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arXiv:hep-th/9203052v1 18 Mar 1992CALT-68-1774DOE RESEARCH ANDDEVELOPMENT REPORTEvaporation of Two Dimensional Black HolesS. W. HawkingCalifornia Institute of Technology, Pasadena, CA 91125andDepartment of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeSilver Street Cambridge CB3 9EW, UKAbstractCallan, Giddings, Harvey and Strominger have proposed an interesting two di-mensional model theory that allows one to consider black hole evaporation in thesemi-classical approximation. They originally hoped the black hole would evaporatecompletely without a singularity. However it has been shown that the semi-classicalequations will give a singularity where the dilaton field reaches a certain critical value.Initially, it seems this singularity will be hidden inside a black hole. However, as theevaporation proceeds, the dilaton field on the horizon will approach the critical valuebut the temperature and rate of emission will remain finite. These results indicateeither that there is a naked singularity, or (more likely) that the semi-classical ap-proximation breaks down when the dilaton field approaches the critical value. Work supported in part by the U.S. Dept. of Energy under Contract no. DEAC-03-81ER40050.February 19922IntroductionCallan, Giddings, Harvey and Strominger (CGHS) 1 have suggested an interest-ing two dimensional theory with a metric coupled to a dilaton field and N minimalscalar fields. The Lagrangian isL =12ge2(R + 4()2+ 42) 12NXi=1(fi)2,If one writes the metric in the formds2= e2dx+dxthe classical field equations are+fi= 0,2+ 2+ 22e2= +,+ 2+ 22e2= 0.These equations have a solution = blog(x+x) c log = 12log(x+x) + log2bwhere b and c are constants and b can be taken to be positive without loss of generality.1A change of coordinatesu = 2blog(x) 1(c + log)gives a flat metric and a linear dilaton field = 0 = 2(u+ u)This solution is known as the linear dilaton.The solution is independent of theconstants b and c which correspond to freedom in the choice of coordinates. Normallyb is taken to have the value12.These equations also admit a solution = c = 12log(M1 e2cx+x). This represents a two dimensional black hole with horizons at x= 0 and singu-larities at x+x= M2e2c. Note that there is still freedom to shift the fieldon the horizon by a constant and compensate by rescaling the coordinates x, buttheres nothing corresponding to the freedom to choose the constant b. In terms ofthe coordinates udefined as before with b =12 = 12log(1 M1e(u+u) = 2(u+ u) 12log(1 M1e(u+u)This black hole solution is periodic in the imaginary time with period 21.One would therefore expect it to have a temperatureT =2and to emit thermal radiation 2. This is confirmed by CGHS. They considered ablack hole formed by sending in a thin shock wave of one of the fifields from the weak2coupling region (large negative ) region of the linear dilaton. One can calculate theenergy momentum tensors of the fifields, using the conservation and trace anomalyequations. If one imposes the boundary condition that there is no incoming energymomentum apart from the shock wave, one finds that at late retarded times utherea steady flow of energy in each fifield at the mass independent rate248If this radiation continued indefinitely, the black hole would radiate an infiniteamount of energy, which seems absurd. One might therefore expect that the backreaction would modify the emission and cause it to stop when the black hole hadradiated away its initial mass. A fully quantum treatment of the back reaction seemvery difficult even in this two dimensional theory.But CGHS suggested that inthe limit of a large number N of scalar fields fi, one could neglect the quantumfluctuations of the dilaton and the metric and treat the back reaction of the radiationin the fifields semi-classically by adding to the action a trace anomaly termN12+.The evolution equations that result from this action are+ = (1 N24e2)+,2(1 N12e2)+ = (1 N24e2)(4+ + 2e2).In addition there are two equations that can be regarded as constraints on the dataon characteristic surfaces of constant x(2+ 2+) =N24e2(2+ + t+(x+),(2 2) =N24e2(2 t(x),where t(x) are determined by the boundary conditions in a manner that will be3explained later.Even these semi-classical equations seem too difficult to solve in closed form.CGHS suggested that a black hole formed from an f wave would evaporate completelywithout there being any singularity. The solution would approach the linear dilatonat late retarded times uand there would be no horizons. They therefore claimedthat there would be no loss of quantum coherence in the