Black holes and conformal mechanics

Blackholesandconformalmechanics

arXiv:hep-th/0104175v1 20 Apr 2001S.Mignemi

DipartimentodiMatematica,Universit`adiCagliari

vialeMerello92,09123Cagliari,Italy

andINFN,SezionediCagliari

WeshowhowthemotionofachargedparticlenearthehorizonofanextremeReissner-Nordstr¨omblackholecanleadtodifferentformsofconformalmechanics,dependingonthechoiceofthetimecoordinate.

Recently,ithasbeenshownthatthemotionofachargedparticlenearthehorizonofanextremalReissner-Nordstr¨omblackholecanbedescribed(throughdimensionalreductiontoadS2)byamodelofconformalmechanics[1],which,forgreatblackholemassreducestothemodelofDeAlfaro,FubiniandFurlan(DFF)[2].Whenquantized,theDFFmodelhasacontinuousspectrumofpositiveenergyeigenstates,butnonormalizablegroundstate.

In[3,4],theabsenceofagroundstatewasinterpretedasduetoawrongchoiceofcoordinatesintheblackholemetric,whichdonotcovertheentiremanifold,andadifferentchoicewasproposed,thatsolvestheproblem.ThenewtimecoordinatecorrespondstoadifferentchoiceofconformalgeneratorsasHamiltonianfortheconformalmechanics,andgivesrisetoa’regularized’versionoftheDFFmodel,possessinganormalizablegroundstate[2].

Althoughanalgebraicproofofthisfactwasgivenin[3,4],itwasnotshownhowtoderivethenewHamiltonianfromthemotionofachargedparticleinthenear-horizonReissner-Nordstr¨ombackground,andinparticulartherelationbetweentheparametersoftheblackholeandthoseoftheregularizedDFFHamiltonianremainedobscure.

Inthisletter,wederiveexplicitlytheregularizedDFFHamiltonianfromthemotionofachargedparticleinasuitablyparametrizedadS2spacetime,inthe’non-relativistic’limit,bytakingintoaccounthigherordercorrectionsintheinversemassoftheblackhole.Wealsodiscussafurtherpossiblechoiceoftimecoordinate,whichhoweverleadstoaspectrumofenergyunboundedfrombelow.

ItiswellknownthattheextremeReissner-Nordstr¨ommetricinthenear-horizonlimit,r/M≫1canbeputintheBertotti-Robinsonform:

ds2=−

󰀍r

󰀂2

dr2+M2dΩ2,

(1)

r

whichisadirectproductadS2×S2.Themotionofatestparticleinthisbackgroundcanbestudiedbyconsideringthe2-dimensionalanti-deSittersection[1],namely

ds2=−

󰀍r

󰀂2

dr2.

(2)

r

ThisprovidesamodelofconformalmechanicsinwhichtheSO(1,2)isometryoftheback-groundspacetimeisrealizedasaone-dimensionalconformalsymmetry.Theso(1,2)alge-braisgeneratedbytheKillingvectorsh=∂t,d=t∂t−r∂r,k=(t2+M4/r2)∂t−2tr∂r,whichobeythecommutationrelations,

[d,h]=−h,

[d,k]=k,󰀎

r

[h,k]=2d.(3)

Defininganewcoordinateq=−2M

ds2=−

󰀑2M

themetric(2)transformsinto

󰀂2

dq2.

q

2

(4)

ThehamiltonianofaparticleofmassmandchargeQinthisbackgroundis[1]

H=

󰀑2M

m2

+q2p2q

2

󰀅

p2q

q2

󰀈

,(6)

wherethecouplingconstantgisgivenby8M2(m−Q).

ItisknownthattheDFFmodelhasnogroundstateanditsspectumiscontinuous.Thisisduetothefactthattheso(1,2)generatorhassociatedtotheHamiltonian(6)isnoncompact,andcanalsobeunderstoodasaconsequenceofthescalinginvarianceoftheDFFmodelwhichdoesnotallowachoiceofascalefortheenergy.Fromtheblackholepointofview,theabsenceofagroundstatecanbeinterpretedasduetotheexistenceofafixedsetfortheKillingvector∂t,correspondingtoitsKillinghorizon[3].Thisproblemcanberemediedbyadoptinggloballywelldefinedcoordinates,

u=arctan

2rt

2

(r−r−1+rt2),

Inthisparametrization,thetimelikeKillingvector∂ucorrespondstothecompactgener-atorofso(1,2),∂u=h+k.Itisknownthatthisgeneratoradmitsadiscretespectrumwithwell-definedgroundstate[2].

