extreme a = p 3 black hole and a singular object with opposite (i.e. neg- CH{1211, Geneve 23, Switzerland

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1 CERN{TH/96{36 hp-th/ BLACK HOLES AS BLACK DIHOLES AND ASSLESS QUADRUHOLES Tomas Otn C.E.R.N. oy Divion CH{, Gnv 3, Switzl Abstact black hols can b undstood as bound stats a (positiv Masslss xtm a = p 3 black hol a singula objct with opposit (i.. ng- mass) mass with vanhing ADM (total) mass but non-vanhing gavitational ativ) Supsymmtic balanc focs cucial fo xtnc th kind ld. bound stats xplains why systm dos not mov at spd in spit bing masslss. W also xplain how supsymmty allows light ngativ mass as long as it nv olatd but in bound stats total fo mass. known masslss black-hol solutions should n b non-ngativ paticula cass \gavitational dipols". W also psnt \gavitational considd quadupols" commnt on possibl ol all s objcts in sting phas tansitions. ERN{TH/96{ buay addss: tomas@suya0.cn.ch oy, bsids lmntay stings, dscibs many intsting pointlik Sting o xtndd objcts with unbokn supsymmtis solitonic natu. to supsymmty, many poptis s objcts can b liably studid Thanks in famwok low-ngy supgavity oy. Extm black hols paticulaly intsting sting oy objcts, most intsting ( a amongst m a phaps masslss ons. i xtnc was mtious) by Stoming in Rf. [] in contxt typ II sting oy conjctud phas tansitions na conifold points whos signal would pcly b duality appaanc s masslss xtm black hols caying Ramond-Ramond which can condnsat in ctain cass []. Maslss black-hol solutions chag low-ngy htotic sting ctiv action w cntly dcovd in Rf. [3] also Rfs. [, 5, 6]). Som most makabl poptis s objcts (s a. i canonical mtic, which can b cast in fom [] ds = D D dt d~x ; () singula whn quals valu constant D. singulaity a singulaity aa sphs adius gos to zo in that cuvatu limit. Th mtic dos not sm to b xtm limit any black non-xtm. mtic. hol If w xp gtt componnt mtic fa away fom singulaity, 3. gavitational ld wak ( mtic asymptotically at) w wh nd cocint gtt =+ D +3D 8 +O : () m, wh m ADM mass. n, tm ADM mass s objcts zo (hnc adjctiv masslss). limit, gtt +, wh Nwtonian gavitational potntial. th fo, All th ally maks nam black hols quit inapopiat fo m but w will stickto it fo momnt.

2 D 3D + 6 ; (3) has wakly pulsiv (instad attactiv) chaact whn acting usual tst paticls [5] g tt < 0 ; > D). If Nwtonian ap- on was valid na singulaity, w could immdiatly say that poximation pulsion gows without bound in its nighbouhood. A mo dtaild analys sms to conm th [5]. spit having vanhing ADM mass, y do not sm to mov at. light. Th mtic dos not admit any light-lik Killing vcto spd must conclud that whol ADM fou-momntum s objcts w vanhs. It n suping how, with zo total ngy actually momntum, somthing instad nothing. Whn y a ightly mbddd in a supgavity oy, y hav half 5. = o N= supsymmtis unbokn low-ngy solutions N dscibing m a also xact solutions sting oy. abov popty implis, as usual, that xt static mtics dscibing 6. many s objcts in quilibium. a canclation btwn gavitational o (scala vcto) focs. Rf. [5], it was obsvd that pulsiv foc that appas at a nit fom s objcts may b intptd as a gavitational intaction with stanc massiv co. Th ltt an invstigation into natu \massiv s \masslss black hols" fo which w will popos a modl. " will stat by stablhing a somwhat hutic analogy btwn lag W Eqs. (,3) multipola xpansions in lctostatics. If w w xpansions ld catd by som chag dtibution connd in a gion udying w had abov xpansions fo lag w would immdiatly say ac chag dtibution has no monopol momnt, that : total chag at Howv, th dos not man that no chag! It just mans that zo. a as many positiv as ngativ chags, but its numb cannot b dducd monopol momnt alon. xtnc tms high od in om to indicat that, in fact, numb positiv o ngativ chags not ms (that why whav a non-tivial ld). o analogy nds h, bcaus dipol high multipol momnta tms not sphically symmtic. Lt us consid 3, though, twochag dtibutions a connd into a gion, with