formation and evaporationof a two dimensional black hole: the radiation would be in a pure quantum state,rather than in a mixed state.In 3 and 4 it was shown that this scenario could not be correct. The solutionwould develop a singularity on the incoming f wave at the point where the dilatonfield reached the critical value0= 12logN12This singularity will be spacelike near the f wave 4. Thus at least part of the finalquantum state will end up on the singularity, which implies that the radiation atinfinity in the weak coupling region will not be in a pure quantum state.The outstanding question is: How does the spacetime evolve to the future of thef wave? There seem to be two main possibilities:1 The singularity remains hidden behind an event horizon. One can continue aninfinite distance into the future on a line of constant 0, would increase away from the horizon and would always be greaterthan its horizon value. This shows that to get a static black hole solution that is5asymptotic to the weak coupling region of the linear dilaton, hmust be less than thecritical value 0. One can then show that both and must decrease with increasingr. This means the back reaction terms proportional to N will become unimportant.For large r one can therefore approximate by putting N = 0. This gives = (2b 1)logr c+1r= 2( (2b 1)r1)2 2e2)Asymptotically these have the solution = logr + log2bK + Llogrr4b+ .where b,c,K,L are parameters that determine the solution. The parameters bandc correspond to the coordinate freedom in the linear dilaton that the solution ap-proaches at large r. The parameter L does not appear in the black hole solutions. Ifit is zero, the parameter K can be related to the ADM mass M of the solution. Theeffects of the back reaction terms proportional to N will affect only the higher orderterms in r1.For h N/12) iscreated by sending in an f wave. One could approximate the subsequent evolutionby a sequence of static black hole solutions with a steadily increasing value of on thehorizon. However, when the value of on the horizon approaches the critical value60, the back reaction will become important and will change the black hole solutionssolutions significantly. Let = 0+, = log + Then N and disappear and the equations for static black holes become+1r=12?2 e2? +1r ?1 e2?+1r?=?2 e2?()2 e2 ?As the dilaton field on the horizon approaches the critical value 0, the term(1e2) will approach 2, where = 0h. This will cause the second derivative of to be very large untilapproaches e hin a coordinate distance r of order 4.By the above equations, approaches 2e hin the same distance. A power seriessolution and numerical calculations carried out by Jonathan Brenchley confirm thatin the limit as tends to zero, the solution tends to a limiting formc, c.The limiting black hole is regular everywhere outside the horizon, but has a fairlymild singularity on the horizon with R diverging like r1. At large values of r, thesolution will tend to the linear dilaton in the manner of the asymptotic expansiongiven before. One or both of the constants K and L must be non zero, because thesolution is not exactly the linear dilaton. Fitting to the asymptotic expansion givesa valuebc 0.4If the singularity inside the black hole were to remain hidden at all times, as inpossibility (1) above, one might expect that the temperature and rate of evolutionof the black hole would approach zero as the dilaton field on the horizon approachedthe critical value. However, this is not what happens. The fact that the black holes7tend to the limiting solutionc, cmeans that the period in imaginary time will tendto4bc. Thus the temperature will beTc=4bcThe energy momentum tensor of one of the fifields can be calculated from theconservation equations. In the xcoordinates, they are:DTf+E= 112(+ + 2+ + t+(x+),DTfE= 112( 2 + t(x)where t(x) are chosen to satisfy the boundary conditions on the energy momentumtensor. In the case of a black hole formed by sending in an f wave, the boundarycondition is that the incoming fluxDTf+Eshould be zero at large r. This wouldimply thatt+=14x2+The energy momentum tensor would not be regular on the past horizon, but this doesnot matter as the physical spacetime would not have a past horizon but would bedifferent before the f wave.On the other hand, the energy momentum tensor should be regular on the futurehorizon. This would imply that t(x) should be regular at x= 0. Converting tothe coordinates u, one then would obtain a steady rate2192b2cof energy outflow in each f field at late retarded times u.8ConclusionsThe fact that the temperature and rate of emission of the limiting black hole donot go