Black Hole Horizons and the Thermodynamics of Strings

UPR-769-T,hep-th/9708090

BlackHoleHorizonsandtheThermodynamicsofStrings∗

arXiv:hep-th/9708090v2 3 Sep 1997MirjamCvetiˇcandFinnLarsenDavidRittenhouseLaboratoriesUniversityofPennsylvaniaPhiladelphia,PA19104

Wereviewtheclassicalthermodynamicsandthegreybodyfactorsofgeneral(rotating)non-extremeblackholesanddiscussuniversalfeaturesoftheirnear-horizongeometry.Wemotivateamicroscopicinterpretationofgeneralblackholesthatrelatesthethermodynamicsofaneffectivestringtheorytothegeometryoftheblackholeinthevicinityofboththeouterandtheinnereventhorizons.Inthisframeworkweinterpretseveralnear-extremeexamples,theuniversallow-energyabsorptioncross-section,andtheemissionofhigherpartialwavesfromgeneralblackholes.

1.Introduction

Inthelastfewyearsaprecisecorrespondencebetweencertainblackholesandquantumstatesinstringtheoryhasbeenfound.Inparticular,thedegeneraciesofboundstatesofD-braneshavebeencalculatedandforlargechargestheresultagreeswiththeBekenstein-Hawkingarealawfortheblackholeentropy[1].TheconfigurationsthatallowforthepreciseargumentsareBPSsat-urated,i.e.theyaresupersymmetricblackholeswithzerotemperature.Itisinterestingtocon-sideralsothenear-BPSconfigurationsbecausetheyexhibitnon-trivialthermodynamicalprop-erties.Thereisagreementbetweenstringtheoryandthemacroscopicblackholepropertiesinthiscase[2,3]also,butthereasoningislesssecurelyfoundedinthemicroscopictheory.Itisthere-foreimportantthatthecountingofmicroscopicstatescanbesupplementedwithcomparisonsofdynamicalprocesses[4–6]:therateofHawkingdecayandthepreciseenergydependenceoftheradiationinthesemi-classicaltheoryisinagree-mentwiththatinthestringmodel.Theseresultsgivestrongsupporttotheidentificationofthe

2

Thiscontributionisbasedonworkpresentedin[11–14].Itisorganizedasfollows.Insec.2itisexplainedhowsomefeaturesofanunderlyingstringdescriptioncanbereadoffdirectlyfromtheclassicalgeometry.Insec.3wemaketheseideasexplicitinthecontextofthestatisticalme-chanicsofthemostgeneralfivedimensionalblackholes.WeshowthatpreviouslyknownresultsarerecoveredintheBPSandnear-BPSlimits;andwediscusssometestsoftheextensiontothegeneralnon-extremecase.Thefourdimensionalcaseismoreinvolvedthanthefive-dimensionalonebut,asdiscussedin[14],mostresultsnever-thelesscarryoverwithonlyminormodifications.AsexplainedabovethemostdetailedtestofthecorrespondencebetweenblackholesandstringsisthatthespectrumoftheHawkingradiationagreesinthetwodescriptions.Insec.4wede-velopthisapproachbydiscussingthefeaturesofthefieldequationofaminimallycoupledscalar.Insec.5weusetheseresultstofindthegrey-bodyfactorsoftheblackholesandhighlighttheirstringtheoryinterpretation.Thisprovidesatestofthegeometricpicturepresentedinsec.2.Insec.6wesummarizethesuccessesandthelim-itationsofthestringdescriptionofnon-extremeblackholes.

2.ThermodynamicsofStringsandGeom-etryofBlackHolesAcharacteristicfeatureofstringsisthattheirspectrumdividesintotwodistinctsectorsassoci-atedwiththeright(R)andleft(L)movingexcita-tions,respectively.Thespaceofstatesthereforedecomposesintoadirectproductoftwospaces.Consequentlytheentropy,calculatedfromthede-generaciesofthesestates,isthesumoftwocon-tributions:S=SR+SL.

