PHYSICAL REVIEW D VOLUME 56, NUMBER 2 15 JULY 1997
Black holes by analytic continuation
D. Amati SISSA, Trieste, Italy and INFN, Sezione di Trieste, Trieste, Italy
J. G. Russo Theory Division, CERN, CH-1211 Geneva 23, Switzerland ͑Received 18 October 1996͒ In the context of a two-dimensional exactly solvable model, the dynamics of quantum black holes is obtained by analytically continuing the description of the regime where no black hole is formed. The resulting spectrum of outgoing radiation departs from the one predicted by the Hawking model in the region where the outgoing modes arise from the horizon with Planck-order frequencies. This occurs early in the evaporation process, and the resulting physical picture is unconventional. The theory predicts that black holes will only radiate out an energy of Planck mass order, stabilizing after a transitory period. The continuation from a regime without black hole formation—accessible in the 1ϩ1 gravity theory considered—is implicit in an S-matrix approach and suggests in this way a possible solution to the problem of information loss. ͓S0556-2821͑97͒00612-7͔
PACS number͑s͒: 04.70.Dy, 04.60.Kz
I. INTRODUCTION the subcritical regime. The obtained outgoing energy- momentum tensor may be continued beyond the threshold— It has often been advocated ͓1,2͔ that the study of scatter- Sec. IV— and we discuss in Sec. V which boundary condi- ing of matter and radiation in a quantum gravity theory tions would reproduce it. Section VI generalizes the should solve the conflict between classical black-hole solu- preceding results to other infalling distributions of interest. tions and quantum mechanics, which leads to information In Sec. VII we summarize the physical picture and discuss loss ͓3͔. The mere existence of an S matrix below the thresh- the origin of the differences with preceding treatments. old of black hole formation would be enough to exhibit, through its analytic structure, eventual thresholds for the cre- II. SEMICLASSICAL DILATON GRAVITY ation of new objects and to describe, through analytic con- tinuation, the physics above them in a unitary framework. The semiclassical action of the RST model ͑which in- By studying a semiclassical solvable model in which the cludes the one-loop quantum anomaly͒ is given by black-hole evolution can be explicitly investigated, we will 1 see that analytic continuation ͑from below the threshold of Sϭ d2xͱϪg eϪ2͓Rϩ4ٌ͑͒2ϩ42͔ black-hole formation to above it͒ completely determines the 2 ͵ ͩ structure of the theory in