Composite black holes in string theory

a r X i v :h e p -t h /9405185v 1 28 M a y 1994EDO-EP-1May 1994Black Hole Thermodynamics from String Theory Ichiro Oda Edogawa University,474Komaki,Nagareyama City,Chiba 270-01,JAPAN ABSTRACTIn this note we consider a stringy description of black hole horizon.We start with a nonlinear sigma model defined on a two dimensional Euclidean surface with background Rindler metric.By solving the field equations,we show that to the leading order the Bekenstein-Hawking formula of black hole entropy can be produced.We also point out a relation between the present formalism and the ’tHooft formalism.To construct a theory of quantum gravity in four dimensions is one of the most difficult and challenging subjects left in the modern theoretical physics since we have so far neither useful informations from experiments nor consistent quantum field theory.Under such a circumstance,it seems to be an orthodox attitude to attack concrete problems with logical conflicts and then learn the fundamental principle from which in order to construct a full-fledged theory.In the case of quantum gravity,as one of such unsolved problems,we have quantum black holes1. In particular,it is widely known that there are at least three problems which remain to be clarified in quantum black hole,those are,the endpoint of Hawking radiation, the information loss paradox and the statistical origin of black hole entropy2.Recently,there have been some progresses on the last problem3−9.Among them,the authers of Ref.[6]made an interesting observation that superstring theory might play an important role in deriving the Bekenstein-Hawking formula of the black hole entropy1,10.On the other hand,in previous works11,’tHooft has stressed that black holes are as fundamental as strings,so that the two pictures are really complementary.In fact,he has demonstrated that by properly taking account of a leading gravitational back-reaction of the black hole horizon,the gravitational shock wave,from hard particles,his S matrix which describes the dynamical properties of a black hole can be recast in the form of functional integral over the Nambu-Goto string action. Although his formalism has some weaknesses,it is extremely interesting from the physical viewpoint since quantum incoherence never be lost and all information of particles entering into a black hole is transmitted to outgoing particles owing to the Hawking radiation through the quantumfluctuations of the black hole horizon.As it is expected that superstring has many degrees of freedom and hairs associated with its many excited states,the’tHooft formalism might also give us a clue to understanding of a huge entropy10and quantum hairs2of a black hole.In this note,we shall simply assume that the dynamics of the event hori-zon of a black hole can be described by the world sheet swept by a string in the Schwarzschild background,and then would like to discuss what physical conse-quences can be derived from this assumption.However,the Schwarzschild metric is rather complicated,so that we shall confine ourselves to the case of the Rindler spacetime.The case of the Schwarzschild metric will be reported in a separate pa-per.We will see that a nonlinear sigma action leads to the well-known Bekenstein-Hawking formula of black hole entropy,S=12 d2σ√gµν(X)can be identified as the background spacetime metric in which the string is propagating.Note thatα,βtakes values0,1andµ,νdoes values0,1,2,3.The classicalfield equations give us that0=Tαβ=−2√δhαβ,=∂αXµ∂βXνgµν(X)−1hhαβgµν∂βXν)−1hhαβ∂αXρ∂βXσ∂µgρσ.(3)In this note we consider the case that the background spacetime metric gµν(X) takes a form of the Euclidean Rindler metricds2=gµνdXµdXν=+g2z2dt2+dx2+dy2+dz2,(4)where g is given by g=1are the two dimensional diffeomorphisms and the Weyl rescaling byx(τ,σ)=τ,y(τ,σ)=σ,G(τ,σ)=1.(6) At this stage,let us impose an”axial”symmetryr(τ,σ)=r(τ),t(τ,σ)=t(τ).(7) From Eq.s(5),(6)and(7),the world sheet metric hαβtakes the formhαβ= g2z2˙t2+˙z2+1001 ,(8) where the dot denotes a derivative with respect toτ.And the remainingfield equations(3)become∂τ(z2˙th)=0,(9)∂τh=0,(10)∂τ(˙zh)−1hg2z˙t2=0,(11)whereh=g2z2˙t2+˙z2+1.(12)Now it is straightforward to solve the abovefield equations.We have two kinds of solutions.One solution is a trivial one given byz=˙z=¨z=0,t(τ)=arbitrary.(13)which corresponds to a world-sheet surface of the Euclidean string just lying on the black hole horizon.The next solution is the solution of”world sheet instanton”described byz (τ)=c 2,t −t 0=1gc 1(τ−τ0),(14)where c 1,c 2,τ0,and t 0are the integration constants,in other words,”the moduli parameters”.To understand the physical meaning of this solution more vividly,it is convenient to rewrite z in terms of the time coordinate variable t .From Eq.(14),we obtainz (t E )=gc 1c 21time t Lz (t L )=gc 1c 21g whose inverse gives us nothingbut the Hawking temperature T H =12π=1¯h ,(17)where S E denotes the Euclidean action,and the path integral is performed under the boundary condition of being periodic in the Euclidean time with periodβ¯h. Then the black hole thermodynamics can be recovered in the limit¯h→0by expanding S E around its saddle point.Thus evaluating the free energyβto the leading term equals to substituting a classical solution into the Euclidean action.In the model just considered,it is easy to calculate the free energy.To do so we shall consider the solution(14)since this solution gives us the thermal temperature whose situation should be contrasted to the case of the other solution(13).The result isF=−1c2+1T A H,(18)where A H= dxdy which corresponds to the area of the black hole horizon if we consider the Schwarzschild black hole.By the formula which gives us the entropyS=β2∂Fc2+1T A H.(20) Note that the black hole entropy is proportional to the horizon area.Moreover,by selecting the string tensionT=1c2+1G,(21)we arrive at the famous Bekenstein-Hawking entropy formula1,10S=1duced by hard particles having a large amount of momenta.Thus let us introduce ”vertex operator”in the original action(1)S E=−Thhαβ∂αXµ∂βXνgµν(X)+ d2σ√√hhαβ∂βXµ)+Pµ,(24)where we have replaced gµνwith theflat metricηµν.This approximation would become good when the black hole mass is large compared to the Planck mass. Moreover,we havefixed the world sheet metric hαβ(τ,σ)to be the metric on S2. Therefore we obtainT∆tr Xµ+Pµ=0,(25) where∆tr=1h∂α(√δXµ(σ).(28)From(25)and(27),we haveXµ(σ),Xν(σ′) =iNotes addedDuring the preparation of this article,we noticed that there is a recent work where the black hole is described by the membrane theory16.AcknowledgementWe are grateful to K.Akama,N.Kawamoto,A.Sugamoto and Y.Watabiki for valuable discussions.REFERENCES1.S.W.Hawking,Comm.Math.Phys.43,199(1975)2.J.Preskill,Physica Scripta T36,258(1991)3.L.Bombellli,R.K.Koul,J.Lee and R.D.Sorkin,Phys.Rev.D34,373(1986)4.G.’tHooft,Nucl.Phys.B256,727(1985)5.M.Srednicki,Phys.Rev.Lett.71,666(1993)6.L.Susskind and J.Uglum,Stanford preprint SU-ITP-94-1,hep-th/94010767.C.Callan and F.Wilczek,IAS preprint IAS-HEP-93/87,hep-th/94010728.D.Kabat and M.J.Strassler,Rutgers preprint RU-94-10,hep-th/94011259.T.Jacobson,Maryland preprint,gr-qc-940403910.J.D.Bekenstein,Nuovo Cim.Lett.4,737(1972);Phys.Rev.D7,2333(1973);ibid.D9,3292(1974);Physics Today33,no.1,24(1980)11.G.’tHooft,Nucl.Phys.B335,138(1990);Physica Scripta T15,143(1987);ibid.T36,247(1991);Utrecht preprint THU-94/02,gr-qc/940203712.L.Susskind,L.Thorlacius and J.Uglum,Phys.Rev.D48,3743(1993);L.Susskind and L.Thorlacius,Phys.Rev.D49,966(1994)13.W.G.Unruh,Phys.Rev.D14,870(1976)14.G.Gibbons and S.W.Hawking,Phys.Rev.D15,2752(1977);S.W.Hawking,From General Relativity:An Einstein Centenary Survey,Cambridge Univ.Press197915.I.Oda,Int.J.Mod.Phys.D1,355(1992)16.M.Maggiore,preprint IFUP-TH22/94,hep-th/940417211。