(1)

TheconditionforthisdivisiontobemeaningfulisthatthecouplingbetweentheR-andL-movingmodesismuchweakerthanthecouplingwithineachsector.Underthesameconditiontwoinde-pendenttemperaturescanbeintroduced;TR,LforfortheR-andL-movingmodes,respectively.Thetemperatureofthecombinedsystemisrelatedto

thatofitspartsas:

TH−1=

1

4G,(3)

N

andtheHawkingtemperatureisrelatedtothesurfaceaccelerationattheoutereventhorizonκ+as:

TκH=

+

+

2

(

A4G),(5)

N

whereA±aretheareasoftheinnerandoutereventhorizons;andtheR–andL–movingtem-peratures:1

κ+

±2π

∂M

)anunfamiliarideathatQ,󰀡J󰀡.

Itistheinnereventhorizonplaysanyroleatall,asitiseffectivelyisolatedfromanoutsideobserver.Theultimatejustificationofthisideacomesfromitsapplica-tions;thecalculationspresentedinthefollowingsectionsindeedsupportthat,atleastformally,the“contributions”fromtheinnerandtheouterhorizonappearonequalfooting.

3.GeneralBlackHolesinStringTheory—theFive-DimensionalExampleAstheworkingexampleweconsiderthemostgeneralfive-dimensionalblackholeintoroidallycompactifiedstringtheory.ItdependsontheADMmass,M,threeU(1)chargesQi,andtwoangularmomentaJR,L.Weparameterizethesephysicalvariablesbythenon-extremalityparam-eterµ,thethreeboostparametersδi,andthe(bare)angularmomentumparametersl1,2intro-ducedthrough:M

=

1

2µsinh2δi

;i=1,2,3,(8)

JR,L=

1

4

(α′)4g2/(R1R2R3R4R5)=

π

3

gα′

(D1−branes)(10)

Q2

=

n2R1R2R3R4R5

R(KK−charge)(12)

1

Theseassignmentswillbeassumedfordefinite-ness,butmanyotherchoicesareequivalentbyduality2.Indeed,dualitytransformationsinthemaximalcompactsubgroupofthefulldualitygroupgeneratethemostgeneralblackhole[16]whentheyactonthegeneratingsolutiondefinedby(7–9)3.(Thisprocedurewasmadeexplicitin[18].)

Fromtheexplicitlyknownsolutionstheareasoftheinnerandouterhorizonscanbereadoff[15,11].Theentropiescalculatedusing(5)become:

SR,L=2π󰀅4µ3(Accordingtoourinterpretation󰀇icoshδi∓󰀇isinhδi)2−JR,L

2

(13)theseexpressionsshouldbeidentifiedwiththeentropiesoftheR–andL–movingmodesoftheunderlyingstringthe-ory.

TheR–andL–movingtemperaturesaresimi-larlycalculatedfromthesurfaceaccelerationsattheinnerandoutereventhorizons,usingeq.(6).Theyare:1

󰀅

󰀇

󰀇

(14)

4

µ3(

i

coshδi∓

i

sinhδi)2

−JR,L

2Thephysicalcontentoftheseformulaebecomes

clearerinvariouslimitingcasesthatweconsiderinthefollowing.TheBPSlimit:

Theextremalcasecorrespondstothelimitwhereδi→∞.Thislimitisonlyregularwhen

4

alsoJRofthethree→0.charges:TheBPSmassisgivenbythesumM=Q1+Q2+Q3;

(15)

sotheblackholecanbeinterpretedasamarginalboundstateofthreekindsofobjects.Thede-generacyofthiscompositeobjectisgivenbythe(exponentialof)theentropy[19]:

S=2π󰀅

n1n2n3−JL2.(16)Intheintermediatestepthequantizationcondi-tions(10–12)onthechargeswereused.Themod-ulicancelout[9,10,20]sothattheentropycan

beinterpreteddirectlyintermsoftheunderly-ingconstituents.NotethatSR=0andS=SL;sointheBPS-limittheblackholeentropydoesnotdivideintotwoterms.Accordingto(5)theR–movingcontributionindeedvanishesinthege-ometricinterpretationbecausethetwohorizonscoincide.