the regime in which black holes are 1 N N formed. The model is the two-dimensional dilaton gravity 2 2 Ϫ1 . Ϫ ͚ ٌ͑ f i͒ Ϫ͓2RϩRٌ͑ ͒ R͔ , ϭ with matter ͓Russo-Susskind-Thorlacius ͑RST͓͒4͔͔, which 2 iϭ1 ͪ 48 represents a toy model for spherically symmetric infalling shells in four-dimensional gravity. Because of quantum ef- ͑2.1͒ fects there is a threshold on the incident matter energy den- In the conformal gauge g ϭ0, g ϭϪ(1/2)e2, the ac- sity under which there is no black-hole formation. ϮϮ ϩϪ tion is simplified by introducing new fields , ⍀, related to We shall adopt the usual boundary conditions below the and by threshold, so the subcritical regime will be as in the RST model. It will then be shown that the corresponding outgoing ϭ4ϩeϪ2Ϫ2, ⍀ϭeϪ2ϩ2. ͑2.2͒ energy-momentum tensor can be straightforwardly continued above the critical incoming energy-density flux. The semi- Then action ͑2.1͒ takes the form classical supercritical treatment that would give rise to the same outgoing radiation requires a boundary at the apparent 1 1 2 2 ͑1/2͒͑Ϫ⍀͒ Ϫ⍀͒ϩ eץϩ⍀ץϪϩץϩץhorizon ͑this is at variance with the standard boundary on the Sϭ d x ͑Ϫ singularity͒. As a result, a very unconventional picture ap- ͵ ͩ 4 pears. In particular, Hawking radiation stops early in the N evaporation process and a stable macroscopic black hole re- 1 Ϫ f i , ͑2.3͒ץϩ f iץ ϩ ͚ mains in the final state, thus avoiding the information loss. 2 iϭ0 ͪ This goes in the direction advocated by Giddings ͓5͔ as a possible solution of the information loss problem. with the constraints ͓corresponding to the gϮϮ equations of In Secs. II and III we briefly review the model of ͓4͔ and motion of action ͑2.1͔͒
0556-2821/97/56͑2͒/974͑9͒/$10.0056 974 © 1997 The American Physical Society 56 BLACK HOLES BY ANALYTIC CONTINUATION 975
1 2ץ⍀͒ϩ ץ⍀ ץϩ ץ ץt ͑xϮ͒ϭ ͑Ϫ Ϯ 4 Ϯ Ϯ Ϯ Ϯ Ϯ
1 N Ϯfi . ͑2.4͒ץϮfiץ ϩ 2 i͚ϭ0
Ϯ The functions tϮ(x ) are determined by boundary condi- tions. Let us consider a general distribution of incoming matter:
1 N ϩ . ϩfiץϩfiץ Tϩϩ͑x ͒ϭ 2 i͚ϭ0
In the Kruskal gauge ϭ⍀, the solution to the semiclassical equations of motion and the constraints is given by
2 ϩ Ϫ Ϫ2 ϩ 2 ϩ Ϫ ⍀ϭϭϪ x ͓x ϩ Pϩ͑x ͔͒Ϫ ln͑Ϫ x x ͒ FIG. 1. Kruskal diagram in the subcritical regime. ϩϪ1M͑xϩ͒, ͑2.5͒ III. SUBCRITICAL REGIME where M(xϩ) and P (xϩ) physically represent the total en- ϩ In order to investigate the analytic continuation of the ergy and total Kruskal momentum of the incoming matter at subcritical regime to a supercritical regime, it is convenient advanced time xϩ: to