ar X iv:g r -q c /9603063v 1 30 M a r 1996Black Hole Entropy from Loop Quantum GravityCarlo Rovelli ∗Department of Physics and Astronomy,University of Pittsburgh,Pittsburgh,Pa 15260,USA(February 7,2008)We argue that the statistical entropy relevant for the thermal interactions of a black hole withits surroundings is (the logarithm of)the number of quantum microstates of the hole which aredistinguishable from the hole’s exterior,and which correspond to a given hole’s macroscopic con-figuration.We compute this number explicitly from first principles,for a Schwarzschild black hole,using nonperturbative quantum gravity in the loop representation.We obtain a black hole entropyproportional to the area,as in the Bekenstein-Hawking formula.In this letter,we present a derivation of the Bekenstein-Hawking expression for the entropy of a Schwarzschild black hole of surface area A [1]S =c k∗e-mail:rovelli@corresponding to a non-spherical deformation of a spherically symmetric event horizon.Imagine that this deformation is located in a certain region of the horizon.Then the future evolution of thefield–for instance the radiation that reaches future infinity–depends on the location of the deformation on the event horizon.Thus we are interested in the quantum states of the geometry on a surfaceΣof area A,where different regions ofΣare distinguishable from each other.At this point the problem is well defined,and can be translated into a direct computation,provided that a quantum theory of geometry is given[8].In loop quantum gravity,the quantum states of the gravitationalfield are represented by s-knots[9].An s-knot is an equivalence class under diffeomorphisms of graphs immersed in space,carrying colors on their links(corresponding to irreducible representations of SU(2)),and colors on their vertices(corresponding to invariant couplings between such representations).The relation between s-knots and classical geometries was explored in[10].If a surfaceΣis given,the geometry ofΣis determined by the intersections of the s-knot with the surface.Given a quantum state and a surface,let i=1...n label such intersections,and p i be the color of the link through i.Generically,no node of the graph will be on the surface;here,we disregard the“degenerate”cases in which a node falls on the surface.Thus, the quantum geometry of the surface is characterized by an n-tuple of n colors p=(p1,...,p n),where n is arbitrary. In particular,it was shown in[4]that the total area of the surfaceΣisA= i=1,n8π¯h Gp i(p i+2).(3)Therefore,the quantum geometry on the surface is determined by the ordered n-tuples of integers p=(p1,...,p n). States labeled by different orderings of the same unordered n-tuple are distinguishable for an external observer.We are thus interested in the number of ordered n-tuples of integers p such that the macroscopic geometry of the surface is the geometry of the surface of the black hole.The geometry of the surface of the Schwarzschild black hole is characterized by the total area A,and by the uniformity of the distribution of the area over the surface.Uniformity is irrelevant in a count of microscopic configurations,because the number of configurations and the number of uniform configurations are virtually the same for large area(for the vast majority of random configurations of air molecules in a room,the macroscopic air density is uniform).Thus,our task of counting microscopic configurations is reduced to the task of counting the ordered n-tuples of integers p such that(2)holds.More precisely,we are interested in the number of microstates(n-tuples p)such that the l.h.s of(2)is between A and A+dA,where A>>¯h G and dA is much smaller than A,but still macroscopic.LetAM=p i(p i+2)=M.(5)First,we over-estimate M(N)by approximating the l.h.s.of(5)dropping the+2term under the square root.Thus, we want to compute the number N+(M)of ordered n-tuples such thatp i=M.(6)i=1,nThe problem is a simple exercise in combinatorics.It can be solved,for instance,by noticing that if(p1,...,p n)is a partition of M(that is,it solves(6)),then(p1,...,p n,1)and(p1,...,p n+1)are partitions of M+1.Since all partitions of M+1can be obtained in this manner,we haveN+(M+1)=2N+(M).(7)ThereforeN+(M)=C2M.(8) Where C is a constant.In the limit of large M we haveln N+(M)=(ln2)M.(9) Next,we under-estimate M(N)by approximating(5)as(p i+1)2−1≈(p i+1).(10) Thus,we wish to compute the number N−(M)of ordered n-tuples such thati=1,n(p i+1)=M.(11)Namely,we have to count the partitions of M in parts with2or more elements.This problem can be solved by noticing that if(p1,...,p n)is one such partition of M and(q1,...,q m)is one such partition of M−1,then(p1,...,p n+1)and (q1,...,q m,2)are partitions of M+1.All partitions of M+1in parts with2or more elements can be obtained in this manner,thereforeN−(M+1)=N−(M)+N−(M−1).(12)It follows thatN−(M)=Da M++Ea M−(13)where D and E are constants and a±(obtained by inserting(13)in(12))are the two roots of the equationa2±=a±+1.(14)In the limit of large M the term with the highest root dominates,and we haveln N−(M)=(ln a+)M=ln 1+√2M.(15)By combining the information from the two estimates,we conclude thatln N(M)=d M.(16) whereln 1+√2<d<ln2(17)or0.48<d<0.69.(18) Since the integers M are equally spaced,our computation yields immediately the density of ing(4), the number N(A)of microstates with area A grows for large A asln N(A)=dA¯h GA.(21)which is the Bekenstein-Hawking formula.The constant of proportionality that we have obtained isc=d4.Notice that the dynamics(the Hamiltonian)does not enter our derivation directly.However,it does enter indirectly by singling out the states with given area A as the ones with the same energy M.This is the usual role of the Hamiltonian in the microcanonical framework.In particular,in a gravitational theory different from GR the relation between the hole’s energy M and its area A may be altered–or even lost.Thus,the relation between number of states and area is purely kinematical,by the relation between this number and the entropy(which is the number of states with the same energy)is theory dependent[11].We leave a number of issues open(which may affect the proportionality factor).We have disregarded the degenerate states in which a node falls over the surface.Also,we have worked in the simplified setting of a black hole interacting with an external system with given geometry,instead of working with a fully generally covariant statistical mechanics [12].Finally,we comment on the relation of our result with Ref.[5].We learned the idea of associating entropy to classical configurations of the geometry–seen as macroscopical states–from Krasnov[13].An earlier attempt to realize this idea along the lines described here failed,yielding an entropy proportional to the square root of the area[7].A crucial breakthrough in[5]was the intuition that intersections are distinguishable.In[5],however,the setting of the problem is substantially different than ours:internal configurations of the black hole are considered, the Bekenstein entropy-bound conjecture and the holographic conjecture are invoked in order to justify the counting considered.Here,we do not need those hypotheses.Furthermore,the entropy-area proportionality is derived in[5]by means of an elegant but complicated argument involving a phase transition in afictitious auxiliary statistical system, while here the combinatorial computation is performed explicitly.In summary:we have argued that the black hole entropy relevant for the hole’s thermodynamical interaction with its surroundings is the number of the quantum microstates of the hole which are distinguishable from the exterior of the hole;we have counted such microstates using loop quantum gravity.We have obtained that the entropy is proportional to the area,as in the Bekenstein-Hawking formula,but with a different numerical proportionality factor.I thank Abhay Ashtekar,Riccardo Capovilla,Luis Lehner,Ted Newman,Jorge Pullin,Lee Smolin,and especially Kirill Krasnov and Ranjeet Tate,for very stimulating discussions.。

a rXiv:h ep-th/961186v123O ct1996MODIFICATION OF BLACK-HOLE ENTROPY BY STRINGS R.Parthasarathy 1and K.S.Viswanathan 2Department of Physics Simon Fraser University Burnaby,B.C.,Canada V5A 1S6.ABSTRACT A generalized action for strings which is a sum of the Nambu-Goto and the extrinsic curvature (the energy integral of the surface)terms,is used to couple strings to gravity.It is shown that the conical singularity has deficit angle that has contributions from both the above terms.It is found that the effect of the extrinsic curvature is to oppose that of the N-G action for the temperature of the black-hole and to modify the entropy-area relation.Recently Englert,Houart and Windey[1]obtained a relation between the entropy and the area of a black-hole that is different from the Bekenstein-Hawking[2,3]relation S=A/4(G=1units)when a conical singularity producing source is present in the euclidean section at r=2M.It is known that a conical singularity at r=2M modifies the euclidean periodicity of the blackhole metric and hence the temperature[4,5,6].It is necessary to take into account the source producing the singularity to maintain euclidean saddle point.The authors in Ref.1,introduced for this purpose an elementary string in the action.The conical singularity arises from the string when it wraps around the horizon and the resulting deficit angle is determined by the string tension.In the evaluation of the free energy,the contribution of the string term exactly cancels that of the Einstein term and so only the boundary terms contribute.As a result one obtains,AS=−gdσdτ,(2) where g is the determinant of the induced metric on the worldsheet.This observation i.e.