The exact decay rate for emission of massless minimally coupled scalar fields
by a non-extremal black hole in 2+1 dimensions is obtained. At low energy, the
decay rate into scalars with zero angular momentum is correctly reproduced
within conformal field theory. The conformal field theory has both left- and
right-moving sectors and their contribution to the decay rate is associated
naturally with left and right temperatures of the black hole.Comment: 11 pages, Latex, References added, to appear in Phys. Lett. 

Bañados

Callan

Carlip

Coussaert

Cvetič

Danny Birmingham

Gubser

Horowitz

Ichinose

Ivo Sachs

Maldacena

Rabenstein

Siddhartha Sen

Steif

Strominger

Physics Letters B

English

arXiv

Crossref

Three-dimensional black holes and string theory

The exact decay rate for emission of massless minimally coupled scalar fields by a non-extremal black hole in 2 + 1 dimensions is obtained. At low energy, the decay rate into scalars with zero angular momentum is correctly reproduced within conformal field theory. The conformal field theory has both left-and right-moving sectors and their contribution to the decay rate is associated naturally with left and right temperatures of the black hole

Birmingham, Danny

Sachs, I.

Sen, Siddhartha

DIAS Access to Institutional Repository

DIAS-STP-97-IO 1 IntroductionThree-Dimensional Black Holes and String Theoryl)oiinv Ba iuiIyl1uiIL1iiivcrsitij Colieqe Dab/ut. I) partuietit of Mathenuttical PhysicsBelfield. Dublin f IrelandIvo Saulis2Dablin Institute fur Adrut red Studies. School of Theoretical Physicsit) Barlinyton Road. Dublin f IrelandSiddliaitlia Seii3Trinity Colleqe Dublin. Department of Mathematics,Dablin 2. IrelandAbstractThe exact decay rate for emission of titassless minimally coupled scalar fieldsby a iion—extreinal black hole in 2 + 1 dimensions is obtained. At low energy.the decay rate into scalars with zero angular momentum is correctly reproducedwithin conformal field theory. The conformal field theory has both left— amid right—nioving sectors amid their contribution to the decay rate is associated naturallywith left and right temperatures of the black hole.,lulv 1997Reeemii lv, it has become clear that a large tInts of extiemmmal timid mieai—extiemmmal blackholes allow for a cumiloimiial held tlmeon ui effective stIimlg tlieav dcsciiptiomi. Exticimialblack holes oftemi eorrespomid to BPS—states of an umniemIving fumidammmental stummig I lieorvAgieermmemit between the l3ekensteimi—Hawkimig entropy amid the coummtimmg of stmimlg statesfor extiemmmal black holes iii five dimmiemisiomis was first uhtanied iii [1]. However, tincorrespommdenee (hues hot seeimi to he lest rid ed to ext remnal black holes. Indeed. tInemmtropv of neam—ext renmal black holes is uftemi coimi phi clv (lescrihed liv au effective si rnigheuiv [2-[ I’Hit aUnt Ui humit, tlit dectiv lulls ((I IIIJIHt\t 011111 lieok lit1 en have also newainnied. flHs involves studyimig the a hsor1>tmon oh Ilanta lu the black hole, and lieuallowing it to evaporate, via IIawkiilg radiation, ark to exti’eniality. In [hi—[9jlow’ energy scattering cross