explore in more detail the subcritical theory of Ref. ͓4͔. Let us assume that the geometry is originally the linear dila- xϩ ϩ ϩ ϩ ton vacuum, and there is an incoming energy density flux Pϩ͑x ͒ϭ dx Tϩϩ͑x ͒, ͵0 ϩ cr ϩ ϩ Tϩϩ(x )ϽTϩϩ(x ), which is different from 0 for x0 ϩ ϩ Ϫ Ϫ Ϫ Ͻx Ͻx1 . Let us define region ͑i͒ as x Ͻx0 , x0 ϭ xϩ 2 ϩ Ϫ Ϫ ϩ ϩ ϩ ϩ Ϫ/( x0 ), and region ͑ii͒ as that between x0 and x1 ͑see M͑x ͒ϭ dx x Tϩϩ͑x ͒. ͵0 Fig. 1͒. In region ͑i͒, the solution is given by Eq. ͑2.5͒, which is completely specified by the asymptotic boundary conditions In the particular case Tϩϩϭ0, Eq. ͑2.5͒ reduces to the fa- miliar linear dilaton vacuum and by demanding a continuous matching with the linear dilaton vacuum in the infalling line. In region ͑ii͒ the bound- eϪ2ϭeϪ2ϭϪ2xϩxϪ. ͑2.6͒ ary ϭcr is timelike and boundary conditions are needed in order to determine the evolution. Continuity along the line Ϫ Ϫ Ϯ x ϭx requires that the solution in region ͑ii͒ be of the In Minkowski coordinates Ϯ, xϮϭϮeϮ , one has 0 form ds2ϭϪd2ϩd2, ϭϪ, ϮϭϮ. Ϫ2 Ϫ, ⍀͑ii͒͑xϩ,xϪ͒ϭ⍀͑i͒͑xϩ,xϪ͒ϩF͑xϪ͒, F͑xϪ͒ϭ0. ͑3.1͒ץϩץ The curvature scalar of the geometry, Rϭ8e can be conveniently written as 0 The ‘‘reflecting’’ RST boundary conditions follow from the 1 requirement of finite curvature on the boundary line. Indeed, eϪ2͒. ͑2.7͒ ץ ץϪ4 ץ ץ͑ Rϭ8eϪ2 ⍀Ј͑͒ ϩ Ϫ ϩ Ϫ from Eq. ͑2.7͒, we see that in order to have finite curvature at the line ⍀Ј()ϭ0 it is necessary that
In this form we see that, generically, there will be a curvature 2 1 singularity at ϭcrϭϪ 2 ln where ⍀Ј()ϭ0. Ϫ͉ϭ ϭϪ , ͑3.2͒ץϩץ As observed in Ref. ͓4͔, there are two different regimes, cr 4 ϩ according to whether Tϩϩ(x ) is smaller or greater than the critical flux: where we have used the equation of motion ͑in the gauge 2 ϪϭϪ . Equation ͑3.2͒ implies, in particularץϩץ ϭ⍀͒ ͓see Eq. ͑2.2͔͒, cr ϩ Tϩϩ͑x ͒ϭ ϩ2 . ͑2.8͒ ⍀ ϭ0. ͑3.3͒ ץ⍀ ϭ ץ x ϩ ͉ϭcr Ϫ ͉ϭcr
Note that the existence of the threshold is a quantum effect. As a result, the function F(xϪ) is determined to be Indeed, is proportional to ប͑here we have set បϭ1͒ and thus Tcr vanishes as ប 0. Using Eq. ͑2.5͒ it can be seen xϪ ϩϩ Ϫ Ϫ1 ϩ → ϩ ˆ that the line ⍀ϭ⍀ ͓ϵ⍀( )͔ is timelike if T (x ) F͑x ͒ϭ ln Ϫ Ϫ2 ϩ Ϫ M͑x ͒, ͑3.4͒ cr cr ϩϩ ͫx ϩ Pϩ͑xˆ ͒ͬ cr ϩ ϩ ϽTϩϩ(x ), and it becomes spacelike as soon as Tϩϩ(x ) cr ϩ ϩ ϩ Ϫ ϾTϩϩ(x ). where xˆ ϭxˆ (x ) is the boundary curve given by 976 D. AMATI AND J. G. RUSSO 56
2 ϩ Ϫ Ϫ2 ϩ Ϫ xˆ ͓x ϩ Pϩ͑xˆ ͔͒ϭ. ͑3.5͒
Finally, in region ͑iii͒, the geometry is matched with the vacuum:
⍀͑iii͒ϭ͑iii͒ϭϪ2xϩ͑xϪϩp͒Ϫ ln͓Ϫ2xϩ͑xϪϩp͔͒,