(1),is important in the light of the recent statisti-cal derivation of Bekenstein-Hawking S=A/4relation in string theory by counting the(microscopic)BPS bound state degeneracy[7,8,9,10,11].As a result of introducing(2)on the horizon,the temperature of the blackhole is increased as T=T H/(1−4µ)leading to an acceleration of the evaporation.The key observation in[1]is the existence of non-trivial solution to the string equations of motion from(2)when the string wraps around the eu-clidean continuation of the horizon,a sphere at r=2M.When the string worldsheet wraps around the sphere,it has non-zero extrinsic curvature and so the simple string action(2)needs a generalization.The Nambu-Goto ac-tion is just the area term while the’energy of the surface’is given by an action involving extrinsic curvature as √(see thefirst line in(12))has the property that there is noflux normal to the surface.On the other hand,the energy momentum tensor for the extrinsic curvature has a non-vanishingflux normal to the surface.Consequently it is conceivable that thisfluxflow might tend to slow down the acceleration of the evaporation,favouring the stability of the black-hole against the effect in[1].Indeed our present study confirms this.The generalized string action[12,13,14]we use in this article is a sum of the N-G action and the one involving extrinsic geometry and is given by,I string=−µ √α20 √2H iαβgαβ,H iαβis the second fundamentalform and i runs from1to D−2.The string worldsheet is considered as a 2-dimensional surface immersed in a D-dimensional space-time.H iαβis then defined by the Gauss equation∂α∂βXµ+˜Γµνρ∂αXν∂βXρ−Γγαβ∂γXµ=H iαβN iµ,(4) where Xµ(σ,τ)are the immersion coordinates of the string worldsheet,˜Γis the affine connection for the(curved)D-dimensional space-time,Γis the affine connection on the string worldsheet,calculated using the induced met-ricgαβ=∂αXµ∂βXνhµν,(5) hµνbeing the metric on the D-dimensional space-time,N iµare the(D−2) normals to the string worldsheet andα20is a dimensionless coupling constant.The present authors have earlier studied[15,16,17,18]the extrinsic geom-etry of strings in some detail including instanton effects in constant mean curvature(H=0)and minimal(H=0)surfaces[18]and recently[19]con-sidered the intrinsic and extrinsic geometrc properties of the string world-sheets in curved space-time background.In this formalism the only string dynamical degrees of freedom are its immersion coordinates Xµ(σ,τ).It is the purpose of this article to consider the effect of introducing the generalized string action(3)in place of(2)when the string wraps around the euclidean3horizon of a Schwarzschild black hole.Wefind the entropy-area relation gets modified asS=A(1−4µ−(α0M)−2).(6)The temperature of the black-hole is found to beT=8π{M(1−4µ)+(α20M)−1}−1.This feature of M dependence for T is shared by quantum one-loop effects[20].The Lorentzian action for gravity coupled to matterfields isI=1−hRd4X−1−h′Kd3X+1−h0K0d3X+I m,(7)where thefirst term is the usual Einstein-Hilbert action,K is the trace of the extrinsic curvature of the boundaryΣof space-time,K0is the same as K but of the boundaryΣ0offlat space-time and I m is given in(3).The boundary terms in(7)arefirst introduced by Gibbons and Hawking[21]and their roles in cancelling the boundary terms arising from the Einstein-Hilbert action and removing the divergence at space-like infinity due to the K-term for asymptoticallyflat space-times are given in[22].It is to be noted here that we have two extrinsic curvatures;one is the trace of the extrinsic curvature of the boundary of the space-time(denoted by K in(7))and the other that of the2-dimensional surface(the string world sheet)immersed in the space-time (denoted by H in(3))which is not a boundary term.The above action(7) admits the usual Schwarzschild blackhole solution corresponding to trivial solution for I m i.e.,zero string area and vanishing extrinsic curvature.The Euclidean continuation of the Schwarzschild solution isds2=(1−2Mr)−1dr2+r2dΩ2.(8)Following Hawking[23],we consider an euclidean section via x=4M(1−2M2,x≥0,0≤τ≤8πM.The periodicity ofτis8πM and this gives,βH=8πM.(9)4In the euclidean formalism of blackholes,the topology of the space-time is R2×S2and with the angular variableτ,the topology becomes D2×S2. There is no cusp now and the deficit angle is zero.The euclidean action following from(7)isI E=−1hRd4X+1h′Kd3X−1h0K0d3X+µ √α20 √2hµνR=−8πTµν,(11) where the energy momentum tensor Tµνfor the generalized string action(3) is given by[19]Tµν=µ d2σδ4(X−X(σ,τ))h(X(σ,τ))√α20[ d2σ√√gδ4(X−X(σ,τ))h(X(σ,τ))gαβ∂αXµ∂βXνH i N iρ](12)In(12)∇αis the covariant derivative with respect to the induced metric gαβon the worldsheet and∇ρis with respect to the background space-time metric hµν.Note that X in(12)is just the a space-time point,whereas X(σ,τ)stands for the string worldsheet dynamical variables.One sees from the4-dimensional Dirac delta function in(12)that Tµν(X)vanishes unless X is exactly on the string worldsheet.Further,theflux of TµνN−G,given by thefirst line in(12)along the normal direction,TµνN−G N iν(X)is zero.This is no longer true for that part of Tµνcoming from the extrinsic curvature action as can be seen from(12).Variation of(10)with respect to the worldsheet coordinates,Xµ(σ,τ)→Xµ(σ,τ)+δXµ(σ,τ)has been evaluated in[19] and by writingδXµ=ξj N jµ+ξα∂αXµ,(13)5as normal and tangential variations,it has been shown that the equation of motion for normal variation is∇2H i−2H i(H2+k)+H j H jαβH iαβ−gαβH j˜Rµνρσ∂αXν∂βXρN jµN iσ=0,(14) where k=α20/µ,i=1,2,˜Rµνρσis the Ricci tensor of the curved background space.The tangential variation has been shown to be just the contracted structure equation of Codazzi.It is clear from(14)that the string worldsheet is non-trivial on the euclidean section alluded to above,as it can admit solutions H i=0.Taking the trace of(11)and integrating over d4X we find,d4X√α20d2σ√h R(X)=A D2√4πD2√α20Ad2σ√α20Ad2σ√Comparison with Eqn.16of[1]shows that the generalized string with ex-trinsic curvature has a non-trivial effect.For a sphere the mean curvature is a constant[24],|H|2=1/r2.The effect of adding the extrinsic curvature action is thus to modify the periodicity ofτfromβH=8πM toβ=βH{1−4µ+(α0M)−2},(19) where r=2M is used.It follows from(19)that in the presence of the extrinsic curvature action,the increase in the temperature of the black-hole due to the N-G action is reduced so that the acceleration of evaporation is slowed down.The calculation of the free energy of the black-hole is made a lot easier in view of(15).Accordingly the contribution from the Einstein term √,(20)16πfrom which the free energy(β−1I Boundary)isME free=={8π(1−4µ−(α20M2)−1}−1,(22)dβand using the relation for entropy S=β2dE f ree{1−4µ+(α0M)−2}24Let us summarize our results.We have argued that when the string worldsheet wraps around the horizon,a2-sphere(r=2M),it has extrinsic curvature and this has been added to the Nambu-Goto term in the form of’energy integral’of the surface.The Einstein equations of motion with the modified energy momentum tensor give the relation(15)from which it follows that the free energy is independent of the string contribution as it is cancelled by the Einstein term.By considering infinitesimal tubular neighbourhood of r=2M,the conical singularity has a deficit angle(18). It is to be noted that the deficit angle depends on the black-hole mass in contrast to the situation in[1].This changes the periodicity ofτand change the entropy-area relation according to(23).When the mean curvature is constant,it has been shown[18]that the string admits instanton solutions. Its effect in the context of QCD was studied in[18].In this article another effect of string instanton is described.In the absence of extrinsic curvature, the string instanton has the effect of raising the global temperature of the black-hole.The extrinsic curvature action has the opposite effect as far as the temperature is concerned(19).It is intresting to note that the inverse temperatureβof the black-hole has M dependence through the N-G action and M−1dependence through the extrinsic curvature action.In[20]a similar feature was obtained from the one-loop quantum effects,M from the classical action and M−1from the quantum correction.The extrinsic curvature action can arise as quantum correction when fermions on the string world sheet are functionally integrated[25].It can be seen that the sign ofσin[20]for spin-halffields is negative agreeing with ourα20>0.The entropy-area relation is modified.The black-hole entropy can be greater that A/4.It is possible to interpret(23)within the S=A/4relation if we introduce an effective area for the horizon asA eff=A{1−4µ+(α0M)−2}2{1−4µ−(α0M)−2}−1.