We study the evaporation process of 2D black holes in thermal equilibrium
when the incoming radiation is turned off. Our analysis is based on two
different classes of 2D dilaton gravity models which are exactly solvable in
the semiclassical aproximation including back-reaction. We consider a one
parameter family of models interpolating between the Russo-Susskind-Thorlacius
and Bose-Parker-Peleg models. We find that the end-state geometry is the same
as the one coming from an evaporating black hole formed by gravitational
collapse. We also study the quantum evolution of black holes arising in a model
with classical action $S = {1\over2\pi} \int d^2x \sqrt{-g} (R\phi +
4\lambda^2e^{\beta\phi})$. The black hole temperature is proportional to the
mass and the exact semiclassical solution indicates that these black holes
never disappear completely.Comment: LaTeX, psfig.sty. 10 pages. 3 figures (included as PS files

Bose

Callan

Cruz

Giddings

J. Cruz

J. Navarro-Salas

Mann

Russo

Strominger

Thorlacius

Wald

Physics Letters B

English

arXiv

We study the evaporation process of 2D black holes in thermal equilibrium when the incoming radiation is turned off. Our analysis is based on two different classes of 2D dilaton gravity models which are exactly solvable in the semiclassical approximation including back-reaction. We consider a one parameter family of models interpolating between the Russo-Susskind-Thorlacius and Bose-Parker-Peleg models. We find that the end-state geometry is the same as the one coming from an evaporating black hole formed by gravitational collapse. We also study the quantum evolution of black holes arising in a model with classical action S = 1/2 pi integral d(2)x root-g (R phi + 4 lambda 2e(beta phi)). The black hole temperature is proportional to the mass and the exact semiclassical solution indicates that these black holes never disappear completely