seetiomis amid (hera rates ui’ a mnassless mnimmmmnallv coupledscalar field were commiputed for a large class of Ibum— amid hve—diiiiemismomial black holesamid agreemmiemmt was l’oumid wit Ii conformal held theory ui effect ive st rimig theory prs I ii:—tions. In each of these cases the result relied on a particular niatchimig of soint iomms. iiia region miear the black hole horizon and ami asymnptol ic region far froni the black hole,For certain ranges of paraimmeters inherent to tIme problem, thus matching agrees with acon formnal field theory deseriptiomi.In this paper. we strmtlv the propagatiomm a mnassltss mnimmimnahlv coupled sealar held iiitime backgroummd geometry of time (2 + 1)—dimmmensiommai Bafmados—Teitelboim—Zammelh blackhole [10]. Tins black hole is described by two paraimmeters, its mnass !tI amid angularmoinemmtum J. 1mm addition, the metric has commstammt negative curvature, amid is tituslocally isomumetric to amiti—de Sitter space. Time special feature here is that die waveequatiomi can be solved exactly. without amiv approximrmations [11]. TIns allows tms todeternmimme exactly time range of emiergy and amigular mnomnemitumn of time scattered field.for which time time decay mate agrees with time commformnal hieltl theory description. \Vefind agreememmt fot’ energies small iii comparison to the size of time black hole, and totime curvature scale of the spacetimne; in additioum, one is restricted to time zero angularmomemmttmmn wave. In this regiomm. however. agreemmmeumt is foumid for all values of .1/amid J. aumd thus time commfornial fleiti theory descriptiomi is not restricted to a mmciii—extrermuah hunmit. Apart from that, we fluid 1)ehlaviour simimilar to that observed in fivedirnensiouis, mianmely that time comiformnai fiehi thmeoi’v has both left— amid righmt—mmiovimigsectors. The correspondimg decay rate is theum written imatnrallv imi terimus of left andright temimperattn’es of time black bole.Similarly to time 5—D black holes, time BTZ black hole is a sohimtioii of st rimig theory[12. 13]. Time strimmg scatteriumg off BTZ black holes has heeim comisidered in [11. 151 (seealso [16]).2 The BTZ Black HoleEmimimil: damimiybii’ollaimmlm.ui’d.ie2Emmmail: ivotnstp.dias.ie(Euilulil semistmimmmtlms.tcd.meGeometrically, thmmee—dimnemisiomial amid—dc Sitteu’ space cami be u’epm’esemmted as the SL(2. R)group space. Isonmetries au’e then represemited liv eletmients of time grormp SL(2. R) xSL (2. R)/Z9 where tile two copies of 5L (2. R) act b left and rigilt multiplication TheBTZ black ilole is obtained as the quotient space SL(2. R)/((pj,. p,)). vllere ((pi PR))denotes a certain finite subgroup of SL(2. R) x SL(2. R)/Z2 generated liv (Pr. PR) [1.We choose Scliwarzsclnld—like coorchnates in \vhlcll tile BTZ metric reads [10. 18]— (.V )2d1 ± /(II — .2 ((/( — Vc’dt)-. (1)(Vitil lapse nid siHit hitictiuiis and clldili ill(tFlV\‘ i(l/ )*The Illet nc (1) is singular when r = r. whereTile 31 = —1. J 0 metric may lie recognized as tilat of ordinary ant ode Sitterspace: it is separated by a mass gap from the 31 = 0. J = 0 “mnassless black hole.whose geometry is discussed ill Refs. [17] and [19]. For convenience, we recall that theHawking temperature TH. the area of the event horizon AH, and the angular velocityat tile event horizon cH are given byr2 —r2 I=+ . . = 27rr. = —. (5)2irL-r 2i7The BTZ black ilole is part of a solution of low energy string theory [12. 