(24) It can be seen that A eff>A and so that A eff represents a’stretched horizon’.In the context of extremal black-holes,the degeneracy associated with extremal black-hole states is smaller than the degeneracy of the elemen-tary string states with the same quantum numbers.To resolve this,Sen[26] postulated that the entropy of the extremal black-hole is not exactly equal to the area of the event horizon,but the area of a surface close to the event horizon;the’stretched horizon’.It is interesting to see that the introduction8of extrinsic curvature action for strings on the horizon favours this idea of ’stretched’horizon for neutral Schwarzschild black-hole as well. AcknowledgementThis work is supported by an operating grant(K.S.V)from the Natural Sciences and Engineering Research Council of Canada.One of us(R.P) thanks the Department of Physics,Simon Fraser University,for hospitality.9REFERENCES1.F.Englert,L.Houart and P.Windey,Phys.Lett.372B(1996)29.2.J.D.Bekenstein,Phys.Rev.D7(1973)2333.3.S.Hawking,Nature.248(1974)30;Comm.Math.Phys.43(1975)199.4.S.Coleman,J.Preskill and F.Wilczek,Nucl.Phys.378B(1992)175.5.F.Dowker,R.Gregory and J.Traschen,Phys.Rev.D45(1992)2762.6.M.Banados,C.Teitelboim and J.Zaneli,Phys.Rev.Lett.72(1994)957.C.Teitelboim,Phys.Rev.D51(1995)4315.S.Carlip and C.Teitelboim,Class.Quant.Grav.12(1995)1699.7.A.Strominger and C.Vafa,”Microscopic origin of the Bekenstein-Hawkingentropy”,HUTP-96/A002,RU-96-01;hep-th/9601029.8.G.T.Horowitz,”The origin of black-hole entropy in string theory”,UCSBTH-96-07;gr-qc/9604051.9.G.T.Horowitz,J.M.Maldacena and A.Strominger,Phys.Lett.383B(1996)151.10.D.M.Kaplan,D.A.Lowe,J.M.Maldacena and A.Strominger,”Micro-scopicentropy of N=2extremal blackholes”,CALT-68-2076,RU-96-88,hep-th/9609204.11.J.M.Maldacena and A.Strominger,”Statistical entropy of4-dimensionalextremal blackholes”,hep-th/9603060.12.A.M.Polyakov,Nucl.Phys.B268(1986)406.13.H.Kleinert,Phys.Lett.174B(1986)335.14.T.L.Curtright,G.I.Ghandour and C.K.Zachos,Phys.Rev.D34(1986)3811.1015.K.S.Viswanathan,R.Parthasarathy and D.Kay,Ann.Phys.(N.Y)206(1991)237.16.R.Parthasarathy and K.S.Viswanathan,Int.J.Mod.Phys.A7(1992)317;1819.17.K.S.Viswanathan and R.Parthasarathy,Int.J.Mod.Phys.A7(1992)5995.18.K.S.Viswanathan and R.Parthasarathy,Phys.Rev.D51(1995)5830.19.K.S.Viswanathan and R.Parthasarathy,”String theory in curved space-time”,SFU-HEP-04-96;hep-th/9605007.20.D.V.Fursaev,Phys.Rev.D51(1995)R5352.21.G.W.Gibbons and S.Hawking,Phys.Rev.D15(1977)2752.22.I.Ya.Arefe’va,K.S.Viswanathan and I.V.Volovich,Nucl.Phys.B452(1995)346;B462(1995)613.23.S.Hawking,”The path integral approach to quantum gravity”,in General Relativity:An Einstein Centenary Survey,Eds.S.Hawkingand W.Israel,(Cambridge University Press,1979).24.T.J.Willmore,Total Curvature in Riemannian Geometry,Ellis Horwood Ltd.1982.25.R.Parthasarathy and K.S.Viswanathan,(to be published).26.A.Sen,”Extremal blackholes and elementary string states”,hep-th/9504147.11。

a r X i v :h e p -t h /9805115v 1 19 M a y 1998SU-ITP-98-??hep-th/9805115May 1997Matrix Theory Black Holes and the Gross Witten TransitionL.Susskind 11Department of Physics,Stanford University Stanford,CA 94305-4060Large N gauge theories have so called Gross-Witten phase transitions which typically can occur in finite volume systems.In this paper we relate these transi-tions in supersymmetric gauge theories to transitions that take place between black hole solutions in general relativity.The correspondence between gauge theory and gravitation is through matrix theory which represents the gravitational system in terms of super Yang Mills theory on finite tori.We also discuss a related transition that was found by Banks,Fischler,Klebanov and Susskind.1.The Gross Witten TransitionAccording to matrix theory[2],there is a duality between Super Yang Mills theory on a spatial d-torus and11dimensional supergravity compactified on a d+1 torus∗.Our knowledge of the two theories sometimes overlaps and this allows us to test the duality.More often,we know something about one theory which leads to predictions about the other.In this paper we will use knowledge about black holes to gain information about Gross Witten type transitions[3]in3+1dimensional super Yang Mills theory compactified on a3-torus of sizeΣ.In particular we will see that such a transition exists and that the transition temperature is a function of the’t Hooft coupling g2ym N which scales likeT3∼1g2ym N(1.1)for large values of g2ym N.Detailed features of the transition could in principle be predicted from classical solutions of supergravity.A similar argument has been given by Witten for the case of compactification on a sphere.In this case the duality is between super Yang Mills theory and supergravity in AdS5×S5[4].We begin with a brief review of Gross Witten transitions.Normally systems of small numbers of degrees of freedom or systems infinite volume do not exhibit sharp phase transitions or singularities in the partition function.However in the large N limit such transitions become possible.In fact the Gross Witten transition wasfirst found in the theory of a single plaquette in lattice gauge theory[3].The plaquette is described by a unitary N×N matrix U.The energy isE=−1g2tr U (1.3)The coupling g is assumed to satisfy’t Hooft scalingg2N=λ(1.4)withλfixed.The integral over unitary matrices can be replaced by an integral over the N eigenvaluesαi=e iθi of U.The measure for the integration is a certain determinant∆whose essential properties we will describe.The partition function isZ= dαexp Nβλand it has sharp edges as N→∞.For smallλthe droplet is small and asλincreases its size increases until at some critical value ofβλ=1origin creates a lattice with regularly spaced eigenvalues that could be described as a crystal.In fact even when the distribution spreads over the circle at high tem-perature,a strong degree of local crystalline order is preserved due to the repulsive forces.In this paper we will present evidence that Gross Witten transitions occur in 3+1dimensional toroidally compactified super Yang Mills theory.In this case the matrix valued variables that replace the plaquette variable U are the Wilson loops around the cycles of the torus.2.Matrix Black HolesIn[1]matrix theory[2]was applied to the behavior of black holes in M Theory compactified on a3-torus of size L.Strictly speaking the theory is compactified on a 4torus but by passing to the large N limit while keeping Lfixed,the11th direction is effectively decompactified.For the problem of black holes the decompactification can be quantified as follows.Begin with a configuration with entropy S.Here we assume that S>>1.For N<S the configuration described in[1]behaves like a10 dimensional near extremal black hole with D0-brane charge.Alternatively it can be thought of as an11dimensional black string wound around the11th direction.It is homogeneous in the11th direction.As N increases,an instability is encountered. The black string breaks and forms a localized blob in the11th direction[6].The blob becomes an11dimensional Schwarzschild black hole.The transition from homogeneous black string to localized black hole occurs at N∼S.In fact one finds that at this point,the free energy of the black string and Schwarzschild black hole are equal.We will return to this transition in sect3but for now we are interested in the”black string”region N<<S where the black hole is uniform in x11.Let us consider the behavior of the black hole as we vary the size of the3-torus L,keeping N large butfixed.It is convenient to think of compactification as periodic identification and to imagine a transverse lattice of black holes with spacing L.It is obvious that for very large L the compactification can be ignored and the black hole is localized somewhere in the torus with a radius much smallerthan L.In this region the black hole behaves as a10dimensional near extremal D0-brane black hole in uncompactified transverse space.It is so far from its periodic images that they can be ignored.On the other hand for small enough L the black hole will merge with its images and a uniform homogeneous configuration on the3 torus will result.In fact a sharp transition must take place[7].To see this consider the horizon.For very large L the horizon will form a small sphere localized in the torus.As L is decreased the horizons of the images will eventually touch and the black holes will merge.The change in topology of the horizon is discrete and signals a singularity in its area and thus in the thermodynamic quantities.The approximate location of the phase transition can be found either by ther-modynamic or by solving the gravitational equations and determining the point at which the horizon changes topology.We will use thermodynamics.The thermody-namic relation between free energy and temperature for a near extremal D0-brane black hole is∗T4N2l6stF=L3∗Throughout this paper all irrelevant numerical constants will be set to13.Gauge Theory Interpretation of the TransitionAccording to Matrix theory the system we are describing is dual to super Yang Mills theory compactified on a3-torus.The parameters of the super Yang Mills theory are a coupling constant g ym and a compactification radiusΣ.We also introduce two M-Theory quantities;R and l11.R is the compactification radius of the11direction and l11is the11dimensional planck length.The various quantities are related byR=l st g st(3.1)l11=l st g1/3standl311g2ym=(3.2)LRPlugging these equations into(2.3)wefind the transition temperature satisfiesΣT c=(g2ym N)−1/2(3.3)Note that the right side of eq(3.3)depends only on the’t Hooft coupling constant g2ym N as we would expect for a Gross Witten type transition.On the left side we the scale invariant function of the temperature that we can construct from the Yang Mills parameters.This reflects the exact conformal invariance of 3+1dimensional super Yang Mills theory with16real supersymmetries.In