofmotionderivedfromtheaction,which presentasolutiondescribingachargedblackhole/brane ℓ2 dr2 r2z ds2 = b dt2+r2dΩ2 , (3) bk r2 − ℓ2z k k,d−1 b = 1+k d−2 2 ℓ2 m +N−1 ρ2jµ−q2dz−−11ℓ2z 1 , (4) k d+z 3 r2 − rd+z 1 2(d 1)(d+z 3)r2(d+z 2) (cid:18) − (cid:19) − j=2 − − − X A′1,t = ℓ−z 2(d+z−1)(z−1)µq2(dz−−11) rd+z−2, (5) A′j,t = ρjµp−q2dz−−11 r2−d−z , (j =2,··· ,N −1) (6) 2k(d 1)(d 2)(z 1) (d−2) A′N,t = ℓ1−z √−d+z− 3 − µ√2(d−1)(z−1) rd+z−4 , (7) p − eφ = µr√2(d−1)(z−1), (8) wherewehavesetΛ = (d+z 1)(d+z 2)/2ℓ2 andµistheamplitudeofthescalarfield. Notice − − − thatinthemetricwehavegeneralizedthespatialsectiontoadmitd 1-dimensionalmaximallysymmetric − manifoldswithpositive,nullornegativecurvatures(k = 1,0, 1respectively). Inthecasek = 1one − − ofthegaugefieldsacquiresaphaseandwedonotconsideritfurther. Thecasek = 0canbeobtainedby takingthelarge-sphere-radiuslimitfromthek =1case[9]. Wewillfocusinthelastcaseinthefollowing, keepingk explicitinsomeexpressions. FromnowonwewillsetN = 3forconcreteness. Inthez 1 → limit,inwhich(1)goestotheAdS metric,thissolutiongoestotheAdS-Reissner-Nordstro¨msolution. d+1 Thismodelhasbeenusedtostudytheconditionsthatstronglyinteractingfermionsshouldobeytosatisfy theARPESsumrules[10]. ThecaveatthissolutionpresentsisthebehavioroftheA gaugefieldsandthescalarφ. Thesediverge 1,3 attheboundary,andthevariationrelativetotheKillingvectorξ = zt∂ r∂ +xi∂ isnotvanishing, D t r i − thereforespoilingtheasymptoticLifshitzsymmetryalgebra. Wewillnottrytomakesenseofthesefields in this note, and therefore the model we use must not be considered as a dual to a non-relativistic field theory,butasanIReffectivetheorythatmustbecompletedintheUV. From the solutions (5) and (7) we can define charges associated to the badly behaved radial electric fields (see equation(15) below). These chargesare fixed in termsof d, z and µ (contraryto the case in equation(6),whereitisgivenbyafreeparameterρ ρ). Thissuggeststhatthesetwofieldsshouldbe 2 ≡ consideredin anensemblein whichtheir chargesarekeptfixed. The originalEinstein-Maxwell-Dilaton actionisappropriateforanensembleinwhichthechargescanvary[9],andthereforeitmustbemodified. Inotherwords,theboundaryconditionforthegaugefieldsimposedattheboundaryofspacetimeisnota Dirichletboundarycondition,butNeumann. Toaccountforthiswehavetosupplementtheactionwitha boundarytermtomakethevariationalproblemwelldefined.Thisisaccountedby 1 1 1 S˜=S− 16πG ddx√−hnµ 2eλ1φA1,νF1µν + 2eλ3φA3,νF3µν , (9) d+1 Z∂M (cid:18) (cid:19) whereh istheboundarymetricandnisaunitvectororthogonaltor=constanthypersurfaces. mn Analternative(equivalent)courseofactionistoperformaLegendretransformoftheactionwithrespect to the electric fields underconsideration[11]. This transformationchangesthe equationsof motion, but Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 3 (3)-(8) is still a solution, only that the fields A and A are not present in the new setup, since the t,1 t,3 solutionhasbeenalreadypluggedintotheLegendretransformedaction. 2 Holographicrenormalization On-shell action As usual in holography, the evaluation of the on-shell action (9) diverges when one evaluates it at the boundary, and a renormalization procedure is needed. Since the publication of the solution (3)-(8), severalpapersappeareddealingwith the holographicrenormalizationof asymptotically Lifshitzspacetimes[2,3,4]. These,however,focusontheProcaaction,andwewillnotconsiderthemin thefollowing.Indeed,therenormalizationwedescribeinthissectioncanbeunderstoodfromtheprocedure forasymptoticallyAdSspacetimeswithafewchanges. To construct the needed counterterm action in asymptotically AdS spacetimes one has to solve a Hamilton-Jacobiequation[12],whichcanbedonesystematicallybyexpandingthenewterminaderiva- tives