Ϫ2 ϩ pϵ Pϩ͑x1 ͒. ͑3.6͒
Ϫ In Minkowski coordinates (xϪϩp)ϭϪeϪ , xϩ ϩ ϭe , this simply becomes ds2ϭϪdϩdϪ, ϭϪ. The outgoing energy density fluxes measured by an out observer can be found from the constraints. They are given by
1 1 ͑i͒ Ϫ TϪϪ͑x ͒ϭ Ϫ 2Ϫ 2 , ͑3.7͒ ͫ͑x ϩp͒ xϪ ͬ FIG. 2. Intermediate regime. 1 4 T͑ii͒ ͑xϪ͒ϭ Ϫ , ͑3.8͒ ϪϪ ͑xϪϩp͒2 ⑀ ˆϩ ϩ 2ϪTϩϩ͑x ͒ Tϩϩ͑x ͒ϭ 2 , ͑3.12͒ xˆ ϩ xϩ
͑iii͒ TϪϪϭ0. ͑3.9͒ whence
Ϫ Ϫ The radiation energy emitted between times 1 and 2 is 1 1 given by the integral 2pϭP ͑xϩ͒ϭ⑀ Ϫ , mϭ⑀ ln͑xϩ/xϩ͒. ϩ 1 ͩ xϩ xϩ ͪ 1 0 0 1 Ϫ Ϫ ͑3.13͒ 2 Ϫ x2 Ϫ Ϫ Eϭ d T Ϫ ϪϭϪ dx ͑x ϩp͒T . ͵ Ϫ ͵Ϫ ϪϪ 1 x1 ϩ ϩ ϩ ϩ For x1 close enough to x0 ͓more precisely, for x1 Ͻx0 /(1 Ϫ Thus the total radiated energies in regions ͑i͒ and ͑ii͒ are Ϫ/e), ⑀Ͼ͔ we can have 1ϩ(p/x0 )Ͼ0 even above the threshold for black hole formation, i.e., with ⑀Ͼ. Ϫ p p ͑i͒ x0 Ϫ Ϫ ͑i͒ EoutϭϪ dx ͑x ϩp͒TϪϪϭϪ Ϫ Ϫ ln 1ϩ Ϫ , ͵Ϫϱ x ͩ x ͪ IV. THE INTERMEDIATE REGIME 0 0 ͑3.10͒ As mentioned before, Eqs. ͑3.10͒ and ͑3.11͒ can be con-
Ϫ tinued above the threshold without encountering any singu- ͑ii͒ x1 Ϫ Ϫ ͑ii͒ Ϫ Ϫ2 EoutϭϪ dx ͑x ϩp͒TϪϪ larity up to pϭ͉x0 ͉, pϵ Pϩ(ϱ), where the logarithmic ͵xϪ 0 singularity appears. We shall call this the intermediate re- Ϫ cr gime, i.e., the case when pϽ͉x0 ͉ and TϩϩϾTϩϩ for some p p ϩ ϭmϩ Ϫ ϩ ln 1ϩ Ϫ , ͑3.11͒ x , as opposed to the ‘‘supercritical’’ regime where p x ͩ x ͪ Ϫ 0 0 Ͼ͉x0 ͉. The former describes small ‘‘Planck-size’’ black ϩ holes, whereas the latter includes macroscopic black holes where mϵM(x ) is the total Arnowitt-Deser-Misner Ϫ 1 ͑the classical picture is approached for pӷ͉x ͉͒. ͑ADM͒ energy of the initial configuration. The coordinate 0 Ϫ ϩ The geometry is exhibited in Fig. 2 for the case where the x0 is related to the time x0 , at which the incoming flux incoming energy density is larger than the critical one in Ϫ 2 ϩ Ϫ begins, by x0 ϭϪ(/ x0 ). For pӶ͉x0 ͉ ͑‘‘low-energy’’ the whole range xϩϽxϩϽxϩ . Region a is defined as ͑i͒ ͑ii͒ 0 1 ͑ ͒ fluxes͒, one has EoutӶm, Eoutϳm, that is, most of the en- Ϫ Ϫ Ϫ Ϫ Ϫ x Ͻx1 , region ͑b͒ as x1 Ͻx Ͻx0 , and region ͑c͒ as ergy comes out by pure reflection on the space-time bound- Ϫ Ϫ Ϫ Ϫ x Ͼx0 . Figure 2 can be understood as a deformation of ary. For pХϪx0 , pϽ͉x0 ͉, the logarithm becomes large Fig. 1. In this process region ͑ii͒ and part of ͑i͒ of Fig. 1 are and negative so that the energy radiated in region ͑ii͒ is nega- superposed into region ͑b͒ of Fig. 2. Region ͑c͒ is part of tive. (c) ͑iii͒ region ͑iii͒, so that T ϭT ϭ0. It is thus convenient to