order to see that this transition is analogous to the Gross Witten transition on a single plaquette,let us recall the description of the black hole(for N<<S) given in[1].It was found that the D0-branes form a regular lattice on the3-torus. Indeed the’t Hooft parameter g2ym N=Nl311theory by the eigenvalues of the Wilson loops around the3cycles.Thus,just as for the single plaquette,the transition is one in which the distribution of Wilson loop eigenvalues begins at high temperatures byfilling the torus with a crystal lattice and passes to a localized blob at low temperatures.However the picture described above only holds over part of the N,L plane.As the temperature is lowered it is possible that another transition intervenes before the Gross Witten transition.This is the transition reported in[1]in which the10 dimensional black hole(11D black string)condenses into a blob in the11direction and becomes an11dimensional schwarzschild black hole.For want of a better name the transition will be referred to as the BFKS transition.The BFKS transition occurs at the point where the entropy is equal to N.The critical temperature for this transition satisfiesΣT bfks=N−1>N(3.5)l911Like the GW transition,the BFKS transition also separates a homogeneous phase from a blob-like phase.At high temperature the black hole system will not only be homogeneous in the3-torus of size L but also the longitudinal compact direction of size R.If the inequality(3.5)is violated then as the temperature is lowered the black object will become localized in the longitudinal direction.This transition is not of the Gross Witten type in the3+1gauge theory but as shown in [1]its existence is easily understood in the D3-brane description of the super Yang Mills theory as a transition which occurs when the thermal wavelength exceeds the size of the effective quantization volume.For a more complete explanation the reader is referred to[1].However in the SYM description it is not clear that this transition is a sharp one.The gravitational description however,makes it clear that a singularity occurs.The point is once again that when the system goes from black string to black hole,the horizon topology suddenly changes.This sudden change signals a singularity in the area and therefore the entropy.REFERENCES1.T.Banks,W.Fischler,I.Klebanov and L.Susskind,hep-th/9709091.2.T.Banks,W.Fischler,S.Shenker and L.Susskind,hep-th/9610043.3.D.J.Gross,E.Witten,Phys.Rev.D21:446-453,19804.Edward Witten,hep-th/98021505.Igor R.Klebanov,Leonard Susskind,Schwarzschild Black Holes in VariousDimensions from Matrix Theory,hep-th/97091086.Gary T.Horowitz,Emil J.Martinec,Comments on Black Holes in MatrixTheory,hep-th/97102177.Gary T.Horowitz,Joseph Polchinski,A Correspondence Principle for BlackHoles and Strings,hep-th/96121468.G.Polhemus,Statistical Mechanics of Multiply Wound D-Branes,hep-th/9612130。

a r X i v :g r -q c /9608008 v 1 2 A u g 1996Pair Creation and Evolution of Black Holes in Inflation ∗Raphael Bousso †and Stephen W.Hawking ‡Department of Applied Mathematics and Theoretical Physics University of Cambridge Silver Street,Cambridge CB39EW DAMTP/R-96/35Abstract We summarise recent work on the quantum production of black holes in the inflationary era.We describe,in simple terms,the Euclidean approach used,and the results obtained both for the pair creation rate and for the evolution of the black holes.Introduction One usually thinks of black holes forming through gravitational collapse,and so it seems that inflation is not a good place to look for black holes,since matter is hurled apart by the rapid cosmological expansion.We will show,however,that it is possible to get black holes in inflation through the quantumprocess of pair creation[1,2].There are two physical motivations that might lead usto expect this:First of all,quantumfluctuations can be very large during inflation, which leads to large density perturbations.Secondly,in order to pair create anyobjects,whether particles or black holes,one needs a force to pull them apart.Think of electron-positron pair creation:unless there is a force pulling them apart,the virtual particles will just fall back and annihilate.But if they are in an externalelectricfield,thefield pulls them apart and provides them with the energy to become real particles.Similarly,whenever one pair creates black holes,one needs to do it on abackground that will pull them apart.This could be,for example,Melvin’s magneticuniverse,where oppositely charged black holes are separated by the backgroundmagneticfield,or a cosmic string,which can snap with black holes sitting on thebare terminals,pulled apart by the string tension.For the black holes we shall consider,the necessary force will be provided by the rapid expansion of space duringinflation.So this expansion,which we naively thought would prevent black holesfrom forming,actually enables pair creation.Inflation In quantum cosmology,one expects the universe to begin in a phasecalled chaotic inflation.In this era the evolution of the universe is dominated bythe vacuum energy V(φ)of some inflatonfieldφ.V starts out at about the Planckvalue,and then decreases slowly while thefield rolls down to the minimum of the potential.During this time the universe behaves like de Sitter space with an effectivecosmological constantΛeff≈V(see Fig.1).Like the scalarfield,Λeffdecreases only very slowly in time,and for the purposes of calculating the pair creation rate,wecan takeΛto befixed[1].Instanton method An instanton is a Euclidean solution of the Einstein equa-tions,i.e.,a solution with signature(++++).Instantons can be used for the de-scription of non-perturbative gravitational effects,such as spontaneous black hole formation.What follows is a kind of kitchen recipe for this type of application.We must consider two different spacetimes:de Sitter space without black holes(i.e., the inflationary background),and de Sitter space containing a pair of black holes. For each of these two types of universes,we mustfind an instanton which can be2(time)large black holes,strongly suppressed01tiny black holes,unsuppressed (e f f e c t i v e c o s m o l o g i c a l c o n s t a n t )Figure 1:The classical evolution of the effective cosmological constant in a typical model of chaotic inflation.We have indicated qualitatively how the nucleation size and pair creation rate of black holes depend on the effective cosmological constant.analytically continued to become this particular Lorentzian universe.The next step is to calculate the Euclidean action I of each instanton.According to the Hartle-Hawking no boundary proposal [3],the value of a wave function Ψis assigned to each universe.In the semi-classical approximation Ψ=e −I ,neglecting a prefactor.P =|Ψ|2=e −2I Re is then interpreted as a probability measure for the creation of each particular universe.(Note that P depends only on the real part of the Eu-clidean action.)The pair creation rate of black holes on the background of de Sitter space is finally obtained by taking the ratio Γ=P BH /P no BH of the two probability measures.One can also think of Γas the ratio of the number of inflationary Hubble volumes containing black holes to the number of empty Hubble volumes.3de Sitter We begin with the simpler of the two spacetimes,an inflationary uni-verse without black holes.In this case the spacelike sections are round three-spheres. In the Euclidean de Sitter solution,the three-spheres begin at zero radius,expand and then contract in Euclidean time.Thus they form a four-sphere of radius ArrayΛ .(1) Schwarzschild-de Sitter Now we need to go through the same procedure with the Schwarzschild-de Sitter solution,which corresponds to a pair of black holes immersed in de Sitter space.The spacelike sections in this case have the topologyS1×S2.This can be seen by the following analogy:Empty Minkowski space has4spacelike sections of topology R 3.Inserting a black hole changes the topology to S 2×R .Similarly,if we start with de Sitter space (topology S 3),inserting a black hole is like punching a hole through the three-sphere,thus changing the topology to S 1×S 2.In general,the radius of the S 2varies along the S 1.The maximum two-sphere corresponds to the cosmological horizon,the minimum to the black hole horizon.This is shown in Fig.3.non-degenerate Schwarzschild-de Sitterblack hole horizon identify identifyS 1S 2cosmological horizondegenerate Schwarzschild-de Sitter Figure 3:The spacelike slices of Schwarzschild-de Sitter space have the topology S 1×S 2.In general (left),the size of the two-sphere varies along the one-sphere.If the black hole mass is maximal,however,all the two-spheres have the same size (right).Only in this case is a smooth Euclidean solution admitted.What we need is a Euclidean solution that can be analytically continued to contain this kind of spacelike slice.It turns out that such a smooth instanton does not exist in general for the Lorentzian Schwarzschild-de Sitter spacetimes.The only exception is the degenerate case,where the black hole has the maximum possible size,and the radius of the two-spheres is constant along the S 1(see Fig.3).The corresponding Euclidean solution is just the topological product of two round two-spheres,both of radius 1/√1The real part of the Euclidean action for this instanton is given by I Re BH=−π/Λ, and the corresponding probability measure isP BH=exp 2πΛ .