series. For a counterterm depending only on gravitational terms this accounts to an expansion in powers of the boundary Ricci scalar R , contractions of the boundary Ricci tensor R Rmn, the (h) (h),mn (h) boundary Riemann tensor, and generic combinations of them. Here we denote the boundary metric as h . Thenumberofcountertermstoadddependsonthedimensionalityofthetheoryunderconsideration. mn Generically,termswithRα (andcombinationsofαboundaryRicciscalars, RiccitensorsandRiemann (h) tensorswithallindicescontracted)willbeneededifthefieldtheorydimensionisd=2+2α. In the present case such terms will regularize a field theory dimensionality d = 3 + 2α z. This − implies that to regularize the theory under consideration for generic z we need to consider an infinite numberofcounterterms. Thisis an impossibletask, since forincreasingvalueofα thecombinationsof curvaturequantitiesincreaseinnumber(see[13]forthegenericexpressionuptosixthorderinderivatives inasymptoticallyAdSspacetimes). Thereisonepropertyofoursolutionwecanusetocircumventthis,though.Focusinginthek =1case, theboundarymetrichasaR Sd 1 topologyandisstatic, implyingthattheboundaryRiemanntensor t − × takes a very simple form: any component with an index along the t direction vanishes, and the spatial partisthatofamaximallysymmetricd 1-dimensionalmanifoldwithpositivecurvature. Inparticular, − this means that any contraction of α boundaryRiemann tensors, Ricci tensors and Ricci scalars will be proportionaltoRα ,implyingthatthecounterterm,expressedpartiallyon-shell,isapowerseriesinR . (h) (h) ForfixeddandzwecantruncatetheR seriesandfindwhatarethecoefficientsweightingeachpower (h) oftheboundaryRicciscalartocanceldivergences. Whenseveralexamplesareworkedout,wecanfinda genericexpressionforthecoefficientinfrontofeverypoweroftheboundaryRicciscalar,forarbitraryd andz.AssumingthatthesecoefficientsarethefirstcoefficientsoftheinfiniteseriesinR ,weinducethe (h) formofthenth coefficientaccompanyingRn intheseries,andre-sumit. Afterthisisdonewefindthe (h) counterterm1 1 d 1 (d 2)ℓ2R S = ddx√ h − 1+ − (h) . (10) ct −8πGd+1 Z∂M − ℓ s (d−1)(d+z−3)2 Thiscountertermpreciselycancelsthevalueoftheon-shellaction(9)inthecasewherenoblackholeis present, meaningthat we could have defined it by covariantizingthe on-shellaction (using he boundary Ricci scalar) in the zero temperature, zero chargesetup. This is how the z = 1 case was foundin [13], andshowsthatthecountertermisvalidinanynumberofdimensionsandforarbitrarydynamicalexponent z 1,despiteitbeingderivedintheinductivewaydescribedbefore. ≥ 1 ThiscountertermintheasymptoticallyAdScase,z=1,wassuggestedin[14]ford=3,andin[13]itwaswrittenforgeneric d. Copyrightlinewillbeprovidedbythepublisher 4 J.Tarrio:LifshitzBlackHoles Forthethermalcaseweobtainafiniterenormalizedon-shellaction,S˜ = S˜ +S +S , ren on shell GH ct − evaluatedatacutoff(andtakentotheboundary) S˜ren = 16πβGVd−1ℓ1+z m(z−2)−2(z−1)rhd+z−1− 2(d−2()d2+(zz−2)3ℓ)22rhd+z−3 . (11) d+1 ! − OnecouldhavealsoconsideredtheLegendretransformedactionwithrespecttothephysicalchargeρ. Itcanbecheckedeasilythatthesamecountertermdoesthejobinthiscase,sincetheLegendretransform isequivalenttotheinclusionofaboundaryterm(seeequation(9)andthecommentsaroundit),whichin ourcasegiveafinitecontribution.Afterthisconsideration,theon-shellLegendretransformedactionreads S˜˜ren = 16πβGVdd+−11ℓ1+z m(2d+z−4)−2(d+z−2)rhd+z−1− 2(d−d2+)2zℓ2rhd3+z−3! . (12) − Internalenergy Thetrick used in the renormalizationofthe on-shellactionis notusefulwhenwe try to renormalizethe stress-energytensor in the field theory. The reason is that we have used the fact that topologyoftheboundarymetricisR Sd 1 