lecture23 ads black holes and thermal gauge theory; equation of state, free energy and stress tensor

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture23:BlackHoleThermodynamicsLecturer:McGreevyDecember2,2008Intoday’slecturewe’lldiscussthelawsofblackhol

2、ethermodynamicsandhowAdSblackholesarerelatedtofinitetemperatureCFTs,andKoushikwillgivearelatedpresentation.1LawsofThermodynamicsRecallfromlasttimethatforablackholeArea∼Entropyκ∼T(1)whereκwasthesurfacegravity.Thenear-horizonmetricisds2∼−κ2ρ2dt2+dρ2+...=κ2ρ2dτ2+dρ2+..

3、.(2)whenwegotoEuclideantimeτ≡it.Ifτhasperiodicityτ∼τ+2π/κthentheeuclideangeometryisregular.RecallthecanonicalensemblethermalpartitionfunctionisZ=tre−H/T(3)thwheree−H/Tpropagatesthesystemwithimaginarytimet=1/iT.Thermalequilibriumisequivalenttoperiodiceuclideantimewi

4、thperiod1/T,soweidentifyκwithtemperatureT.Thelawsof(stationary)blackholethermodynamics,analogoustotheusuallawsofthermodynamics,are:•0th(thermalequilibrium):κisconstantovertheeventhorizon.Thismeanstemperatureisconstantinspaceandtime.Thusstationaryblackholesareinther

5、malequilibriumwithconstanttemperature.Johnthinkstheproofofthe0thlawdoesn’tdependontheshapeoftheblackhole,aslongasitsastationarysolution.1•1st(conservationofenergy):κdE=dM=ΩdJ+ΦdQ+dA(+PdV)(4)8πGΩdJisthechangeinrotationalenergy,ΦdQistheelectricalenergy,andκdA=TdSis8π

6、Gheatexchange.Thislawrelatesthechangeintheenergy(orequivalentlymass)tochangesinvariouspropertiesoftheblackhole.ThelasttermdescribingmechanicalworkPdVisn’tpresentforblackholesbutISforblackbranes...•2nd(entropyincreases):Thisistheareatheoremforablackholeweprovedlastl

7、ecture,A˙≥0,sinceS=A.(ProofoftheexactrelationbetweenSandAinalaterlecture.)4�G•3rd(absolutezeroentropy):κ(orratherT)cannottakentozeroinafinitenumberofsteps.Thisdoesn’tmeanthatS(T=0)=0,butitdoesprobablymeanatT=0thereisaminimuminentropy.TheselawsfollowfromEinstein’sEqu

8、ation,theenergyconditionwediscussedlastclass,andassumingwehavestationaryblackholes.1.13rdlawSincewediscussedthe2ndlawlasttime,andthe0thand1stlaws