(4) Let us interpret this result.The cosmological constant is positive and no larger than order unity in Planck units.This means that black hole pair creation is sup-pressed.WhenΛ≈1(early in inflation),the suppression is week and one can get a large number of black holes.However,by Eq.(2),they will be very small(Planck size).For smaller values ofΛ(which are attained later in inflation),the black holes would be larger,but their creation becomes exponentially suppressed(see Fig.1). This result,which was obtained from the no boundary proposal,is physically very sensible.Tunnelling proposal According to Vilenkin’s tunnelling proposal[5],the wave function is given by e+I,rather than e−I.If we tried to apply this prescription to our problem,the signs would get reversed in all the exponents,and we would get the inverse result forΓ.Thus black hole creation would be enhanced,rather than suppressed.Even worse,the bigger the black holes were,the more likely they would be to nucleate.As a consequence,de Sitter space would be catastrophically unstable. This prediction is obviously absurd.Thus,the consideration of cosmological black hole pair creation provides strong evidence in favour of the no boundary proposal.Classical evolution What happens to black holes that have been pair createdduring inflation?In the above instanton solution they would just retain their con-√stant size r BH=1/Λeff. As the inflatonfield rolls down,Λeffdecreases,and the black hole grows slowly,be-coming quite large by the end of inflation.This growth can be explained by the First Law of black hole mechanics,which states that the increase in a black hole’s6horizon area,multiplied by its temperature,is equal to four times the increase in its mass.The mass increase comes from theflux of energy-momentum of the inflaton field across the black hole horizon,as thefield rolls down.Quantum evolution There are some quantum effects on the evolution which we have not yet taken into account.It is well known that both the black hole and the cosmological horizon emit radiation.The temperature of each horizon is approxi-mately proportional to its inverse radius.In the instanton solution the radii of the two horizons will be equal,and,therefore,also their radiation rates.The black hole loses as much mass due to Hawking radiation as it gains from the incoming cosmo-logical radiation,and it would seem to be stable.Because of quantumfluctuations, however,the radius of the two-spheres will vary slightly along the one-sphere.Then the black hole will be smaller and hotter than the cosmological horizon.It starts to lose mass and evaporates.Only if it was created very late in inflation would it be massive and cold enough to grow classically and survive into the radiation era. But such black holes are highly suppressed.The tiny,hot black holes created early in inflation will all evaporate immediately.Therefore there will be no significant number of neutral black holes after inflation ends.Magnetically charged black holes There also are instantons that correspond to the creation of magnetically charged black holes.Such black holes cannot evaporate altogether,because there are no magnetically charged particles they could radiate. Therefore they are still around today.A detailed calculation shows,however,that they are so suppressed,and so strongly diluted by the inflationary expansion,that there won’t even be a single charged primordial black hole in the observable universe. (This is a sensible prediction,since we don’t observe any.)In dilatonic theories of inflation,however,their number could be significantly larger;this is currently being investigated.Summary Semi-classical calculations indicate that tiny black holes are plentifully produced at the Planck era.The creation of larger black holes is exponentially suppressed.During inflation,the black holes can grow classically,but will mostly evaporate due to quantum effects.Magnetically charged black holes cannot evap-orate,but their number today is exponentially small.Generally,in the context of cosmological pair creation of black holes,the no boundary proposal gives physically sensible results,while the tunnelling proposal does not seem to be applicable.7References[1]R.Bousso and S.W.Hawking:The probability for primordial black holes.Phys.Rev.D52,5659(1995),gr–qc/9506047.[2]R.Bousso and S.W.Hawking:Pair creation of black holes during inflation.Preprint no.DAMTP/R–96/33,gr–qc/9606052.[3]J.B.Hartle and S.W.Hawking:Wave function of the Universe.Phys.Rev.D28,2960(1983).[4]P.Ginsparg and M.J.Perry:Semiclassical perdurance of de Sitter space.Nucl.Phys.B222,245(1983).[5]A.Vilenkin:Boundary conditions in quantum cosmology.Phys.Rev.D33,3560(1986).8。

a r X i v :h e p -t h /9108001v 1 14 A u g 1991UCSBTH-91-39July,1991Exact Black String Solutions in Three DimensionsJames H.Horne and Gary T.Horowitz Department of Physics University of California Santa Barbara CA 93106-9530jhh@ gary@ ABSTRACT:A family of exact conformal field theories is constructed which describe charged black strings in three dimensions.Unlike previous charged black hole or extended black hole solutions in string theory,the low energy spacetime metric has a regular inner horizon (in addition to the event horizon)and a timelikesingularity.As the charge to mass ratio approaches unity,the event horizon remains but the singularity disappears.1.IntroductionIn a recent paper[1],it was shown that string theory has a rich variety of solutions describing extended objects surrounded by event horizons.In particular there are black string solutions in ten dimensions characterized by three parameters:the mass and axion charge per unit length,and the asymptotic value of the dilaton.These solutions were obtained by solving the low energy string equations of motion.Although this is sufficient to establish the existence of exact solutions with these qualitative features,it was not clear how to construct directly the conformalfield theory with these properties.Witten has recently shown[2]that a simple gauged WZW model[3,4]yields a two dimensional black hole.This raises the possibility of using similar constructions tofind exact conformalfield theories corresponding to higher dimensional black holes or extended black holes.(The conformalfield theory associated with an extremal limit of the charged blackfivebranes has recently been found[5].)In this paper we will show that a simple extension of Witten’s construction yields three dimensional charged black strings.These solutions are also characterized by three parameters:the mass M and axion charge Q per unit length,and a constant k related to the asymptotic value of the derivative of the dilaton.The low energy metric,antisymmetric tensor,and dilaton take the form: ds2=− 1−M Mr dx2+ 1−M Mr −1k dr22ln kAlthough the three dimensional black strings are most naturally described in terms of the string metric(the metric appearing in the sigma model),it is also of interest to consider the rescaled Einstein metric(with the standard Einstein-Hilbert action)*.We will see that the Einstein metric also describes a black string in an asymptoticallyflat spacetime.But it is not static!There is still a timelike symmetry outside the event horizon,but it resemblesa boost at infinity rather than a time translation.2.Derivation of Black String SolutionsWe now describe the conformalfield theory construction which yields the black strings. Since our target space is going to have Lorentz signature,we will use a Lorentz metric ds2=2dσ+dσ−on the world sheetΣ.If g is an element of a group G,then the ungauged Wess–Zumino–Witten action can be writtenL(g)=k12π B Tr(g−1dg∧g−1dg∧g−1dg),(2)where B is a three manifold with boundaryΣ.We are interested in gauging a one dimensional subgroup H of the symmetry group of eq.(2),with action g→hgh.We can make this global symmetry local by introducing a gaugefield A i which takes values in the Lie algebra of H.Ifǫis an infinitesimal gauge parameter,then the local axial symmetry is generated byδg=ǫg+gǫ,δA i=−∂iǫ.(3) This local axial symmetry is a symmetry of the gauged WZW actionL(g,A)=L(g)+k*Since the two dimensional Einstein action is a topological invariant,this is not possible for the two dimensional black hole.This gives the ungauged actionL(g)=−k2π Σd2σlog u(∂+a∂−b−∂−a∂+b)+12π Σd2σA+(b∂−a−a∂−b−u∂−v+v∂−u+4ck∂+x)+4A+A−(1+2c28π Σd2σλv2∂+u∂−u+λu2∂+v∂−v+(2−2uv+2λ−λuv)(∂+u∂−v+∂−u∂+v)π Σd2σ1−uv2π Σd2σc2t/√ˆr−(1+λ),v=−e−√k(1+λ)π Σd2σk∂+ˆr∂−ˆrˆr ∂+t∂−t+ 1−λλˆr (∂+x∂−t−∂−x∂+t).(11) This describes a string propagating in a spacetime with metricds2=− 1−1+λˆr dx2+ 1−1+λˆr −1kdˆr2*A related construction with a compactified x has been used to obtain two dimensional charged black holes[10].and an antisymmetric tensorfieldB tx= 1+λ 1−1+λ12H2+8kˆr2.