positiv ngativ chag spctivly, sph- not symmtic concntic, such that total (nit) chags a qual ically opposit). If fall-o chag dnsity both dtibutions dint, (but nt chag dnsity () dint fom zo vywh but its intgal ov whol spac zo. n, nt chag containd in a sph adius Q() = Z S 3 d 3 x() ; () a function that gos to zo whn gos to innity. Applying Gauss' law gts following dpndnc on fo lctic ld on E() Q() : (5) if, fo instanc, Q(), n E() 3 lctostatic potntial Now, '. tm appas only whn Q() Q 0 + :::, in that cas a nt chag in whol spac. moal th modl that a total zo chag non-tiviality at lag dtancs a compatibl with sphical symmty if chag ld not connd. Th would imply fo ou objcts with zo ADM dtibution that y could b composd two concntic \chag dtibutions" with mass signs vanhing total \chag". \chag" gavity opposit gavitational ld itslf cais gavitational chag. n, fo ngy analogy to wok, it not ncssay to hav whol spac lld with th ngativ-mass matt. Th could b localizd in a gion in such positivway that gavitational lds poducd would not compnsat ach o a any point but innity. at with abov modl, it maks sns, n, to idntify D Compaing ctiv mass at a dtanc fom black hol, much in sam spiit an Rf. [5]. Obsv that th ctiv mass ngativ vywh but at innity, it vanhs. wh a a numb dicultis with abov hypos (namly that black hols a composit, not lmntay, objcts, mad som positiv masslss som ngativ mass objcts, bcaus two \chag dtibutions" mak sns indpndntly). Ft all, it usually thought that ngativ should masss in intaction with positiv masss always lad to masslss objcts 3 I am indbtd with Jog Russo fo a hlpful dcussion in which h poposd th modl to m.

3 at spd light. agumnt gos as follows: opposit masss fl oving focs, but a ngativ mass acclats in diction contay to foc pulsiv w nd up with positiv mass acclating till spd light followd d ngativ mass i.., aft som tim, a masslss systm that movs at light. Th systm dos hav positiv ngy, although st mass d Its oigin intaction ngy btwn objcts ( st mass zo. would cancl). gis agumnt would not b valid if was ano intaction btwn Th objcts such that sulting foc on ach m zo. would s conguations dscibing s two objcts in quilibium. Masss static ngis would cancl w would hav a masslss (zo-ngy) taction at st. On o h, xtnc additional chags would stm why no annihilation btwn positiv ngativ masss in plain chags caid by objcts do not add up to zo. ditional nxt diculty would b poducing such a static solution dscibing a mass objct in quilibium with a positiv mass objct (an xtm black gativ If a no-foc condition btwn m holds, it asonabl to xpct that l). solution will b supsymmtic. fact that on masss ngativ obstacl fo having supsymmty as long as total ADM mass not 5. no solution has nough unbokn supsymmtis, solution should If at last und static ptubations mtic. systm could stabl, b unstabl und tim-dpndnt ptubations, but th su should b ill fu aft w nd solutions. vstigatd w should look fo supsymmtic, xtm, multi-black-hol solutions. n, cntly, a makabl on has bn found by Rahmfld in Rf. [8] fo oy abov agumnt shows that th systm would b quit unstabl any ptubation poduc a unaway solution whos nd would b a masslss stat. uld It tmpting to idntify dint constants that appa in multi-black-hol solutions as 5 dint masss s objcts., though, no igoous way to assign a valu to ach individual black hol. only on asymptotically at gion only on ass mass, total mass, can b dnd. On can study initial-data sts dscibing N non- DM black hols which a not in quilibium in m a N + asymptotic gions tm individual total ADM masss can b dnd (s, fo instanc, [7] fncs d masss tun out to b mntiond constants plus intaction ngy tms. in). tms vanh fo static multi-black-hol solutions n it physically asonabl s constants with masss. Th idntication not igoous, though. It idntify among o things, that on can smoothly continuously go to xtm limit sums, spac mtics, which not known. Ano agumnt fo assigning a mass to ach hol basd in taking all o to innity, but, in that cas, spac would not ack at ADM mass black hol lft cannot b dnd. Still it asymptotically asonabl to mak th idntication w will us it to mak hutic asonings ysically ich will b justid by sults. 