13]. Tilelow energy string effective action isI Jd.n G R t 4V5V”o H1’0). (6)where o is the chilaton. is an antisvimnnet nc kalb—R anlondi field. wlHcil Ill threecli nensions mmiust be llOp0lt jollal to the vol lmmmle forum c0,. amid k is tile coslllologicalconstant. It was observed in [12. 13] that. ill three dimnensiolls. tile illl5atzH0, 7f 1) 0. k = 2 (7)reduces the equations of motion of (6) to the Einstein field equations satishecl by (1). Infact. there is a correspomldlillg exact solution of string theory, the SL(2. R) \VZ\V modelwit.ll an appropriately chosen central cilarge describes the propagatioml of strings. Byquotienting out tile discrete group ((pi.ps)) by Illeamls of am orbifolcl constructioml. omwobtains a tileorv tilat may be sllowml to he au exact string t ileoretical represelital loll oftile BTZ black llole [12. 13].3 The Wave Equationit lois 1)1(11 kllowml for 501110 111111 I hut ilw 1111111110db (ii( pled ocular held -ojilut lull CullI salved -‘xaetlv 10 I lie Iiulckgrulllld geulliel lv ((I I lu /11 / Idack hole [11 lii is willnih cv 0 (let (ill 1111’ tIll Swill II lug -u sn SC 1 (II (11(1 IvIC rate II lu li 1 In hiexactly. StlhsIlItitloll of tile llotni( (1) 11110 11w duvurll(111 i.i(placmallU =—dqq’Plends to tile scalar v’u’e equal iouIf)w 11 erei.e..2)(3)(‘clZ Jr÷IV.i2=,2_,,:_,:..4=3f_. (1(1)Tilis suggests the allsatz1’(r. t, p) = R(r. w. trl)e ()(( ((1(5 (11)ieadiiig to tile radial equation for R(r)1 9r ..Juir 4,u2OR(i)+ (— + .i i.) PrR() + i ( — +(12)Cilamlgimlg variables to .1 = r2. the radial equation hecoilles(.t: — .r )(,r — .r÷)0 R(.r) ± [2.r— .c — .r P. R(.i) K(.r) R(1) 0. (131\\iuereV t2 / V .J,u 1i1(4f2 —1-———). lh)We imitrochuce a further cilange of variables h delimlimig2 3(15) rftd11w i’adial equation then hietwniies \vlleie X = .r .c - 1. lien. using 11w fact, that ohs real. we hod tlw total flux )w’Iuchiis independent of ) to be given by— :)i) H):) - (I :. ( B) H):) 0. (16) mii). 2l)(1 ±,In oi’der ii (‘Oiiif(Ul I III’ 1 ii jil ioU (TI (oIl, 01 1100(1 (1 (liVide 2 I (V I he( Ii ) B1 —— ( —j ) ( IT) n iit flux ii niliiiii \ liii uiul ii 1 I i ui ii I 0 6 II o u 0L1 I 0 , ‘hI 11 / solution of tlw loran [20]“11w livpergeoiiietric form of (16) beconws explicit upon relmioving 11w pole in the last.terni through the ansatz :O( () (,(J=yl):) 0 = (18) where p /r. A second linearly nidepenclent solution of (12) at inhoity is t lien given\Ve then havez(1—z)!J(c) t (2n ± 1)(1 * z)Oq(z) + (.4 t Bi)g(z) 0. (19) 02() (in!J’ ± .iu((!J) log(j). (26)In the neighbourhood of the horizon. z = 0. two linearly independent, solutions are thengiven by F(a. b. c. o) and z “CF(a c + 1, b —“ c + 1. 