(16) Thusˆr=0is a curvature singularity.As suggested by eq.(16),the difficulties atˆr=λandˆr=1+λcan be completely removed by an appropriate change of coordinates.In fact, the original u,v,x coordinates in eq.(9)are well behaved at uv=0which corresponds toˆr=1+λ.(The u,v,x coordinates are not well behaved at uv=1(ˆr=λ).This is a direct consequence of the fact that our gaugefixing breaks down there.Note that unlike the case of the two dimensional black hole,the spacetime is nonsingular where the gauge fixing breaks down.)We will see thatˆr=1+λis an event horizon.The solution is clearly invariant under translations of both t and x,and for largeˆr the metric is asymptotically flat.Thus the solution represents a straight,static,black string.We now wish to reexpress the free parametersλand a in terms of the physical mass per unit length and axion charge per unit length of the black string.First note that the overall scaling for the t and x coordinates isfixed by the condition that asˆr goes to infinity, the metric components g tt and g xx approach unity.It is not possible to similarlyfix the overall scaling of the coordinateˆr since the metric asymptotically approaches kdˆr2/8ˆr2.It will be convenient tofix the scaling ofˆr so that the dilaton is exactlyΦ=ln r+12. In other words we setˆr=re−a 2(17) in eqs.(12),(13),and(15).This has the virtue that the metric now depends on two parameters which we will see are simply related to the physical mass and charge per unit length.The fact that the metric depends on both the mass and charge is of course the familiar situation with higher dimensional black holes and black strings.The axion charge is computed as follows.In n dimensions,it follows from the ac-tion(14)that the n−3form K=12λ(1+λ)8k k/2)and H=0.To calculate the mass,one extremizes the action(14) to obtain the metricfield equation,linearizes this expression about the asymptotic solu-tion(19),and integrates the time-time component of this equation over a constant time surface.Since the integrand is a total derivative,the result can be expressed as a surface integral at infinity.The antisymmetric tensorfield appears quadratically in the metric field equation and vanishes in the background,so it does not explicitly appear in the for-mula for the mass.We therefore only need to keep track of the metric and dilaton.Their contributions to thefield equation areeΦ Rµν−12(∇Φ)2−4/k .(20)We now linearize this expression.Since k appears in the background solution,it cannot be changed by the perturbation*.We have chosen our radial coordinate so that,in our2from the one used in ref.[1]*It is as meaningless to compare the masses of two solutions with different k,as it is to compare the masses of two Kaluza-Klein solutions with different compactifications.solutions,Φdepends only on k.So to calculate their mass,we do not need to include a perturbation ofΦ.We need only perturb the metric gµν=ηµν+γµν.Integrating the time-time component of the linearized form of eq.(20)over a spacelike surface yields the following formula for the total mass:M tot=12r dt2+ 1−Q2r −1 1−Q28r2(23)with antisymmetricfield strength and dilatonH rtx=Q/r2Φ=ln r+12.(24)When Q=0,H vanishes and our black string solutions become a simple product of dx2 and the two dimensional metricds2=− 1−M r −1k dr2kr +Q2r+Q2with|Q|≤M.The metric components are ill defined at r=0and r=r±≡M±along the string at infinity becomes timelike inside the inner horizon.Equivalently,thetime coordinate in region I offig.1is t,the time coordinate in region II is r,and the timecoordinate in region V is x.†For this reason it is not possible to represent all aspects of the causal structure by a two dimensional diagram.Nevertheless,most features are faithfullyindicated byfig.1with r+=M and r−=Q2/M.Each point now represents a line inspacetime.However it is the line in the x direction for r>Q2/M and in the t direction forr<Q2/M.Unlike Reissner–Nordstr¨o m,the black string solution also contains the regionslabeled VII and VIII infig.1which correspond to naked singularities.This is becausethese regions correspond to r<0or uv<−(1+λ)which is certainly part of the originalgauged WZW model.We now consider geodesics in the black string solutions.This is particularly simpledue to the two conserved quantities associated with the two translational symmetries.Letξµbe tangent to an affinely parametrized geodesic,and let E=−ξ·∂/∂t,P=ξ·∂/∂xdenote the conserved quantities.Then geodesics satisfyk˙r2r P2M−E2Q24eλM(E2−P2),(28) whereλis an affine parameter.If P2=E2,then r=λ2.Null geodesics can reach the singularity,but only if P2M2−E2Q2>0.Otherwise,the null geodesics reach a minimum value ofE2Q2−P2M2r min=†This shows that it is not possible to compactify the x direction and view this as a two dimensional solution.and(29),the resolution is that geodesics with P=0cross the point labeled p infig.1 (recall this is really a line in spacetime).At p the Killing vector∂/∂x is not only null,but actually vanishes.We have seen that the three dimensional black string is qualitatively very similar to the Reissner–Nordstr¨o m solution.This analogy appears to extend to Hawking evaporation. We can define a Hawking temperature for the black string by analytically continuing t=iτin eq.(1).The horizon r=M is a regular point only if we identifyτwith periodπMπM2k.(30)Therefore,the temperature vanishes as Q→M.(This is also true of Reissner–Nordstr¨o m.) Thus,if the charge cannot be radiated away,the black strings would settle down to|Q|= M.B.The extremal limit:|Q|=MWhat does the extremal configuration look like?If one sets|Q|=M in eq.(1)one obtainsds2=(1−M/r)(−dt2+dx2)+(1−M/r)−2k dr2/8r2.(31) Notice that this extremal metric is not only static and translationally invariant,it is also boost invariant along the string.(Higher dimensional extended black holes are also boost invariant in the extremal limit[1].)Atfirst sight,the global structure of the metric(31) appears to be analogous to the extreme Reissner–Nordstr¨o m metric with a single horizon at r=M and a singularity at r=0.However this is misleading.The proper continuation across the horizon is not to let r become less than M.This is most easily seen by returning to the geodesic equation(27).When|Q|=M this equation becomesk˙r2˜r2+M (−dt2+dx2)+k d˜r2One can easily verify that geodesics now cross the horizon˜r=0from positive to negative values of˜r so the metric(34)and not(31)describes the correct extension across the horizon.But the region˜r<0is identical to the region˜r>0,and the metric(34)is nonsingular!Nevertheless,˜r=0is still an event horizon.Thus one has the unusual situation of a spacetime with an event horizon but no singularity*.The global structure is described infig.3.Observers in this spacetime who cross the event horizon are not able to return,but(fortunately for them)find themselves in another asymptoticallyflat region of spacetime which is identical to the one they started in.What happens to the region V infig.1,near the singularity,as|Q|approaches M? Setting˜r2=M−r,the metric in this region becomesds2=˜r22˜r2(35)which is singular at˜r=±√k(M2−Q2)1/2,ˆt=(1−Q2/M2)1/4t,ˆx=(1−Q2/M2)1/4x andthen taking the limit|Q|→M†.The resulting metric isds2=ky2.(36)To see that this is indeed anti-de Sitter spacetime(albeit in unusual coordinates)one can calculate the curvature andfind Rµν=−48/k whereǫis the volume form.In ten dimensions,the extremal limit of the black string solutions[1]is of particular interest.It agrees precisely with the solution found by Dabholkar et al.[13]describing thefields outside of a fundamental macroscopic string.Dabholkar et al.also found thefields outside of a fundamental macroscopic string in any dimension.In three dimensions,their solution is*ds2=1˜y .(37)This solution,like the ten dimensional analog,has a singularity at˜y=0and no horizon. It does not resemble the extremal limit of our black string.However,in order to compare the two solutions,we must take into account the fact that we have included a correction to the central charge proportional to1/k which was not included in ref.[13].If one wants to view the three dimensional black string,by itself,as a solution to critical string theory, it is necessary to include this modification to the central charge.However if one wants to add an internal conformalfield theory,then the central charge need not be modified.To compare the two solutions,we must take the limit as k→∞.Since large k means that the metric is rescaled by a large factor,the limit k→∞is usually thought to yield a flat metric.However,if one starts with a singular solution,one can take k→∞staying close to the singularity and obtain a nontrivial limit.More precisely,set˜y=(k/8)1/2r/M,˜t=(k/8)1/4x,˜x=(k/8)1/4t in eq.(31).Then in the limit k→∞,the solution agrees exactly with eq.