Cruz Muñoz, José Luis

Navarro Salas, José

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-th/9607155v1  18 Jul 1996FTUV/96-41IFIC/96-49hep-th/9607155Black Hole Evaporationby Thermal Bath Removal ∗J.Cruz† and J.Navarro-Salas‡Departamento de F´ısica Teo´rica and IFIC, Centro Mixto Universidad de Valencia-CSIC.Facultad de F´ısica, Universidad de Valencia, Burjassot-46100, Valencia, Spain.AbstractWe study the evaporation process of 2D black holes in thermal equilibriumwhen the incoming radiation is turned off. Our analysis is based on two dif-ferent classes of 2D dilaton gravity models which are exactly solvable in thesemiclassical aproximation including back-reaction. We consider a one parame-ter family of models interpolating between the Russo-Susskind-Thorlacius andBose-Parker-Peleg models. We find that the end-state geometry is the sameas the one coming from an evaporating black hole formed by gravitational col-lapse. We also study the quantum evolution of black holes arising in a modelwith classical action S = 12π∫d2x√−g(Rφ+ 4λ2eβφ). The black hole temper-ature is proportional to the mass and the exact semiclassical solution indicatesthat these black holes never disappear completely.PACS:04.60+nKeywords: Black holes, Back-reaction, Symmetries, Solvable Models∗Work partially supported by the Comisio´n Interministerial de Ciencia y Tecnolog´ıa andDGICYT.†cruz@lie.uv.es‡jnavarro@lie.uv.es1 IntroductionOne of the main features of 1+1 dilaton gravity [1] (see the reviews [2] [3] [4]) isthat the formation and semiclassical evaporation of a black hole can be studiedin an exact analytical setting. The Russo-Susskind-Thorlacius (RST) model [5]predicts, with a natural choice of boundary conditions, an evaporation processending in the linear dilaton vacuum in a finite proper time. Recently, Bose-Parker-Peleg [6] have proposed a different one-loop quantum modification ofthe CGHS model [1] describing a semiclassical evaporation with a semi-infinitethroat as the remnant geometry. In both models the matching between theevaporating black hole solution and the static end-state geometry produces athunderpop carrying a small amount of negative energy. The same physicalpicture of black hole evaporation exhibits the one-parameter family of modelsinterpolating between the RST and BPP models [7]. In this paper we considera different scenario to study the process of black hole evaporation in thesetheories. Instead of analyzing the formation and subsequent evaporation of ablack hole we shall consider the dynamical evolution of a black hole in thermalequilibrium when the incoming flux decreases. We shall find that the aboveconventional picture is maintained. The evaporation leads to the same remnantgeometry as the process initiated with the gravitational collapse although nowthe energy of the thunderpop turns out to be independent of the mass of theinitial state.We shall also study a similar process but within a different theory of 1+1dilaton gravity. It was shown in Ref [8] that the unique theories admitting spe-cial conformal symmetries are the models with a classical exponential potentialS =12π∫d2x√−g(Rφ+ 4λ2e2βφ).The classical CGHS action can be recovered for β = 0 through the field re-definitions gµν = g¯µνe−2φ¯, φ = e−2φ¯. In contrast with the CGHS theory, theblack holes arising from the above theory (β 6= 0) have a non-constant temper-ature proportional to the mass. Due to the presence of a special symmetry(gµν −→ e−βǫgµν , φ −→ φ+ ǫ), it is possible to construct a semiclassical the-ory invariant under the above symmetry. So, in this theory one can also findanalytic solutions describing the evaporation of a black hole in thermal equilib-rium due to the decrease of the incoming thermal flux.2 Evaporation in the BPP-RST modelsThe one-paramater family of models [7] interpolating between the BPP andRST models is given by the actionS0 + SP +N24π∫d2x√−g[(1− 2a) (∇φ)2 + (a− 1)φR], (2.1)where S0 is the classical CGHS action, SP is the Polyakov term and a is an arbi-trary real paramater. For a = 12 we recover the RST model and the BPP model1requires a = 0. The equations of motion, in conformal gauge, are equivalent to∂+∂− (ρ− φ) = 0 , (2.2)∂+∂−(e−2φ +Na12ρ)+ λ2e2(ρ−φ) = 0 (2.3)e−2φ(4∂±ρ∂±φ− 2∂2±φ)− N12((∂±ρ)2 − ∂2±ρ)+N12((2a− 1) (∂±φ)2 + (a− 1) ∂2±φ− 2 (a− 1) ∂±φ∂±ρ)+T f±± −N12t± = 0 , (2.4)where t± (x±) are the boundary terms arising from the Polyakov action. InKruskal gauge (ρ = φ), they reduce to∂+∂−(e−2φ +Na12φ)+ λ2 = 0 , (2.5)∂2±(e−2φ +Na12φ)+ T f±± −N12t± = 0 . (2.6)In the absence of matter(Tf±± = 0), the general solution of (2.5) (2.6) is easilyfound to bee−2φ +Na12φ = −λ2x+x− + N12∫ x+ ∫ x+t+(x+)+N12∫ x− ∫ x−t−(x−)+ bx+ + cx− + d , (2.7)with b, c, d arbitrary constants. Let us consider a solution with b = c =t± (x±) = 0 , d = Mλe−2φ +Na12φ = −λ2x+x− + Mλ, (2.8)corresponding to a black hole with a curvature singularity (we require Mλ− α > 0)α =Na24(1− log Na24)= −λ2x+x− + Mλ, (2.9)and an apparent horizon (∂+φ = 0) located atx− = 0 . (2.10)If we evaluate the flux of radiation< Tf±± >= −N12((∂±ρ)2 − ∂2±ρ+ t±), (2.11)2in past and future null infinity using asymptotically flat coordinates(λx± = ±e±λσ±)we find -due to the anomalous transformation properties oft±- the constant thermal value< Tfσ±σ±>=Nλ248. (2.12)So that (2.8) represents a radiating black hole in thermal equilibrium withincoming radiation at temperature T = λ2π (i.e, the analogue of the Hartle-Hawking state).Now we shall try to find out a solution without incomig flux which can becontinuously matched to (2.8) through the null line x+ = x+0 . We shall assumethat the asymptotically flat coordinate σ+ is the same as for the initial solu-tion. With this hypothesis we must take tσ+ = 0 as the no incoming radiationcondition, which implies tx+ =14(x+)2for x+ > x+0 . Therefore, assuming nowthe boundary conditionstx+ =14 (x+)2θ(x+ − x+0), (2.13)tx− = 0 , (2.14)as the new input for (2.7) and demanding continuity to the metric and itsderivative through x+ = x+0 , we obtain (for x+ > x+0 ) the dynamical solution(k = N12)e−2φ +Na12φ = −λ2x+ (x− +∆)− k4(logx+x+0+ 1)+Mλ, (2.15)where ∆ = − k4λ2x+0.The new asymptotically flat coordinates {σ˜+, σ˜−} areλx+ = eλσ˜+, (2.16)− λ (x− +∆) = e−λσ˜− . (2.17)The equality of σ+ and σ˜+ confirms our previous assumption and thus guaran-tees that solution (2.15) has vanishing incomig radiation.The curvature singularity of (2.15) isα =ka2(1− log ka2)= −λ2x+ (x− +∆)− k4(logx+x+0+ 1)+Mλ, (2.18)whereas the apparent horizon− λ2x+ (x− +∆) = k4, (2.19)recedes as the black hole evaporates and intersects the curvature singularity atthe point (see Fig.1)x+int = x+0 e4k (Mλ−α) , (2.20)3x−int =k4λ2x+0[1− e− 4k (Mλ −α)]. (2.21)We notice that the evaporation time differs from that obtained in the processoriginated by gravitational collapse [7].Following the conventional reasoning [5] [6] we shall try to implement thecosmic censorship conjeture by matching the evaporating black hole solution(2.15) with a stable (not-radiating) solution on the null line x− = x−int. ¿From[7] we know that the stable solutions of (2.5) (2.6) aree−2φ +Na12φ = −λ2x+ (x− +∆)− k4log(−λ2x+ (x− +∆))+ C , (2.22)corresponding to the boundary conditions tx± =14(x±)2. Imposing continuityto the metric through x− = x−int we obtain the value of the constant CC =k4(logk4− 1)+ α . (2.23)The remnant geometry (2.22) coincides exactly with the final state resultingfrom the evaporation of a black hole formed by collapse of a shock wave, asdiscussed in [7]. However, in the present process the energy of the thunderpopemanating from the intersecting point is independent of the initial state andtakes the limiting value Ethunderpop = −λk4 .3 Evaporation in the exponential modelsIn this section we shall investigate the evaporation process of black holes withina different theory but with boundary conditions similar to that considered inthe previous section. Our starting point is the semiclassical action of the expo-nential models [8]S =12π∫d2x√−g(φR+ 4λ2eβφ − 12N∑i=1(∇fi)2)+SP +Nβ96π∫d2x√−g(φR+ β (∇φ)2), (3.1)where SP is the Polyakov action. The local counterterm of (3.1) can be ob-tained from the Polyakov effective action respect to the metric invariant underthe special conformal symmetry. This way one ensures the invariance of thesemiclassical action and so the existence of a free field. The equations of motionof (3.1), in conformal gauge, are∂+∂−(ρ− β2φ)= 0 , (3.2)∂+∂−[(1 +Nβ24)φ+Nρ12]+ λ2e(2ρ+βφ) = 0 , (3.3)4−(1 +Nβ12)∂2±φ+ 2(1 +Nβ12)∂±φ∂±ρ+N12[(∂±ρ)2 − ∂2±ρ+β24(∂±φ)2 + t±(x±)]= T f±± . (3.4)In Kruskal gauge (2ρ = βφ) these equations can be rewritten as∂+∂−2ρ = −λ2βγe4ρ , (3.5)e2ρ∂2±e−2ρ = βγ(Tf±± −N12t±)(3.6)where γ = 11+Nβ12.We see that 2ρ satisfies the Liouville equation (3.5). Its general solution canbe expressed asρ =14log∂+A∂−B(1 + λ2βγAB)2, (3.7)with A (B) an arbitrary function of the x+ (x−) coordinate. Inserting (3.7) intothe constrained equations (3.6) we obtain Ricatti differential equations withrespect to log ∂+A and log ∂−B.14(∂+ log ∂+A)2 − 12∂2+ log ∂+A = βγ(Tf++ −N12t+), (3.8)14(∂− log ∂−B)2 − 12∂2− log ∂−B = βγ(Tf−− −N12t−). (3.9)Taking T f±± = t± = 0, one obtains the solutionds2 =−dx+dx−λ2βγC+ Cx+x−. (3.10)For C < 0 (and λ2β < 0) [8] the above metric represents a black hole with asingularity curveλ2βγC+ Cx+x− = 0 , (3.11)and an apparent horizon located atx− = 0 . (3.12)Proceeding in a similar way as in the past section we can calculate the flux ofingoing and outgoing radiation in the asymptotically flat coordinates {σ±}±√|C|x± = e±√|C|σ± , (3.13)getting< Tfσ±σ±>=N |C|48, (3.14)which corresponds to a radiating black hole in thermal equilibrium with