Thestringtheorycalculationthatleadsto(16)takesasitsstartingpointthesuperconformalfieldtheory(SCFT)withthetargetspace[1]:C=(T4)n1n2/Σn1n2,

(17)

andleveln3.HereΣkisthepermutationgroupofkobjects.Itactsontheproductmanifoldinanorbifoldconstructionandintroducestwistedsec-torsthatcontributefractionallytothemomen-tum[21–23].Intheblackholelimititisthesec-torwiththemaximalfractionationthatprovidesthemostimportantcontribution.Thecontribu-tionofthissectorcanbecapturedbytheeffectivelevel:

NL=n1n2n3−JL2

,

(18)

ofasuperstringwithtargetspaceT4,ormore

generallyaSCFTwiththecentralchargec=6(ˆcπof󰀃=4).Indeed,(16)isrecoveredusingS≃2an6N,validatlargeN.Althoughtheconcepteffectivelevelisapproximateingeneralitbecomespreciseintheblackholeregime.Itisusefulbecauseitcapturesthesymmetrybetweenthethreechargesrequiredbyduality.

TheangularmomentumistheU(1)compo-nentofthelocalSU(2)world-sheetcurrentof

theN=4SCFT[19].Theprojectionontothesectorwithaspecificangularmomentumismul-tiplicationbyanoperatorwiththeappropriateU(1)world-sheetcharge,andthescalingdimen-sion(conformalweight)ofthisoperatorisrespon-siblefortheeffectivesubtractionofJL2

ineq.(18).ExtremeKerr-Newmanlimit:

Arelatedanalysiscanbeappliedtotheex-tremeKerr-Newmantypeblackholesolutions.Thislimitisachievedbytaking(l1ThenSR=0,again,andthusS=−l2)2SLwhich→µ.takestheS=2π

󰀅

form:

δ3

4µe

∓2−JR,L2

.

(20)

Thecorrespondingentropiesindeedagreewitheq.(13)inthelimitwheretwoboostsarelarge.Thisverifiestheroleoftheinnerhorizoninthedilutegaslimit.Onelargeboost:

Thecasewhereonlyoneoftheboostsislarge,say,δ1≫1,canbemodeledsimilarly[25,26]:the

chargethatcorrespondstothelargeboostactsasabackgroundthatisinert,exceptthatitin-ducesthefractionation.Theexcessenergy,∆M,isdistributedamongthestatesinthespectruminawaythatissimilartothefundamentalstringwithbothmomentumandwindingcharge.Con-sideringfirstthestandardrelationofperturbativestringtheory:

Npert

R,L=∆M2)2=µ2cosh2(δ2Wealsotakeinto−(Q2account±Q3thefractionation∓δ3)(21)

andtheprojectionontoaspecificangularmomentumsector.Thentheeffectivelevelsbecome:

NR,L=Q1µ2cosh2(δ2∓δ3)−JR,L

2

.(22)

Thecorrespondingentropiesreproduceeq.(13)inthelimitwhereoneboostislarge,thusverify-ingtheroleoftheinnerhorizonintheregimeofonelargeboost.

Notethatthiscaseincludesgeneralnon-extremeblackholesintheinfinitemomentumframe.Thismayberelevantforthedescrip-tionofblackholesintheframeworkofM(atrix)-theory[27].

Thegeneralcase:

ThegeneralexpressionfortheBHentropy(13)canbeaccountedforquantitativelybyanon-criticalstringwithcentralchargecR=cL=6,excitedtotheeffectivelevels:NR,L=

1

5

6

TL,(25)

whereListhe“volume”—length—ofthegas.