2 — c. z). where where I is a constant. Up to second order in p. we haveb = 2n Oi (ii) = p u (p) = I— ( L p Iog(p)tib = B1. (20) -c = 1 —i— 2w. The distinction between ingoing and otitgoiiig waves is coniplicated by a lie fact t hat t 1wBTZ—spacetin’ie is not asvniptoticalhv flat.. For a taclivon field. nigoilig and oucgoiaigNot.e that C = + b t 1. waves have been defined iii e.g. [15]. A naive extrapolation to niassless Odds is nutsensible as the resulting ingoing and outgoing waves are given by Oi and u an (2T)which both have vanishing flux. I-low-ever, we can cichne ingoing and outgoing waves4 rp DeCay T{ate to be complex linear coIrlI)niatid)iis of u and u which have positive and miegat ive flux.respectively. This leads toWe choose the solution which has ingoing flux at the horizon . namnel.R1 41 R.41,(l-R(c) = z°F(u.b.c’.V). (21) .2jTo see t los. we note i hat the conserved flux 11w (12) is given. lip to an irrelevant where c is sonic positive chimawnsionless comist oil . wlachi we t ike to he intlepeiolenl ofnorniahisation. by time frequency . We note 1 lie coniparisomi lore w-mi Ii t lie near region helnivioui’ oh 11winigoing and outgoing solutions in t lie four—i lniwnsmonal case )eqn . (2.22) iii ) . FOeingoing flux is correspondiiglyF (RL9’rR RA)rR). (22)‘2 “ .= 87w).4Hwhere ..\ = r . I lie flux can be evaluated by noting that4 5The asymptotic behaviour of (21) for large r is readjU available [21). and we canthen inatih tins to (28) to determine the coeflicients :11 and .4g. We findl(n t b 1)- i)F(b-i- 1)F(ui-b-t-I) flog(_\ /(2)±c e-r 1)+c(b--1)--t(1)—c’i7 Z JTh }where i is I lie (Ilgainnia function. We can estnnate 11w relative importance of the IOUterms ni (30) as killiws. Firstly, we note that,, 2 (o.b=— “° =2. (31)Using F(z + 1) 2F(:). we findF(0) 1 F(e + 1)F(b + U]2ci =——A--- (33)1)]2In order to relate the partial wave cross section to I 1w plane wave cross sect iouwe need to divide ci”° by . [22]. We hndF(o I )F(bF)o S ÷ Iwe 1101 f (150 II I ioU 1 ) .4/ 1 .2 . I lie i o IIluil-eNlie111l) Ia lo huh is he siven 0F°‘=TffAJ.[ ‘c”]F(u+ I)F(bw f)1—I= 42(2‘TT’ F (i + F (i± ‘TT)Hwhere the left and right temperatures are defined b4 ——F(u ± b± I)—!of1u)r.. o(o22 ,it2)T(a ± 1 )F(b ± 1) 1 2 — ±[iog( /j2)Furthermore.t(ti±I)±t’(b1)— t(i) — i(2)]}. (32)= 2(U— r.)(w — iii). b= 2(r+ r) ii,). (33)If itt = 0 and o << nnn( _L . the difference .4 — .4 in (32) is small compared to thesum -t + .4g. so that1 F((I+b+1)2F(a± 1)F(b+ 1) (31)TIns approximation means that. the Compt.on wavelength of the scattered particle ismuch bigger that. the size of the black hole and the scale set by the curvature of theanti—dc Sitter space. Note that the logarithnnc term in (32) ensnres that the term inbraces is finite for all values of l/r2, i.e.. all values of 1f and .1. In partictular, theextreme hunt .1 = ±31(. or eqtuuvalentlv _\ . = 0. is well defined. Hence. for iii = ü andj small in the sense defined above. the approxiniat ion (3—1) should he valid. For in 0on the other hand. there is no obvious choice of black hole paranuet ers for which theterm in braces in (32) is small. and hence (3.1) is modified for io 0.Let us consider the in = 0 wave and assume .