3 dscibd by following action 6 S = Z dx p g R (@) +(@) +(@) + h (+) (F () ) + ( ) (F () ) + (+) (F () ) + ( ) (F () ) io : (6) action a tuncation low-ngy ctiv action htotic Th [9]. paticula, fou-dimnsional dilaton. tuncation not sting constnt, though, only som solutions a indd solutions compltly complt action. Th impotant fom point viw sting oy supsymmty (s Rf. [0]). W will igno s sus in th pap, simply concntat on obtaining masslss black-hol solutions though, abov action with mtic (). solution givn in tms fou indpndnt hamonic functions H () ;K () ;H () ;K () (@i@ih =0; i =;;3) ds = U dt U d~x ; U = H () K () H () K () ; = H() H () F () ti = c F () ti = c K () K () ; = H() K () ; ~ () F ti = d () H ; ~ () F ti = d () H H () K () ; = H() K () () ; K wh constants c () ;c () ;d () ;d () tak valus H () K () ; () ; (7) K ~ F () = (+ )? F () ; ~ F () = ( +)? F () ; (8)? F Hodg dual F. Usually, H's K's a chosn to b positiv, that, all constants in xtictly 6 H w follow convntions notation Rf. kn:ko.

4 H =+ X n qn j~x ~xnj ; K=+X n pn ~xnj ; (9) j~x non-ngativ constants to avoid ocunc singulaitis in mtic, fo any positiv o ngativ valu constants on gts a solution t, with som ngativ q's o p's a what w a aft. Baing th lutions following Rf. [8], lt us consid, fo simplicity, solutions fom ind, () = + q j~x ~xj ; K() = + p j~x ~xj ; H () + q j~x ~x3j ; K() = + p (0) j~x ~xj : = H all q's p's but on vanh, solution an a = p 3 xtm Whn black hol if non-vanhing constant positiv. n, if sval laton a positiv, on can consid that abov solutions dscibs as nstants a = p 3 black hols in quilibium. ADM mass systm givn any m = (q + p + q + p) : () would b positiv. Whn coodinats all black hols coincid on d a =;= p 3;0 xtm dilaton black hols (dpnding on how many con- ts vanh) fo abov solution, cosponding xtm ants black hols can b thought as dscibing xtnal ld a bound laton p \lmntay" a = 3 black hols [8]. at now allow fo ngativ constants on immdiatly ss fom abov w If fomula that on could gt solutions with total ADM mass vanhing o ass W a intstd in fom. y can b thought as dscibing gativ. xtm a = p 3 dilaton black hols in quilibium amongst m with ual o objcts with ngativ mass 7. m simplst masslss combination phaps q = q = q; p = p = 0 a hol-anti-black hol pai o dihol. H it cla why w hav somthing ack nothing with zo ngy. On o h, Ricci scala a stad a = p 3 xtm black hol, with mtic ngl q + = ds q + dt d~x ; () W stss again that no igoous way tlling what mass ach individual 7 although, physically, it cla that must b som ngativ mass. jct, R = q + ) 3 : (3) (q q ( mass) positiv, singulaity at= 0. Whn q ngativ, Whn singulaity at = jqj. two cosponding \chag dtibutions" should cancl at innity, fo all agumnts givn abov apply to th only W should gt a masslss, nonotivial, point-lik objct with vanhing cas. mass whn two massiv objcts a plac at sam point,, in ADM substituting H's into mtic placing both black hols in fact, point w cov masslss black-hol mtic ()with D = q! sam sam asoning as in Rf. [8] w would conclud that known Following black hols a ctiv ld bound stat a pai objcts with masslss masss, o a dihol. opposit simpl masslss combination q = q = q; p = p = p. If Ano lctic chags q a placd at sam point two magntic chag two a placd tog at a dint point, sulting solution dscibs two p dihols in quilibium masslss ds = q p p q dt d~x : () fou chags a placd at sam point on gts a quaduhol. If Whn q, its mtic taks a vy simpl fom p= q = ds dt q d~x : (5) mo masslss solutions a possibl s a, phaps most Although ons, at last to pov ou point. intsting conclusion, w hav xhibitd massls xtm black-hol solutions that can considd as bound stats positiv ngativ-mass objcts satfying a b condition. no-foc dicult to avoid idntifying s massls black hols with thos which, It to Stoming [], bcom masslss whn a typ II sting oy compactid accoding on a Calabi-Yau thfold na a conifold singulaity CY moduli which can, in som cass, condnsat [], giving to a phas tansition. spac Th has bn poposd in Rf. [6]. Howv, w hav sn that maslss black hols found in Rf. [3] a ally composit objcts y do