Thus,using(13)forSand(14)forT,thelengthLoftheeffectivestringbecomes4:

L=2πµ2(󰀂

cosh2δi(26)

i

−

󰀂sinh2δi).i

Thislengthscaleisindependentoftheangu-larmomenta,whichprovidesanothertestofthemodel.Namely,instringtheoryangularmo-mentaareimplementedasprojectionsontheHilbertspaceofstatesanddonotaffectthelengthofthestring.Moreover,itissatisfyingthatthesamestringlengthLisfoundintheR–andL–movingsectors.LreducestoL=2πn1n2RintheBPS-limitbutthegeneralexpression(26)isalsovalidfornon-BPSblackholes.ThefactthattheeffectivestringlengthLincreaseswiththesizeoftheblackholeispotentiallyimportantfortheissueofinformationloss.Indeed,forlargeblackholesthescaleLismuchlargerthanthePlanckscale,wherestringeffectsareusuallyas-sumedtobecomeimportant.

Theseconsiderationsalsoapplytothestaticcaseswithoneormoreδi=0,includingtheSchwarzschildsolutionwithallδi=0.However,intheseexamplesSR=SLandTR=TL=TH;sothecharacteristicstringfeaturesofthethermo-dynamicsareabsent.Inthegeometricinterpreta-tionthisisaconsequenceofthedegeneratelimitofthevanishinginnerhorizonarea.However,thispresumablydoesnotimplyanylimitationinthestringdescription.Aplausibleinterpretationissimplythat,inthiscase,thereisanequilibriumbetweenthetwosectors.

6

4.TheWaveEquation—MinimallyCou-pledScalarFieldAdetailedtestofthecorrespondencebetweenblackholesandstringsisthespectrumofHawk-ingradiation.Aspreparationforthiscalculationweconsidergeneralpropertiesofthewaveequa-tionforaminimallycoupledscalarfield:

1

−g

∂µ(√2(r2+

+r2−)

2

andx=−1

4)∂xΦ0

+

1

R

x−1

κ)2

(30)

+−mΩR

κ+

L

−

1

(ω

2

κ+mΩ+

L

∂

r3

∂r

+ω2)Φ0=0.

(31)

ThisistheradialpartoftheKlein-Gordonequa-tioninflatspace.Thus,theterm14Mω

2

canbeinterpretedasthe

Coulomb-typescreeningduetothegravitationalfield.Atlargexthistermissuppressedrelativetoflatspacebyonepowerofx∼r2asexpectedforaCoulombpotentialinfivedimensions.

ThevariableΛistheeigenvalueoftheangularLaplacian.Ittakestheform:Λ=n(n+2)+O(ω2),

(32)

wherethecorrectionsO(ω2)areduetotherota-tionofthebackgroundandarediscussedin[13].

Thistermisalsosuppressedbyonepowerofx∼r2,asexpectedforanangularmomentumbarrier.

Thetermsconsideredsofararemanifestationsofthelongrangefieldsandtheflatasymptoticspace.Theremainingtwotermsdivergeattheouter(x=12)horizons,re-spectively,andsoarespecificforblackholeback-grounds.Themodesclosetotheouterhorizonbecome(takingmR=mL=0,andthusignoringtheeffectsofangularvelocities,ΩR,L,forsimplic-ity):Φ0∼(x−

1

2κ+

e−iωt.

(33)

Thebranch-cutaroundx=

1

2π,

awellknownresultfoundin

severaldifferentwaysintheseventies.

Themodesclosetotheinnerhorizonsimilarlybecome:Φ0∼(x+

1

2κ−

e−iωt.

(34)

Thesignificanceofthesemodesislessclearbe-causetheinnerhorizonisnotexpectedtohaveanyeffectsontheasymptoticobservers.How-ever,thereisastrikingparallelbetweenthetwohorizons.Itisthereforenaturaltosuspectthatthebranch-cutaroundx=−1

2π

combines

withtheHawkingtemperature,THtwonewtemperaturesT−1−1,and−1

forms

beattributedtheR–R,L=TandL–movingH±T−whichcanstringmodes,respectively.

HiddenSupersymmetry:

Forrotatingblackholestherearenotsuffi-cientlymanyconservedquantitiesthatthesepa-rationofvariablescanbeguaranteed.Itisthere-foreasurprisethatwewereabletodoso5.Ananalogoussurpriseiswellknowninthecontextoffour-dimensionalKerr-Newmanblackholes.Theretheseparationofvariablesisaconsequenceofconservedfermionicchargesthatarerelatedtothemorefamiliarconservedbosonicchargesbythesupersymmetryalgebra[30].Itisbe-lievedthatthesignificanceofthesupersymmetryinthiscontextisdifferentfromitsroleinparti-clephysics.However,itwouldbeinterestingtoreexaminethisquestioninlightoftherelationbetweenblackholesandsuperstrings.Stringsymmetries:

Thewaveequationintheregionclosetothehorizonshasasimpleformthatmaybeuniver-sal,astheidenticalequationappearsinbothfourandfivedimensions[14].Wenowexpressthisstructuremathematically.

WeintroducethedimensionlessRindlertimeτthatisaregulartime-likecoordinateclosetotheouterhorizon.Themonodromyaroundthecoordinatesingularityisencodedbyanimagi-naryperiod2πiTH

ω

.Withtheseauxiliaryvariablesthe

radial1

equation(30)(withouttheflatspaceterm

4)∂x

−

1

∂22

τ+

1

∂2

2

σ

(35)

withtheeigenvalue

1

5We

wouldliketothankG.Gibbonsforpointingthisout.

7

similardescriptionhaspreviouslyappearedinthecontextofexactconformalfieldtheoriesthatde-scribetwo-dimensionalblackholes,butthere-lationwiththepresentresultsisnotclear(seee.g.,[31]).

ThecompactgeneratorsR3andL3ofthetwoSL(2,R)R,Lgroupsarediagonalandtheireigen-valuesare:R3=14πTR,(36)L3

=

1

4πTL

.

(37)

Notethatthenaturalvariablesofthegrouparethesumandthedifferenceofτandσ,ratherthanτandσthemselves.Theauxiliaryvariablesτandσarelocalized“times”closetoeachofthetwohorizons.Thus,inadefinitesense,itisthesumandthedifferenceofthetwohorizonsthataresingledoutbythegroupstructure.Thisresultissatisfyingbecauseitispreciselythesecombinationsofthesurfaceaccelerationsthatweassignmicroscopicsignificance.

Thewaveequationexhibitsanobvioussymme-trybetweeninnerandouterhorizontermsthatcanbeexpressedintermsofthegroupgenera-torsasanautomorphismofthealgebrathattakesR3→R3andL3→−L3.InstringtheorytheT-dualitysymmetryactsinpreciselythisway.Itisintriguingthatsymmetriesthatarecloselyassociatedwithstringtheoryarerealizedexplic-itlythroughthewaveequationinthegeneralblackholebackground.However,wemustem-phasizetheirprecisesignificanceremainsunclear.5.TheGreybodyFactors

Wenowconsiderthesolutionsofthewaveequationdiscussedinsec.4,following[6,26,29,13]Atlargexweconsidertheradialequationinflatspaceandweincludetheeffectsofthelongrangefieldsfromtheblackhole.ThesolutiontothisapproximateequationisaBesselfunc-tion.Nextweconsidertheequationwithonlytheunperturbedenergy1

8

sult1

dependsonthevalueofthepotentialterms

(eω/TH2

2T−1)(e

−ω

1)L

d4k

eω/TH−11

2

2TL

Itisimportanttonotethat−1)(thee

ω

(2π)4

Bosedistribu-tionwiththeHawkingtemperaturecanceledout.ThereforethefinalresultdependsonlyonthequantitiesTR,LandLthathavesignificanceintheeffectivestringdescription.Infactthereisanexplicitmicroscopicinterpretationofthisfor-mula:theemissionistheresultofatwo-bodypro-cess[2,5,6,13].TheBosedistributionsarephasespacefactorsoftheR–andL–movingquantapropagatingonthestring.Moreover,theampli-tudefortheannihilationoftwoquantacollidinghead-to-headonastringoflengthLcanbecal-culated,usingonlytheNambu-Gotoformofthestringaction.Theresultofthiscalculationisidenticaltoeq.(39).Thustheagreementbe-tweenthespace-timecalculationandthemicro-scopicinterpretationinvolvesthefunctionalde-pendenceontheenergy,andallnumericalfactorsagreeaswell.