InordertoobtaintheHamiltonianforachargedparticlewhichisconformalintheappropriatelimit,wemakeanotherchangeofcoordinates,whichcaststhemetricintheform,inspiredby(4),

ds2=−A2(x)du2+A(x)dx2.Thisisobtainedbydefining

x=

󰀌

dv

M

intermsofwhichthemetrictakestheform

󰀄󰀍

v

ds2=−

M

󰀏2

+1

󰀆−1

dv2.

(7)

Unfortunately,thisisanellipticintegral.Inordertoobtainaclosedformforthemetric,weexpandtheintegralforM/v≪1(near-horizonlimit),keepingtrackofthefirstordercorrectionsinM/v.Itresults

󰀐󰀑󰀄󰀏4󰀆3M23

x≈−2M1−,1−

vx22M

3

󰀉2

+1

󰀃3/4.

andthen

ds2≈−

󰀑2M

5󰀍x

󰀂2󰀄

1+

1

2M

󰀏4󰀆

dx2.

(8)

x

TheHamiltonianforachargedparticlemovinginthisbackgroundisgivenby

H≈

󰀑2M

5󰀍x

󰀑

m2+

x2p2x

5󰀍x

p2x

x2

2

+ω2x2,

󰀂

(10)

whereg=8M2(m−Q),ω2=(m−Q)/10M2.ThishastheformoftheregularizedDFF

Hamiltonian,whichwasintroducedin[2]inordertoobtainadiscreteenergyspectrumwithnormalizablegroundstate.Theparameterωactsasaninfraredcutoff,whichbreaksthescalinginvariance,settingthescalefortheenergy.IntheapproximationofgreatM,ω2≪g.In[3,4]wasarguedthat(10)shouldberelatedtothemotionofachargedparticleinthebackground(7),butnoexplicitderivationwasgiven,andinparticularthevalueofωwasleftundetermined.Thespectrumofenergycannowbeobtainedfromtheresultsof[2]andreads:󰀅󰀐󰀐

1

2n+1++En=

5m

r−r−1+rt2

Inthesecoordinatesthemetrictakestheform

ds2=−

󰀄󰀍ρ

󰀏2

−1

.

M

󰀆−1

dρ2,

(11)

andthetimelikeKillingvector∂τcorrespondstothenon-compactso(1,2)generatord.Thisparametrizationhasnotbeenconsideredpreviouslyinthiscontext.

Asbefore,onecandefineanewcoordinate

σ=

󰀌

dρ

M

󰀉2

−1

󰀃3/4≈−2M

󰀒

ρ

󰀑

1+

3M2

intermsofwhichthemetrictakestheform

ds2≈−

󰀑2M

5󰀍σ

󰀂2󰀄

1−

12M

󰀏4󰀆

dσ2.

(12)

σ

TheHamiltonianforachargedparticlemovinginthisbackgroundisgivenby

H≈

󰀑2M

5󰀍σ

󰀑

m2

+σ2p2σ

5󰀍σ

p2σ

σ2

2

−ω2σ2,

󰀂

(14)

wheregandωaredefinedasbefore.WehaveagainaregularizedDFFmodel,butnow

theharmonicpotentialhasthewrongsign,leadingagaintotheabsenceofagroundstateandtoaspectrumunboundedfrombelow[2].ThisisinagreementwiththefactthattheHamiltoniancorrespondsinthiscasetoanon-compactgeneratord.Fromtheblackholepointofview,thiscanagainberelatedtothepresenceofahorizonatρ=M.

References

[1]P.Claus,M.Derix,R.Kallosh,J.Kumar,P.K.TownsendandA.VanPreoyen,

Phys.Rev.Lett.81,4553(1998);[2]V.DeAlfaro,S.FubiniandG.Furlan,NuovoCimento34A,569(1976);[3]G.GibbonsandP.Townsend,Phys.Lett.B4,187(1999);[4]R.Kallosh,hep-th9902007.

[5]D.ChristensenandR.B.Mann,Class.QuantumGrav.9,1769(1992);M.Cadoni

andS.Mignemi,Mod.Phys.Lett.A10,367(1995).

5