5 cospond to on-paticl, but to two-paticl stats. It could wll b that t massls black hols a also two paticl stats. fact that oming's black hol cais minimal Z chag may not b an obstacl fo th. = abov masslss black hols also cay minimal chags ( mo than on h ld, but s still hav to b diagonalizd und supgavity). It also () to compa masslss dihols with Coop pa in BCS oy stibl spit many dincs tha analogis a vy supconductivity. paling. hav not studid wh s solutions a solutions full lowgy W sting ctiv action vn at lowst od in 0, that, how y can b in tn-dimnsional htotic sting low-ngy ctiv oy. bddd cla (it known [3, 5]) that som m can b mbddd into it, in many ways. Knowldg mbdding ncssay to study i ssibly poptis. Again, claly, som m a supsymmtic, psymmty a no-foc condition dos not gant unbokn supsymmty by itslf []. t cannot, howv, igno an impotant su: how can supsymmty b W with objcts with ngativ mass? ADM mass a masslss black mpatibl zo, to stat with, no poblm in admiting that compos- l objct could b supsymmtic. Howv, unbokn supsymmty objct common scto unbokn supsymmtis its composit Killing spino (whos xtnc a ncssay condition unokn mponnts: supsymmty, s fo instanc Rf. []) has to satfy all constaints psnc ach componnt imposs (s, fo instanc, Rfs. [3, ]). at : componnts hav to admit Killing spinos mslvs. hat cas at hs, sms to b a poblm with thos constitunts that ngativ mass. Ctainly, y cannot b supsymmtic. Supsymmty v a positivity bound on mass [5, 6] which would b violatd. Howv, plis can still admit Killing spinos (whos xtnc not a sucint condition y supsymmty). fact, it asy to s by dict calculation that hav p 3multi-black-hol mtics (fo instanc) always admit Killing spinos fo = choic hamonic function V (s, fo instanc Rf. [0]). Th function y to b stictly positiv to avoid ocunc singulaitis, but th takn not usd in nding Killing spinos 8 ct lvl supsymmty algba, xtnc Killing spinos At Th may look stang to ad that knows that Killing spino tchniqus (Nst 8 a usd to pov positivity mass mo stictiv bounds [7]. nstuctions) in all cass a additional assumptions in fom inqualitis that owv, tnso has to satfy. y a pobably violatd in cass ngativ gy-momntum ass. 5 that ctain supsymmty chags annihilat stat. What dos th mans fo ngativ mass stats? Fo an appopiat choic supsymmty man bas, N xtndd supsymmty algba can b wittn in th way [6] n m ;S?n o () () S n m ;S?n o () () S = mn (m + jznj) ; = mn (m jznj); (6) n =[N=] all o anticommutatos vanh. Sinc opatos in wh l.h.s. s quations a positiv, w hav bounds m + jznj 0 ; (7) m jznj 0: (8) mass supsymmtic objcts satuat on st bounds Eq. (7) Positiv satfy all os. satuation on st bounds associatd xtnc a supsymmty chag that annihilats cosponding to That chag associatd to Killing spino. st chags stat. non-tivially in a way constnt with supsymmty algba on act i action on it gnats (shotnd) supmultiplts []. stat a ngativ mass objct admitting Killing spinos must b a supsymmty Fo chag that annihilats cosponding stat. A supsymmty bound scond typ Eq. (8) satuatd 9 but all bounds st typ a violatd. n, a no o supsymmty chags to complt algba, cannot build supmultiplts stat cannot b said supsymmtic. on a bound stat with a positiv mass supsymmtic objct, can b in masss chags, if total mass in not ngativ, compnsations sinc both componnts admit Killing spinos, composit objct can b supsymmtic. fobids xtnc olatd ngativ-mass objcts, but it Supsymmty not fobid i xtnc in non-ngativ mass bound stats, just as quaks dos not xt in olation at low ngis. do Obsv that quadatic fom bound, which nough to hav an xtm solution 9 quations motion Killing spinos can b satd by ngativ mass objcts.