Themicroscopictwo-bodyinterpretationoftheemissioncanbeemployedasamodelwhenevertheformofthesemi-classicalresultisofthetype(38).Asufficientconditionforthisisthelow

energyrequirementMω2rprobe+−ris−)ω2solarge≪1;thatthusthethe≪targetwave16.Thisimplies(22cannotlengthbeofpos-theitivelyidentifiedasablackhole.Ontheotherhand,theconditionalsoimpliesLω4presumablyimpliesthatthetargetcannot≪1.beThisun-ambiguouslyidentifiedasastringeither.

(i)Animportantspecialcaseisthedilutegaslimit[6],definedinsec.3asδ1,2thelow-energyconditionMω2frequenciesω∼TR∼TL.Therefore≪≫11.issatisfiedInthiscasethecalcu-forlationissensitivetotheBosedistributionfactorsineq.(39).Thisverifiesindetailthatthestringtemperatureshavebeencorrectlyidentifiedinthedilutegasregime.

(ii)Anotherinterestingexampleistheregimeofrapidlyspinningblackholes[13],whichisob-tainedbytuningthebareangularmomental1,2,

definedineq.(9),sothatl2=0andµ−l2

µǫ2

1=ergy≪conditionµ.Asinisthesatisfieddiluteforgasω∼case,TRthelowen-factorsaresignificant.However,∼nowTLsotheBosetherearenoconditionsontheboostsδi;sothefunc-tionaldependenceofthetemperaturesTL,RandthestringlengthLonallthreeboostsδiistestedindetail.

(iii)Asafinalexample,considerthelimitofverylowenergieswheretheuniversalabsorption

cross-sectionσ(0)

abs=A+isvalidforallblackholes.Thetwo-bodymodelstillapplies[33]butthetestaffordedbythecalculationoftheemissionratesisweakerbecauseitdoesnotinvolvethefunc-tionaldependenceonω.However,theagreementstillinvolvesthedependenceontheindependentblackholeparametersµ,δi,andl1,2.Thisresultthusprovidesevidencethat,atlowenergies,theeffectivestringmodelappliestoallblackholes.Matchingonnon-zeropotential:

Thetwo-bodyannihilationsconsideredsofararethesimplestdecayprocesses,butmorecom-plicatedonesgiveimportantadditionalinforma-tion.Recallthatoneoftheconditionsfortheva-lidityofthetwo-bodyformoftheemissionrateisthatthepotentialvanishesintheregionwherethe

asymptoticsolutionandthenearhorizonsolutionarematched.Thisconditioncanbereplacedwiththemilderassumptionthatthetwosolutionscanbematchedinaregionwherethepotentialisanyconstant[26,29,13].Inthiscasethefinalresultfortheemissionrateintothen-thpartialwavebecomes:

Γ(4π(n+1)2󰀉2h−1

emn)

(ω)=×(40)

×Gh

TR

󰀄8π

ω2

󰀆d4k󰀆

(2πT)

2h−1e

−

ω

4πT

)|2

2

≡astheFouriertransform2)

has

aninterpretationofthecanonicallynormalizedthermalGreen’sfunctionsforaconformalfieldwiththescalingdimensionshR=hL=h[34]:

Gh󰀈

πTR

TR(z)=

9

2

+1

sothattheemissionvertexoperatorhasafreeconformalfieldtheoryrealizationoftheschematicform:

V∼:∂X(z)∂X¯(¯z)[Sa(z)]n[S¯a˙(¯z)]n:,

(43)

where::denotesthenormalorderingoftheoper-ators,Xarecoordinatesofthestringintheinter-naldirections,andSa,S¯a˙areworld-sheetfields

withconformaldimensions(hR,hL)=(1/2,0)and(hR,hL)=(0,1/2),respectively,andwhoseindices(a,a˙)specifyquantumnumbersofthespace-timespinors7.Theemissionisthere-foreinterpretedasamany-bodyprocessthatin-volves1bosonandnfermionsinbothR–andL–movingsectors.Lorentzinvarianceimpliesthatthecouplingofthisvertexoperatortotheout-goingfieldinvolvesnderivativesactingontheoutgoingfield.Thisgivesrisetoafactorofω2nintherateand,rememberingthenormalizationω−1oftheoutgoingwave,thecompletefrequencydependenceof(40)canbeaccountedforqualita-tively[29,35,36,14].Thus,forlargepartialwavenumbers,themicroscopicdescriptionintermsofaneffectivestringmodelaccountsfortheemis-sionratesinanarbitraryblackholebackground.Animportantunresolvedproblemremainsthecalculationoftheoverallnumericalcoefficientin(40).However,thisissueisnotspecifictothenon-extremecase.Itisexpectedthatthiscoef-ficientiscalculableintheBPS-limitandthatitagreeswiththeclassicalresult.Ifthisisborneoutitiswillalsobepossibletomodelthegeneralnon-extremeblackhole.

Thevertexoperators(43)arefermionicwhennisodd.Thereforethephase-spacefactorsas-sociatedwiththeinitialstateareexpectedtobe

10

oftheFermi-Diractype.InthiscasetheGreen’sfunctions(41)areproportionaltogammafunc-tionswithargumentswhoserealpartsarehalf-integral,whichindeedgivefactorsoftheFermi-Diracform(eω/2T+1)−1[29].Thisisaninter-estingexamplewheretheblackhole“looks”likeastringwithfermionicdegreesoffreedom.6.Conclusion

Weconcludebysummarizingtheaccomplish-mentsandtheshortcomingsoftheeffectivestringmodelfornon-extremeblackholes,startingwiththeformer:

•Theablesmodelintherelateseffectivethethermodynamic(weaklycoupled)vari-stringdescriptiondirectlytogeometricalfeaturesoftheblackholespace-time.•Theinagreementextremeandwithnear-extremethemodel.Thislimitsresultareprovidesastrongmotivationthatageneralnon-extremeblackholecanbemodeledbyaneffectivestringmodelaswell.

•Two-bodyquantitatively.processesSpecificallycanbethisaccountedresultgivesforamicroscopicinterpretationoftheuniversallow-energyabsorptioncross-section.•Many-bodyqualitatively.

processescanbeunderstoodDespitethesesuccessesitisappropriatetocon-cludewithsomecaution.Theunderstandingofnon-extremeblackholespresentedhereleavesmuchroomforimprovement:

•Thedamentaldetailedstringconnectiontheory,withalongawithspecificthefun-de-taileddescriptionoftheunderlyingSCFT,isnotclear.

•Thelimiteddescriptiontotheblackoftheholestringregimespectrumbecauseweisemployanapproximatenotionofaneffec-tivelevel.

•Currentoryarenotmodelssensitiveofblacktotheholesnontrivialinstringcausalthe-structureofblackholespace-times.

Itisnotclearwhethertheseobstaclescanbeover-comewithfurtherdevelopmentsoftheideasandtechniquesthatarepresentlyknown.

AcknowledgmentWewouldliketothankG.Gibbons,S.Gubser,G.Horowitz,S.Mathur,E.Verlinde,andH.Verlindefordiscussions.TheworkwassupportedbyDOEgrantDE-FG02-95ER403andNATOcollaborativegrantCGR949870(M.C.).WewouldliketothanktheAs-penCenterforPhysics(M.C.)andNORDITA(F.L.)forhospitalitywhilethemanuscriptwasbeingprepared.REFERENCES

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