‘ << nnn(i. ) so that (3-I) is valid.Then tIme partial wave absorptioui cross section is given liv= T (i±hz). (3±and we have used AH 2irr+.We now compare tins decay rate with the conformal field theory predict ion. -Isexplained in [5. 7. 23], in the effectiye string theory picture such decays are describedby a coupling of the spacetilne scalar field to an operator with dimension I in tieconformal field theory. both iii the left— and righit—nuoving sector. TIH5 might have heeiexpected since the \VZW—model of which the B TZ- I lack hole is a classical solutioncontains both chiralitmes [13] .alculation closely a nalogous to that presented iii 7shows the emission rate is given by2T 2 /2,smmih(2irTg+). smmili(2cilacijPerfornnng the u integrals we t lien obtain, till to an undet ermnimie( I miumneriraf i’oust. a tot.F 1t2,0Tf,rer_ (i 1T) F (i÷1z.) ) It))w’Imere L2 has been included for dimnemosmomoaf reasons. amid oo fact or I/’ os-onuiL, hotime nornmahisation of the outgoing scalar [7]. Hence. 1 lie (‘F T—predi(.’t ion ) 10) and I liesennclassical decay rate (37) agree. Note that the above amoalvsms is valid For all yahoo’sof 41 and J. subject to the low energy restrict ion. Thus. for t lie t horee—dimitemisionalblack hole the CFT description is itot restricteol to tfw miear—extremne. or near- B115’limit we recall that an extreme BTZ—I ohack hole is also a BPS configuration. as shiooomi1,, F(o)F(b)6ill [24] [9] S. Giiiiser all(1 I. I\leballov. Foui’Dineiisioiia1 GievhoIv Factois aiid t 1w ElketiveNote also that siipersviiiiiiet.1v has hot I)1ve(.l a role iii the calculatioiis 1)r(se11tec1 Stiiiig. Pliys. Rev. Lett. 77 (1996) 4491. 1iept1i/96O9OT6.lieie. In the four-diiiieieiuiia! ease.suIwLs11ln1et1y a)pea1e(1 iiidiuect ly iii deter- [1 0] i . I3añaIs. C. Teitellmiiii. alici .J . Zaitelli. T1i- 131ak 1I()Ie [ii IILU(e I)i1[Iensii1111111II1, the i 1e itit ( uiitoi iii 1 fieh I t I1( oi I low ci iicii iii au tiE iii i Ia 5) ( ( j 11I1( PII\ S h( \ Let t 69 (1 992 ) I ‘ 19 lL 1 t ii /J2( )3-diiiieiisioiial ae. SiiiiilaiIv. iii live (IiIllensioIIs. hosoiis and ieiiiiioiis iiiiist 1)roI)agate.(HI the ellettive stiiig iii uI(lel I u le(i)v( I he leeav rates for even aiid odd angular iiio [1 1 I. Ii:Iiiiioe I11( V. Stoli. Llltlt)1)l(-S UI -uolar FIel( oh Iliree I)iiiwiisiujoii IiwkIUClltèI alike. taiii. tlii- fi1s iii life (Il1U(I1SI(J11S )t(al1SE of the l(StIittlUIl II! ::::O. (iii I1t)l(S. Niiu1. 1livs. 1-117 (19D) 3—10. lop—i Ii U1121 1 1.lie other 10111(1. ii lois heeii sIiovii [2 1 that t lie BTZ-hlak hole a(11111t 01 siipcivin110 tIX Ill ilo \11(1Il ii ( iiIi 11 U 01(1 tO( 11l)1lIl1I1etIl( ifliiitli Iliiuu If_ti [12 \ k luio i \1i ii ul ili FIii Piii i I Ii 1 RI ol 11u1 PIi h I ) H I i ). , . •)(J.lu lit L0119 U IIh1V H \It\\I(l lu IuiLi1d iii it t I I )-i1S siIpIIaII\IL\- hULl 1 1100 1 (1)0(1 1 I hL I uiIi V Li 0 (0 tlit B I /-h1wk IjuIv jt I lu )Il(•L I .3 ( . ui o ii , i itt I ). \\ ll Si iiuu I I I ott ii I i I Ii - I Jiiit ring [13 . Iliis s1igitsts I hat an iii1sis siiiiilar to that pieseoted here iiiiglii go B1ak I lole. PI1\S. Be\. lJ1 I . t9 ç 199.1 1 .32N. 101)—I ii J.U2 I 2b.tI11t)IlLl1 11)1 1 lit 1)1O(k si liog . \. (jlleStlt)I1 that ailses nat iiia1I here is lio tlit Ilow CiiCik\ (1( ( ix i ites oinp ile I oi Ilk t irlooii tli ir[lertioii ( oeth( lelils Ii xe lo [14 k (1ioiu1 11(1 L I iis Ii 11 iol iiiI 111101 1110 (11(1 SI iiit Sr ttl ii iiobtained in [25j for the black st ring. and in [1-1. 