to zero, establishes a contradiction with the idea that the black hole settlesdown to a stable state.Of course, this does not tell us what the semi-classicalequations will predict, but it makes it very plausible that they will lead either toa naked singularity, or to a singularity that spreads out to infinity at some finiteretarded time.The semi-classical evolution of these two dimensional black holes, is very similarto that of charged black holes in four dimensions with a dilaton field 5.If onesupposes that there are no fields in the theory that can carry away the charge, thesteady loss of mass would suggest that the black hole would approach an extremestate.However, unlike the case of the Reissner-Nordstom solutions, the extremeblack holes with a dilaton have a finite temperature and rate of emission. So oneobtains a similar contradiction. If the solution where to evolve to a state of lowermass but the same charge, the singularity would become naked.There seems no way of avoiding naked singularity in the context of the semi-classical theory. If spacetime is described by a semi-classical Lorentz metric, a blackhole can not disappear completely without there being some sort of naked singularity.But there seem to be zero temperature non radiating black holes only in a few cases.For example, charged black holes with no dilaton field and no fields to carry away thecharge.What seems to happening is that the semi-classical approximation is breakingdown in the strong coupling regime. In convential general relativity, this breakdownoccurs only when the black hole gets down to the Planck mass.But in the twoand four dimensional dilatonic theories, it can occur for macroscopic black holeswhen the dilaton field on the on the horizon approaches the critical value. When thecoupling becomes strong, the semi-classical approximation will break down. Quantumfluctuations of the metric and the dilaton could no longer be neglected. One couldimagine that this might lead to a tremendous explosion in which the remaining mass9energy of the black hole was released. Such explosions might be detected as gammaray bursts.Even though the semi-classical equations seem to lead to a naked singularity,one would hope that this would not happen in a full quantum treatment.Quitewhat it means not to have naked singularities in a quantum theory of gravity is notimmediately obvious. One possible interpretation is the no boundary condition 6:spacetime is non singular and without boundary in the Euclidean regime. If thisproposal is correct, some sort of Euclidean wormhole would have to occur, whichwould carry away the particles that went in to form the black hole, and bring in theparticles to be emitted. These wormholes could be in a coherent state described byalpha parameters 7. These parameters might be determined by the minimumizationof the effective gravitational constant G 7,8,9. In this case, there would be no loss ofquantum coherence if a black hole were to evaporate and disappear completely. Or thealpha parameters might be different moments of a quantum field on superspace10.In this case there would be effective loss of quantum coherence, but it might bepossible to measure all the alpha parameters involved in the evaporation of a blackhole of a given mass. In that case, there would be no further loss of quantum coherencewhen black holes of up to that mass evaporated.I was greatly helped by talking to Giddings and Stominger who were working alongsimilar lines. I also had useful discussions with Hayward, Horowitz and Preskill. Thiswork was carried out during a visit to Cal Tech as a Sherman Fairchild Scholar.References1. Callan, C.G., Giddings, S.B., Harvey, J.A., Strominger, A. Evanescent BlackHoles UCSB-TH-91-54.2. Hawking, S.W. Particle Creation by Black Holes, Commun.Math.Phys.43,199 (1975).3. Banks, T., Dabholkar, A., Douglas, M.R., OLoughlin, M. Are Horned Particlesthe Climax Of Hawking Evaporation? RU-91-54.104. Russo, J.G., Susskind, L., Thorlacius, L. Black Hole Evaporation in 1+1 Di-mensions SU-ITP-92-4.5. Garfinkle, D., Horowitz, G.T., Strominger, A. Charged Black Holes in StringTheory, Phys. Rev D 43, 3140.6. Hartle, J.B., Hawking, S.W. Wave Function of the Universe Phys. Rev. D28,2960-2975 (1983).7. Coleman, S. Why There Is Nothing Rather Than Something: A Theory Of TheCosmological Constant Nucl. Phys. B310 (1988), 643.8. Preskill, J. Wormholes In Spacetime And The Constants Of Nature.Nucl.Phys. B323 (1989), 141.9. Hawking, S.W. Do Wormholes Fix The Constants Of Nature? Nucl. Phys.B335,155-165 (1990).10. Hawking, S.W. The Effective Action For Wormholes. Nucl. Phys. B363, 117-131 (1991).11