toexpresscurvatureinvariantscomposedbytheRiemann t − × tensor,theRiccitensorandtheRicciscalaraspowersoftheRicciscalaralone. However,thecalculation oftheboundarystressenergytensorinvolvesavariationoftheactionwithrespecttotheboundarymetric, andthisresultwillbesensitivetotheprecisecombinationofcurvaturetensorsthatwehave.Despitethese considerations,letuswriteheretheBrown-Yorktensorasderivedbyconsideringthecounterterm(10) 1 d 1 (d 2)ℓ2R (h) T = K Kh + − 1+ − h . (13) mn 8πGd+1 mn− mn ℓ s (d 1)(d+z 3)2 mn! − − Inprinciple,T shouldhavecorrectionsinvolvingtermsproportionaltocontractionsofcurvaturetensors mn withtwofreeindices(forexampleR orR Rpq ).Asanexample,thelengthyexpressionone (h),mn (h),pmqn (h) getsconsideringthemostgenericcountertermuptosixderivativesoftheboundarymetriccanbeseenin [15]. Now, thestaticity ofoursolutionimpliesthatanycomponentofthe Riemanntensorwithan indexin the time direction vanishes. This in turn means that the tt component can still be evaluated, since the correctionsto(13)cancelforthiscomponent.WiththeevaluationofT wecancalculatetherenormalized tt energydensityinthefieldtheorystraightforwardly,obtaining V m(d 1) E = dd−1x dethijkmξnTmn = 16πdG−1 ℓ1+−z , (14) Z d+1 p with deth involving only the spatial direction components of the boundary metric, ξ = ∂ a Killing ij t vectorandk the unitvectororthogonalto t = constantsurfaces. Thisresultcoincideswithe the Komar massresultusedin[8]bysubtractingareferencebackground. 3 Thermodynamics Thetemperaturecanbedefinedfromtheinverseoftheeuclideantimebytheabsenceofsingularitieswhen performinga Wickrotation,givinganexplicitexpressionT = T(r ,ρ,µ)thatcanbefoundin[8]. The h massparameterm 0canbetradedbytheradiusofthehorizon,whichisdefinedbyb (r ) = 0. The k h ≥ entropyisgiven,asusual,bytheBekenstein-HawkingformulaS =Vd 1rhd−1/4Gd+1. − ThechargecarriedbythegaugefieldA canbeobtainedfromtheGausslaw,andthechemicalpotential 2 associatedtothischargeinthefieldtheoryisobtained,asusualintheholographiccontext,fromthevalue ofthecorrespondinggaugefieldattheboundary.Thesecalculationsgive Q= 16πG1 eλ2φ∗F2 = V1d6−π1Gℓz−1ρ , Φ=A2,t(∞)= ρdµ+−qz2dz−−311rh3−d−z , (15) d+1 Z d+1 − Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 5 wherewehaveusedtheusualboundaryconditionA (r )=0. 2,t h AnanalogouscalculationforthechemicalpotentialassociatedtothebadlybehavedfieldsA andA 1 3 impliesa divergentvalue. However, as we work in an ensemble in which the chargeassociated to these fields is kept fixed, explicit factors of these chemical potentials will not appear in the thermodynamic relations. Thermodynamicpotentials Withtherelationsgivenaboveitisstraightforwardtochecktherelation W =TI =E TS QΦ, (16) ren − − withT thetemperature.Actually,thedifferentialthermodynamicrelationsinthegrand-canonicalensemble (inwhichQisallowedto varyandthepotentialΦisfixed)arealso satisfied. We thenconcludethatthe renormalizedon-shellaction(11)(timesthetemperature)correspondstothefreeenergyofthefieldtheory inthegrand-canonicalensemble. FortheLegendretransformedaction(12)onefinds F =T(I˜ I˜T=0)=∆E TS , (17) ren− ren − and looking at differential relations the identification with the free energy in the canonical ensemble is realized.Thisconfirmstheeducatedguessin[8],wherethesepotentialsweredefinedfromther.h.s.ofthe previoustwoequations,withoutgoingthroughtherenormalizationprocedure. In(17)wehavesubtracted theT =0valuetomakecontactwiththeanalysisinthatpaper. Here∆E =E E istheenergywith ext − thevaluecorrespondingtotheextremalsolutionsubtracted. Phasediagrams Weshowthedependenceofthephasediagramswiththedynamicalexponentinfigure 1,forboththecanonicalandgrand-canonicalensembles. For z < 2 the situation is analogous to the