6 autho would lik to thank Lu Alvaz-Gaum, Rnata Kallosh Russo fo hlpful dcussions. H would also lik to thank M.M. Fnz g h suppot. fncs A. Stoming, Masslss Black Hols Conifolds in Sting oy, ] Phys. B5 (995) 96. Nucl. B.R. Gn, D.R. Moon A. Stoming, Black Hol Condnsation ] Unication Sting Vacua, Nucl. Phys. 5 (995) K. Bhndt, About a Class Exact Sting Backgounds, Nucl. Phys. B55 3] 737. (995) R. Kallosh, Duality Symmtic Quantization Supsting, Phys. Rv. D5 ] (995) R. Kallosh A. Lind, Exact Supsymmtic Massiv Masslss 5] Hols, Phys. Rv. D5 (995) Whit R. Kallosh A. Lind, Supsymmtic Balanc Focs Condnsation 6] BPS Stats, Rpot SU-ITP-95-6 hp-th/955. T. Otn, Tim-Symmtic itial{data Sts in -D Dilaton Gavity, Physical 7] Rviw D5 (995) J. Rahmfld, Extmal Black Hols as Bound Stats, Rpot CTP-TAMU- 8] hp-th/ /95 J. Mahaana J.H. Schwaz, Non-Compact Symmtis in Sting oy, 9] Phys. B390 (993) 3. Nucl. R.R. Khui T. Otn, Supsymmtic Black Hols in N = 8 Supgavity, 0] Rpot CERN-TH/95-38, Mc-Gill/95-6 hp-th/9577. R.R. Khui T. Otn, A Non-Supsymmtic Dyonic Extm Rsn- ] Black Hol, Rpot CERN-TH/95-37, Mc-Gill/96-0 Nodstom hp-th/9578, (to b publhd in Physics Ltts B). G.W. Gibbons, Aspcts Supgavity ois, (th lctus) in: Supsymmty, ] Supgavity Rlatd Topics, ds. F. dl Aguila, J. d Azcaaga L. Iba~nz, Wold Scintic, Singapo, 985, pag 7. 6 T. Otn, Elctic-Magntic Duality Supsymmty in Stingy Black [3] Phys. Rv. D7 (993) Hols, T. Otn, SL(,R){Duality Covaianc Killing Spinos in Axion{Dilaton [] Hols, Physical Rviw D5, (995) 790{79. Black E. Wittn D. Oliv, Supsymmty Algbas that clud Topological [5] Phys. Ltt. 78B (978) 97. Chags, S. Faa, C.A. Savoy B. Zumino, Gnal Massiv Multiplts in Extndd [6] Supsymmty, Phys. Ltt. 00B (98) 393. G.W. Gibbons C.M. Hull, A Bogomolny Bound fo Gnal Rlativity [7] Solitons in N =Supgavity, Phys. Ltt. 09B (98) 90.