(37).C.The solutions with|Q|>MWe now consider the solution when|Q|>M.The metric(1)appears to change signature at r=Q2/M.But this is just another indication that an incorrect extension is being used.The correct extension can again be found by considering the motion of geodesics.It corresponds to setting˜r2=r−Q2/M.In terms of t,x,˜r the metric becomesds2=−Q2−M2+M˜r2Q2+M˜r2dx2+Mk2k/(Q2−M2).The resulting spacetime is completely nonsingular.Notice that the identification changes the structure of the spacetime at infinity from R3 to R2×S1.A spacelike surface t=constant now resembles an infinite cigar.This is reminiscent of the form of a Euclidean two dimensional black hole.In fact if we take thelimit M →0,Q →0keeping Q 2/M =m fixed,one finds that (38)reduces to exactly the product of −dt 2and the two dimensional Euclidean black hole discussed in ref.[2].This insight helps us to resolve another aspect of these solutions.The conformal field theory construction described in the previous section only yields the solutions (1)with |Q |<M .However,the fields (38)with |Q |>M also solve the low energy string equations and its natural to ask what is the exact conformal field theory that they correspond to.The answer is a slight modification of the construction in sec.2.One again starts with SL (2,R )×R but now puts a timelike metric on R .One then gauges a translation of R together with the subgroup of SL (2,R )generated by 01−10 .The result is exactly the solutions (38)with |Q |>M .In the limit that R is not gauged,one obtains the two dimensional Euclidean black hole cross −dt 2.Investigations of the two dimensional Euclidean black hole have shown that this non-singular space is dual to the Euclidean negative mass solution which has a curvature sin-gularity [14,15,16,17,18,19].In other words,these two different geometries are equivalent as conformal field theories.Although this sounds intriguing,the physical interpretation of this result remains unclear.The reason is that in the Euclidean context,the duality involves string winding modes in Euclidean time.In the Lorentzian context,both the original conformal field theory and its dual contain all six regions of the black hole,so the spacetime metric does not change at all!*In three dimensions there is an analog of this duality with a clear physical interpretation and a rather striking conclusion.By the usual two dimensional arguments,the spacetimeds 2=−dt 2+ 1−M 8r 2+ 1−M r −1k dr 2r dθ2(40)(where the spatial metric is now the negative mass Euclidean solution).The duality is now the familiar one involving winding modes in a spacelike direction.There is no need for further analytic continuation.This shows that the curvature singularity in eq.(40)is not seen by strings.String propagation in this spacetime is completely equivalent to string propagation in the nonsingular geometry (39).This duality can be extended to the general solution with |Q |>M ,and will be discussed in detail elsewhere [20].4.ConclusionsWe have been describing the black string solutions in terms of the metric which appears in the sigma model.This is the metric that the strings couple directly to and is the most natural one to use in string theory.However to compare with results in general relativity, it is sometimes useful to rescale this string metric by a power of the dilaton to obtain a metric with the standard Einstein action.Since conformal transformations do not change the causal structure,both the original string metric and the new Einstein metric will have the same horizons.But since the dilaton is growing linearly at infinity one might think that the Einstein metric will not be asymptoticallyflat.This is incorrect.In three dimensions,the Einstein metric˜gµνis related to the string metric by˜gµν=e2Φgµν.Thus asymptotically,the Einstein metric approaches˜ds2=r2(−dt2+dx2)+k*In higher dimensions,if one starts with aflat string metric and a linear dilaton,and rescales to the Einstein metric,one againfinds that the curvature falls offlike r−2.Thus higher dimensional analogs could not be considered asymptoticallyflat in the usual sense.similar to the Reissner–Nordstr¨o m solution.They have an event horizon,an inner horizon and a timelike singularity.When|Q|=M the spacetime has an event horizon but no singularity.(Another way to take the extremal limit,which does not preserve the boundary conditions,yields anti-de Sitter spacetime.)When|Q|>M,both the horizon and the curvature singularity disappear.To avoid a conical singularity one must compactify one of the directions.A limiting case yields just the product of time and the two dimensional Euclidean black hole.Since our black string solutions are three dimensional and have a linear dilaton at infin-ity,they presumably are not of direct physical interest.Their importance is twofold.First, they illustrate that a wide range of causal structures(including some having no analog in general relativity)can occur in string theory.Indeed,wefind it surprising that a simple conformalfield theory construction can result in such nontrivial spacetime structure.This encourages the hope that an exact conformalfield theory describing higher dimensional black holes and black strings will soon be found.Second,like the two dimensional black hole,they provide an important test of whether gravitational collapse will lead to singu-larities in string theory.It has been shown that string theory does have exact solutions which are singular[21].However the known singular solutions do not have event horizons and hence do not describe gravitational collapse.It is still not clear whether the two dimensional black hole is singular in string theory. (Recall that although one has an exact description of the conformalfield theory in terms of a gauged WZW model,the spacetime metric(25)is only the lowest order approximation to the geometry.Higher order corrections can be large near the singularity.)It has been argued[2]that even though the conformalfield theory may be regular at r=0,it does not make sense to consider signals propagating from r>0to r<0for two reasons.One is that,in the two dimensional black hole,the surface r=0is spacelike on one side and timelike on the other.Thus if signals can propagate across,there would appear to be a violation of causality.The other is that r=0appears to be unstable in that generic perturbations blow up there.For our three dimensional black strings(with0<|Q|<M) the surface r=0is timelike on both sides,so no causality problems should arise.On the other hand,general arguments suggest that the inner horizon r=Q2/M is now unstable. So once again,it appears to be impossible to propagate signals from large positive r to large negative r.AcknowledgementsWe wish to thank S.Giddings,N.Ishibashi,M.Li,A.Steif,A.Strominger,and especially D.Garfinkle for helpful discussions.J.H.H.would like to thank the Aspen Center for Physics,where this work was begun.This work was supported in part by NSF grant PHY-9008502.References1.G.Horowitz and A.Strominger,“Black Strings and p-Branes,”Nucl.Phys.B360,197(1991).2.E.Witten,“On String Theory and Black Holes,”Phys.Rev.D44,314(1991).3.K.Gawedzki and A.Kupiainen,Phys.Lett.B215,119(1988);Nucl.Phys.B320,625(1989).4.I.Bars and D.Nemeschansky,“String Propagation in Backgrounds with CurvedSpace-time,”Nucl.Phys.B348,89(1991).5.S.Giddings and A.Strominger,“Exact Black Fivebranes in Critical SuperstringTheory,”UCSB preprint,UCSBTH-91-35,July1991.6.G.Gibbons,Nucl.Phys.B207,337(1982).7.G.Gibbons and K.Maeda,Nucl.Phys.B298,741(1988).8.B.Ivanov,“Black Holes and the Heterotic String,”ICTP preprint,IC/89/3,January1989.9.D.Garfinkle,G.Horowitz and A.Strominger,“Charged Black Holes in String The-ory,”Phys.Rev.D43,3140(1991).10.N.Ishibashi,M.Li and A.Steif,“Two Dimensional Charged Black Holes in StringTheory,”UCSB preprint,UCSBTH-91-23,July1991.11.S.Hawking and G.Ellis,The Large Scale Structure of Space-Time(CambridgeUniv.Press,Cambridge)1973.12.S.Chandrasekhar and J.Hartle,Proc.Roy.Soc.Lond.A384,301(1982).13.A.Dabholkar,G.Gibbons,J.Harvey and F.Ruiz,“Superstrings and Solitons,”Nucl.Phys.B340,33(1990).14.A.Giveon,“Target Space Duality and Stringy Black Holes,”Berkeley preprint,LBL-30671,April1991.15.E.Kiritsis,“Duality in Gauged WZW Models,”Berkeley preprint,LBL-30747,May1991.16.A.Tseytlin,“Space-Time Duality,Dilaton,and String Cosmology,”to appear inProceedings of the First International Sakharov Conference on Physics,May1991.17.R.Dijkgraaf,E.Verlinde and H.Verlinde,“String Propagation in a Black Hole Ge-ometry,”Princeton preprint,PUPT-1252,May1991.18.E.Martinec and S.Shatasvili,“Black Hole Physics and Liouville Theory,”EnricoFermi preprint,EFI-91-22,May1991.19.I.Bars,“Curved Space-Time Strings and Black Holes,“USC preprint,USC-91-HEP-B4,June1991;“String Propagation on Black Holes,”USC preprint,USC-91-HEP-B3,May1991.20.J.Horne,G.Horowitz,and A.Steif,to appear.21.G.Horowitz and A.Steif,Phys.Rev.Lett.64,260(1990);Phys.Rev.D42,1950(1990);Phys.Lett.B258,91(1991).Figure CaptionsFigure1:The global structure for both the Reissner–Nordstr¨o m solution and for the black string when0<|Q|<M.The jagged lines represent singularities,and r=r±representhorizons.For the Reissner–Nordstr¨o m solution,r=r±≡M±。