AdS-Reissner-Nordstro¨m case discussed in [9] (which correspondsto the z = 1 case in this note). There is a line of first orderphase transitions in the grand- canonical ensemble separating the thermal Lifshitz solution (no black hole but periodic euclidean time) valid at low values of the temperature and the chemical potential, from the black hole setup. For large values of the chemical potential and vanishing temperature the dominating solution is a finite-entropy extremalblackhole,whichisexpectedtodecayintoanunspecifiedgroundstate.Inthecanonicalensemble thereisalineoffirstorderphasetransitionsbetweensmallandlargeblackholes,terminatingatacritical pointabovewhichwehaveacrossover. AtQ = 0wehavetheanalogoustotheHawking-Pagetransition [16]. Theshadowedregioninthisdiagrammarkselectricallyunstablesetups. Forz = 2thesituationinthegrand-canonicalcasestaysunchanged,butinthecanonicalcasetheline of firstorderphase transitionsgetsreducedto justthe criticalpointatQ = 0, i.e., to the Hawking-Page transition.Theelectricinstabilityalsodisappearsforthisvalueofthedynamicalexponent. Finally, for z > 2 the phase diagrams simplifies in both ensembles. In the grand-canonicalone the first order transitions disappear and the black hole setup (extremalblack hole setup in the T = 0 case) dominateseverywhereinthephasediagram,exceptatacriticalvalueofthechemicalpotential(forT =0) wherethedominatingsolutionisLifshitzspacetime. Inthecanonicalensemblewedonotfindanyphase transition. 4 Conclusions Wehavestudiedthethermodynamicsofthesolutionpresentedin[8]usingholographicrenormalizationto calculatethefreeandinternalenergiesofthesystem,relatedtotheon-shellactionandthettcomponent oftheboundaryBrownYorktensor,respectively. Thefactthatacountertermcanbegivenforgenericvaluesofdandz ispossibleduetotheextremely simpleboundarytopologyofthesolution,whichallowstore-sumtheinfiniteseriesofcounterterms. The Copyrightlinewillbeprovidedbythepublisher 6 J.Tarrio:LifshitzBlackHoles F F F F F c BH c F c BH BH Lifshitz Lifshitz T T T Q Q Q Tcrit THP T THP=Tcrit T T Fig.1 Sketchofthephasediagramsobtainedherefordifferentvaluesofthedynamicexponentz < 2(left),z > 2 (right)andz =2(middle)inthegrand-canonical(top)andcanonical(bottom)ensembles. mostimportantlimitationofthisprocedurebeingtheimpossibilitytocalculatethespatialcomponentsof theboundarystress-energytensor. InthisrenormalizationprocedureitisfundamentalthatwehaveperformedtheLegendretransformation, or equivalently the addition of the boundary term in (9), canceling the divergent behavior of the gauge fieldsA andA intheon-shellaction. Weexpecttorevisittheroˆleoftheboundary-divergentfieldsand 1 3 toprovideanextendedanalysisoftherenormalizationproceduresomewhereelse. Acknowledgements IwouldliketothankStefanVandorenforhiscomments,collaborationandinsightin[8]. This research is supported by the Netherlands Organization for Scientic Research (NWO) under the FOM Foundation researchprogram.IwouldliketothankalsotheFrontofGalician-speakingScientistsforencouragement References [1] S.Kachru,X.LiuandM.Mulligan,Phys.Rev.D78(2008)106005[arXiv:0808.1725[hep-th]]. [2] S.F.Ross,Class.Quant.Grav.28(2011)215019[arXiv:1107.4451[hep-th]]. [3] M.Baggio,J.deBoerandK.Holsheimer,arXiv:1107.5562[hep-th]. [4] R.B.MannandR.McNees,JHEP1110(2011)129[arXiv:1107.5792[hep-th]]. [5] S.A.Hartnoll,Class.Quant.Grav.26(2009)224002[arXiv:0903.3246[hep-th]]. [6] I.AmadoandA.F.Faedo,JHEP1107(2011)004[arXiv:1105.4862[hep-th]]. [7] M.Taylor,arXiv:0812.0530[hep-th]. [8] J.TarrioandS.Vandoren,JHEP1109(2011)017[arXiv:1105.6335[hep-th]]. [9] A.Chamblin,R.Emparan,C.V.JohnsonandR.C.Myers,Phys.Rev.D60(1999)064018[hep-th/9902170]. 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