cr ϩ2 ϪϪ ϪϪ Note that it is possible to have TϩϩϾTϩϩϭ/x and split the integral ͑3.10͒ as Ϫ yet pϽϪx0 . This means that the threshold given by the singularity of the logarithm in Eqs. ͑3.10͒ and ͑3.11͒ is not in ͑i͒ p p ͑a͒ ͑ib͒ general the threshold for black-hole formation. To see this EoutϭϪ Ϫ Ϫ ln 1ϩ Ϫ ϭEoutϩEout , ͑4.1͒ explicitly, let us consider the simplest case in which the in- x0 ͩ x0 ͪ coming energy-density flux is constant in Minkowski coor- dinates, so that in Kruskal coordinates it reads where 56 BLACK HOLES BY ANALYTIC CONTINUATION 977
Ϫ p p ͑a͒ x1 Ϫ Ϫ ͑i͒ EoutϭϪ dx ͑x ϩp͒TϪϪϭϪ Ϫ Ϫ ln 1ϩ Ϫ , ͵Ϫϱ x ͩ x ͪ 1 1 ͑4.2͒
xϪ E͑ib͒ϭϪ 0 dxϪ͑xϪϩp͒T͑i͒ out ͵Ϫ ϪϪ x1 Ϫ p p ͑1ϩp/x0 ͒ ϭϪ Ϫ ϩ Ϫ Ϫ ln Ϫ . ͑4.3͒ x0 x1 ͑1ϩp/x1 ͒ The first integral gives the energy radiated in region ͑a͒ of Fig. 2. The second integral contributes to the radiation in region ͑b͒. The total energy in region ͑b͒ is obtained by ͑ii͒ Ϫ Ϫ adding Eout. Since now x1 Ͻx0 , it is convenient to write this integral in the following way:
Ϫ ͑ii͒ x0 Ϫ Ϫ ͑ii͒ EoutϭϪ dx ͑x ϩp͒͑ϪTϪϪ͒ ͵xϪ 1 FIG. 3. Apparent horizon in the supercritical regime. p p ϭmϩ Ϫ ϩ ln 1ϩ Ϫ , ͑4.4͒ correspond to the analytic continuation of the subcritical re- x0 ͩ x0 ͪ gime. We will now show that the other option, namely im- ϩ Ϫ ϭ -ϩ⍀(x ,x ) 0 ͑the apparץ so that posing boundary conditions at ent horizon͒, will reproduce the results that were previously Ϫ obtained by a simple continuation of the subcritical formulas. ͑b͒ ͑ib͒ ͑ii͒ x0 b E ϭE ϩE ϭϪ dxϪ͑xϪϩp͒˜T͑ ͒ , ͑4.5͒ Let us assume that the incoming supercritical energy- out out out ͵Ϫ ϪϪ x1 ϩ ϩ density flux Tϩϩ(x ) starts at x0 , and it is turned off at a later time xϩ ͑a more general situation is discussed in Sec. ˜T͑b͒ ϵT͑i͒ ϪT͑ii͒ , 1 ϪϪ ϪϪ ϪϪ VI͒. In region ͑a͒ the geometry will be given by Eq. ͑2.5͒. Ϫ Ϫ ϩ⍀ϭ0 becomes timelike for x Ͼx1 , andץ The boundary ͑b͒ p p Eoutϭmϩ Ϫ ϩ ln 1ϩ Ϫ . ͑4.6͒ boundary conditions are needed in order to determine the x1 ͩ x1 ͪ evolution of the geometry in region ͑b͒͑see Figs. 2 and 3͒. Ϫ ͑a͒ ͑b͒ Continuity along the line x1 requires that Clearly, EoutϩEoutϭm, so that the whole incoming energy has been radiated ͓see Eqs. ͑3.10͒ and ͑3.11͔͒. This means ͑b͒ ϩ Ϫ ͑a͒ ϩ Ϫ Ϫ that these black holes evaporate completely. ⍀ ͑x ,x ͒ϭ⍀ ͑x ,x ͒ϩF͑x ͒, ͑5.3͒ It should be noticed that in the region b i.e., in the ͑ ͒͑ Ϫ region in causal contact with the apparent horizon͒ the with F(x1 )ϭ0. We need to generalize the expression ͑3.4͒ for the case when there is some energy stored in the geom- TϪϪ arising in the RST formalism does not coincide with the straightforward continuation of the subcritical formulas etry by the time the boundary becomes timelike. The form of Ϫ given by Eq. ͑4.5͒. Indeed, in RST the energy-momentum F(x ) in the subcritical regime suggests the choice ͑the gen- (i) eral structure will be clear in Sec. VI͒ tensor keeps being TϪϪ until the geometry is matched with the vacuum. Although in both cases the original energy is xϪϩ Ϫ2P u completely radiated, the structure of the out-going energy- Ϫ ϩ͑ ͒ Ϫ1 Ϫ1 ϩ F͑x ͒ϭ ln Ϫ Ϫ2 ϩ ϩ M͑u͒Ϫ M͑x1 ͒, density flux in the two models is different. ͩ x ϩ Pϩ͑x ͒ͪ 1 ͑5.4͒ V. APPARENT HORIZON AS A BOUNDARY Ϫ ϩ ϩ with u(x ) given by the branch x0 ϽuϽx1 of the solution In the subcritical regime the boundary conditions ͑3.3͒ or, to the equation equivalently, 2 Ϫ Ϫ2 Ϫ u͓x ϩ Pϩ͑u͔͒ϭ. ͑5.5͒ ⍀ϭ⍀cr , ͑5.1͒ We will find that given the boundary condition 5.3 with ͒ ͑ ϩ⍀ϭ0, ͑5.2͒ץ Eqs. ͑5.4͒ and ͑5.5͒, then Eqs. ͑4.2͒, ͑4.5͒ are reproduced ͑in can be implemented simultaneously on some line. Above the particular, this means that this boundary condition conserves threshold the line defined by Eq. ͑5.1͒ is necessarily different energy͒. The formulas for the outgoing fluxes will be identi- from the line defined by Eq. ͑5.2͒. The usual choice is to cal to those obtained by direct extrapolation from the sub- define the boundary line by ⍀ϭ⍀cr , since it is on this line critical regime. that the curvature is singular. This leads to the black-hole Let us note that the matching between regions ͑a͒ and ͑b͒ evolution described in ͓4͔ which, although it reproduces the is smoother than in the case of ͓4͔, i.e., there is no outgoing standard Hawking model of gravitational collapse, does not shock wave: 978 D. AMATI AND J. G. RUSSO 56
͑b͒ ϩ Ϫ 2 ϩ Ϫ Ϫ Ϫ dF Ϫ ⍀ ͑x ,x ͉͒ ϱϭϪ x ͑x ϩp͒ T ͑x ϩ⑀͒ϪT ͑x Ϫ⑀͒ϭϪ ␦͑xϪx ͒ϭ0, → ϪϪ 1 ϪϪ 1 dxϪ 1 m f ͑5.6͒ Ϫ ln͓Ϫ2xϩ͑xϪϩp͔͒ϩ , since ͓see Eqs. ͑5.4͒ and ͑5.5͔͒ ͑5.11͒
dF 1 1 ϩ Pϩ͑x2 ͒ Ϫ ϭ Ϫ Ϫ2 Ϫ Ϫ m ϭM͑xϩ͒ϩ ln 1Ϫ , xϩϵu͑Ϫp͒. dx ͫx ϩ Pϩ͑u͒ x ϩpͬ f 2 ͩ 2p ͪ 2 ͑5.12͒ Ϫ Ϫ vanishes at x ϭx1 . This is a static geometry with ADM mass equal to m f . In the 2 ϩ Ϫ A. Intermediate regime whole of region ͑b͒, where Ϫ x (x ϩp)Ͼ, the loga- rithmic term can be neglected and the geometry is essentially Let us first consider the intermediate regime. In region the same as the classical black-hole geometry. The logarith- Ϫ Ϫ Ϫ ͑c͒, defined by ϪpϾx Ͼx0 , the geometry is matched with mic term is only significant close to the line x ϭϪp, where the linear dilaton