ingoingradiation at temperature T =√|C|2π . The black hole mass can be calculated5immediately [8]: M = 2βπ√|C|. Therefore, the Hawking temperature turnsout to be proportional to the mass: T = β4M . The existence of 2D blackholes whose mass is proportional to their temperature was noted in Ref. [9].At this point, the next step would be to solve equations (3.8) and (3.9) withthe modified boundary conditions (2.13) (2.14) and impose continuity to theresulting metric and its derivative through x+ = x+0 . However this leads to anexcesively complicated analytical treatment. In order to simplify the analysiswe shall admit that, in addition to the discontinuity in the boundary functiontx+, there is an incomig shock-wave at x+ = x+0Tf++ =12x+0 βγδ(x+ − x+0). (3.15)With this assumption, and in the limit Nβ >> 1, the solutions of (3.8) and(3.9), for x+ > x+0 , lead to the metricds2 = −(x+0x+) 12λ2βγC+Cx+0 x−(log x+x+0+ 1)dx+dx− . (3.16)The most relevant feature of this solution is that the singularity curveλ2βγC+ Cx+0 x−(logx+x+0+ 1)= 0 , (3.17)and the horizonλ2βγC+ Cx+0 x−(logx+x+0+ 1)+ 2Cx+0 x− = 0 , (3.18)do not intersect each other at any finite point (see Fig.2). Moreover, in contrastwith the BPP-RST models the relation of the asymptotically flat coordinate σ˜+of (3.16) and the Kruskal one x+, which regulates the incomig flux, is not thesame as for the initial solution (3.10). In fact, in past null infinity, this relationbecomesdx+√|C|(x+0 x+) 12(log x+x+0+ 1) = dσ˜+ . (3.19)This tell us that the thermal bath has not been actually removed. Using (3.19)we can calculate the value of < T fσ˜+σ˜+> in past null infinity for x+ > x+0< Tfσ˜+σ˜+> = −N12 |C|x+04 (x+)(logx+x+0+ 1)2− 18 (x+)2− 316 (x+)2 logx+x+0+ 3log x+x+0+ 12 . (3.20)6We see that the flux goes to zero at late times (x+ −→∞) whereas at x+ = x+0< Tfσ˜+σ˜+> |x+0= −N12 |C|4− 2916(x+0)2 . (3.21)At this point it seems natural to choice the point x+0 which makes (3.21) tocoincide with the constant value of < T fσ+σ+>= N48 |C| of the initial solution(i.e, x+0 =√298|C|). In this way the external flux starts to decrease at x+ = x+0(see Fig.3), vanishes at retarded time x+1 and for x+ > x+1 becomes a smallnegative energy flux vanishing again at infinity.In conclusion, the semiclassical solution implies that the black hole evapo-ration never ceases. One can argue that the non-vanishing incoming flux forx+ > x+0 prevents the complete evaporation§. However after x+ > x+1 theflux becomes negative and produces the opposite effect and, despite of this,the apparent horizon never meets the singularity. This qualitative behaviour ofthe solution can be understood in terms of thermodynamic arguments. In theevaporation process the temperature slows down because it is proportional tothe mass and the third law of thermodynamics [10] does not permit to achievethe absolute zero temperature (i.e, the complete evaporation).After completing this work we were informed that the black hole solution(3.10) also appears in Ref. [11].The authors wish to thank M. Navarro and C. F. Talavera for useful discus-sions.References[1] C. G. Callan, S. B. Giddings, J. A. Harvey and A. Strominger, Phys. Rev.D45 (1992) 1005.[2] A. Strominger, Les Houches Lectures on Black Holes, hep-th/9501071[3] S. G. Giddings, Quantum Mechanics of Black Holes, hepth/9412138[4] L. Thorlacius Black Hole Evolution, hep-th/9411020[5] J. G. Russo, L. Susskind and L. Thorlacius, Phys. Rev. D46 (1993) 3444;Phys. Rev. D47 (1993) 533.[6] S. Bose, L. Parker and Y. Peleg, Phys. Rev. D 52 (1995) 3512 ; Phys.Rev. Lett. 76 (1996) 861 ; Validity of the Semiclassical Aproximation andBack-Reaction, WIS-MILW-96-th-10[7] J. Cruz and J. Navarro-Salas, Phys. Lett. B375 (1996) 47§To obtain an evaporating solution with a vanishing incoming flux an adequate ansatz fort± is required.7[8] J. Cruz, J. Navarro-Salas, M. Navarro and C. F. Talavera, Symmetries andBlack Holes in 2D Dilaton Gravity, hep-th/9606097[9] R. B. Mann, A. Shiekh and L. Tarasov, Nucl. Phys. B341 (1990) 134.[10] R. M. Wald, ”Quantum Field Theory in Curved Spacetime and Black HoleThermodynamics”, University of Chicago Press, Chicago (1994)[11] R. B. Mann, Nucl. Phys. B418 (1994) 231.8x++xxx-+intint-xoFig.1 Kruskal diagram of an evaporating black hole in the BPP-RST modelsx xx ++o-Fig.2 Kruskal diagram of an evaporating black hole in the exponential model9x +++of1x xT< σ+ >σ+Fig.3 Flux of incoming radiation associated with the boundary condition tx+ =14(x+)2θ(x+ − x+0)in the exponential model10

Black hole evaporation by thermal bath removal

Crossref

http://dx.doi.org/10.1016/0370-2693(96)01004-0

Black Hole Evaporation by Thermal Bath Removal

Black Hole Evaporation by Thermal Bath Removal

Abstract

Similar works

Full text

Available Versions

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

Crossref