15] ir the case of the black hole. — 1( 1) 0. s. tt. .3_u 19J 11[l5. [. NalsuliHie. N. Sakai. and M. Saw. String ScllItcriilg off lie 2 I)-diiiteioniuai1{olatmg Black Hole. .\imi. Plis. Lett. All) 1996) I 197. ini—Iii,’9d03l3.AcknowledgementsD B would like to di ink tht I litoi Di isioli tt CERN lot liospit lhtl duiiiig thc [lb S H on C I-I L intl I H ‘ire Hlw 1 in. i oh 111011 bout t.2—i ) iluint ioioui IIconupietioii of this work. hole. Pits. Lett. B322 (199-1) 182.[17] M. Bai’iados. ‘ii. Henneaux. C. 1’iitelboiui. and 2. Ziiui1Iu. (lioiiiet rv of I lie 2—Il3lack Flole. Pbs. Rev. 13-18 (1993) 1506. gr—qc/9302012.References[18] S. Carlip. The (2—i—I )—Diniensional Black Hole. Class. Qnant. C ray. 12 (1995) 2853,[1] A. Strominger and C. Vafa. Microscopic Origin of the Bekenstein-Hawking En- gr-qc/95060i9.tropv. Phys. Lett. B379 (1996) 99. hep-t.li/9601029. . . .—‘ [19] A. Steif, Snpergeoinetry of Three Dnneitsioiial Black Holes. Phvs. Rev. 053 (1996)[2] See, for example. C. Horowilz. Quantum States of Black Holes. gr-qc/91040i2. 5521. liep-tli/9504012.and references therein. . ,[20] AL. Rabensteni, Introduction to Oi’dz’nai’g L)z/jcrciitial bquutio as. Acadenuc[3] C. Horowitz and A, Stroininger, Counting States of Near-Extremal Black Holes. Press. New York. 1966.Phvs. Rev. Lett. 77 (1996) 591. hep-th/9602051. [21] M. Abrainowitz anti LA. Stegun, ed. Handbook of Aia//icinatical fraiiction.s. Dovcr.[4] C. Cahlan and 1. Maldacena. D—brane Approach to Black 1-hole Quantum ‘ilechan— New York. 1972.ics. NucI. Phvs. B172 (1996) 591. liep-th/9602043. [22] 5. Das. Cl. Gibbons and S. ‘ilathiur, Univcrsafitv of i.ow Lncrgv Absorption Cross-[5] S. Das and S. Matliur. Coniparing Deca Rates En Illack Holes and D—branes. Sections for Black Holes. Phvs. Rev. Lett. 78 (1997) -117. ltep—t 1/9009052.Nuch. Phvs. B478 (1996) 561. liep-t.li/9606185.[23] S. Cubser. Can the Efiective String See Higher Purl al Waves’!. he1 (—thi/’90 1105.[6] 2. ‘iialdacena and A. Stronnnger. Black Hole Gi’eybody Factoi’s aiid D—hrane Spec—ti OS( opi Phi Hr i io (1997) bb i lit p I hi/0609020 [24] 0 C ouss lul intl ‘ii hit nnc iux Snp isi unto lii of tin ç2 t 1) Bunt noon il 11111 IHoles. Phvs. Rev. Lett. 72 (199—1) 183. lup—tli/93 1019 I.[7] .1. Nlaldacena and A. St rontinger. Universal Low—Energy Dvnanncs for Rotating131 irk hole, hop tli/970201 [25] N SE tsos C onloi Hi ilk Lx itt Results 1i 81)2 H) 80)1 1) Oi I I) (uuModels. NucI. Plivs. 13389 (1993) 42-1. hep—thi/92000 1$.[8: ‘ii. Cveti9 and F. Larsen. Geiiei’al Rotating Black Holes in String Theory: Grey—body Factors and Event Horizons. hep-thu/9705192: Grevbodv factors for RotatingBlack Holes iii Four Diniensioiis. Iiep—thi/9i060i18 9

Three-Dimensional Black Holes and String Theory

Three-Dimensional Black Holes and String Theory

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