vacuum: i.e., there is a singularity. However, this is beyond the boundary at the apparent horizon. ͑c͒ xϩ,xϪ ϭϪ 2xϩ xϪϩp Ϫ ln Ϫ 2xϩ xϪϩp . ⍀ ͑ ͒ ͑ ͒ ͓ ͑ ͔͒ Let us check that energy is conserved. We now obtain by ͑5.7͒ explicit integration The energy-momentum tensor in the different regions are xϪ p p ⍀ϩt (xϪ), t (xϪ)ϭ/(xϪ ͑a͒ 1 Ϫ Ϫ ͑a͒ 2ץfound to be ͓T ϭϪ ϪϪ Ϫ Ϫ Ϫ EoutϭϪ͵ dx ͑x ϩp͒TϪϪϭϪ Ϫ Ϫ ln 1ϩ Ϫ , ϩp)2] ϱ x1 ͩ x1 ͪ ͑5.13͒ 1 1 T͑a͒ ͑xϪ͒ϭ Ϫ , ͑5.8͒ Ϫp ϪϪ ͫ͑xϪϩp͒2 xϪ2ͬ E͑b͒ϭϪ dxϪ͑xϪϩp͒T͑b͒ out ͵Ϫ ϪϪ x1 du Ϫ ͑b͒ Ϫ 2 ͑c͒ Ϫ p ͑1ϩp/x1 ͒ TϪϪ͑x ͒ϭ ϪϪ 2 , TϪϪ͑x ͒ϭ0. ͑5.9͒ ϩ dx xϪ ϭmϪM͑x2 ͒ϩ Ϫ ϩ ln ϩ 2 , x1 ͓1ϪPϩ͑x2 ͒/ p͔ In particular, we note that, since uЈϭ2(/u2 ͑5.14͒ ϪT (u))Ϫ1Ͻ0 ͑the flux is above the critical flux͒, the ϩϩ so that outgoing flux in region ͑b͒ carries negative energy. Since in region ͑a͒ the solution was not modified, one has 2xϩp (a) (i) ͑a͒ ͑b͒ ϩ 2 TϪϪϭTϪϪ ͓see Eq. ͑3.7͔͒. Now we note the surprising re- EoutϩEoutϭmϪM͑x2 ͒ϩ ln ϭmϪm f , lation ͓see Eqs. ͑3.7͒, ͑3.8͒, and ͑4.5͔͒ ͩ ͪ ͑5.15͒ ͑b͒ ͑i͒ ͑ii͒ ˜͑b͒ TϪϪϭTϪϪϪTϪϪϵTϪϪ . ͑5.10͒ 2 ϩ Ϫ2 ϩ where we have used the relation x2 ͓pϪ Pϩ(x2 )͔ ϭ. Thus energy is indeed conserved, and the total radiated Thus the outgoing energy momentum tensor in this theory energy is positive definite, since ϩ⍀ϭ0 as a boundary coincides with the extrapolationץ with of the subcritical energy momentum tensor beyond the ϩ ϩ x1 ϩ ϩ threshold for black-hole formation, indicating that it is the mϪM͑x2 ͒ϭ dx x TϩϩϾ0, ͵xϩ theory defined with the boundary at the apparent horizon that 2 represents the analytic continuation of the subcritical regime. 2 ϩ 2 ϩ 2 ϩ Ϫ and ln( x2 p/)Ͼ0. Indeed, x2 p/Ͼ x0 p/ϭp/͉x0 ͉, with p/͉xϪ͉Ͼ1 in the supercritical regime. B. Supercritical regime 0 Let us estimate the mass m f of the remaining black hole. Ϫ Ϫ Let us now proceed by considering the case pϾ͉x0 ͉ ͑Fig. For a ‘‘macroscopic’’ black hole, i.e., with pӷ͉x0 ͉,itis 1 ϩ ϩ 3͒. The energy-momentum tensor in region ͑b͒ can either be clear that M(x2 ),Pϩ(x2 ) will not differ much from ϩ ϩ ϩ ϩ obtained by analytic continuation or by using Eqs. ͑5.3͒ and M(x1 ),pϵPϩ(x1 ), since x2 Хx1 ͑see Fig. 3͒.Wecan ͑5.4͒, and it will be given by Eq. ͑5.9͒, just as in the inter- ϩ therefore anticipate that m f ХM(x1 )ϵm. This means that mediate regime. The final ϱ geometry for a timelike ob- very little energy has been radiated and the final black hole → ϩ Ϫ server is obtained by taking the limit x ϱ and x Ϫp will have a mass similar to the total imploding energy. This ϩ→Ϫ → in Eqs. ͑5.3͒ and ͑5.4͒͑recall 2ϭlnx /͉x ϩp͉͒: is very different from the standard picture of Hawking evaporation. To be explicit, let us consider two extreme cases, namely the case of a constant energy-density flux fall- 1Note that possible discontinuities in T (xϩ) produce disconti- ing in for a long time, and the case of a shock-wave collapse. ϩϩ ϩ nuities in the derivative of the curve representing the apparent ho- Using Eqs. ͑3.12͒ and ͑3.13͒ we find for the former x2 ϩ rizon ͑such discontinuities can of course be present in all regimes͒. ϭx1 (1Ϫ/⑀), and 56 BLACK HOLES BY ANALYTIC CONTINUATION 979
ϩ ϩ ⑀ x1 ⑀ ⑀ x0 E͑a͒ ϩE͑b͒ϭ ln Ϫ1 ϩ͑⑀Ϫ͒ln . ͑5.16͒ aϵ , yϵ , y ϭ1ϪaϪ1, bϵa͑y Ϫ1Ϫ1͒. out out ͩ xϩ ͪ ⑀Ϫ xϩ 0 0 1
͑ii͒ ͑b͒ For xϩ/xϩӷ1 we get It can be easily seen that the minimum value of Eout, Eout 1 0 given by Eqs. ͑5.18͒–͑5.20͒ is Ϫ, and it occurs at the point yϭ0 and aϭ1 ͑corresponding to an incoming flux equal to the critical flux lasting forever͒. Thus mϪm ϭ m. f ⑀ ͑b͒ EoutуϪ. ͑5.21͒ Since pӷ͉xϪ͉ implies ⑀ӷ, this is a small quantity. The 0 This is essentially the same bound as appears in the RST total radiated energy in the opposite limit of a shock wave model. Although we have proved Eq. ͑5.21͒ for a constant can be found by using Eq. ͑5.15͒ and the fact that, for a 3 ϩ incoming flux, a similar bound can be obtained in the general shock wave, pϭm/ x0 . This gives ͑b͒ case. Consider the general expression for Eout in the super- critical regime ͑which includes the case of macroscopic m black holes͒. It is convenient to write Eq. ͑5.14͒ in the form m ϭmϪ ln . ͑5.17͒ f ͑b͒ ϩ p EoutϭmϪM͑x2 ͒ϩ Ϫ Ϫ ln 1ϩ 2 ϩ While the radiated energy logarithmically increases with x1 ͩ px1 ͪ m, the ratio m f /m 1asm ϱ. ϩ ϩ → → Ϫ ln͑x1 /x2 ͒. ͑5.22͒
C. Outgoing energies From the inequalities The energies radiated in region ͑i͒ of Fig. 1 and in region xϩ xϩ ͑a͒ of Figs. 2 and 3 are positive definite, since they are the mϪM͑xϩ͒ϭ 1 dxϩxϩT Ͼ 1 dxϩxϩTcr 2 ͵ ϩ ϩϩ ͵ϩ ϩϩ integral of a positive-definite quantity ͓see Eq. ͑3.7͔͒.We x2 x2 have also seen in the previous subsection that the total radi- ϩ ϩ ated energy is positive definite. In the subcritical regime—as ϭ ln͑x1 /x2 ͒, mentioned in Sec. III—the energy in region ii can be posi- ͑ ͒ Ϫ tive or negative, depending on the characteristics of the in- p/x1 ϾϪ, and coming flux. This will be clear from the examples that we give below. As pointed out after Eq. ͑5.9͒, the energy E͑b͒ is out Ϫ ln 1ϩ ϾϪ ln 1ϩ negative definite, being the integral of a negative-definite ͩ 2pxϩͪ ͩ 2pxϩͪ 1 0 quantity ͓see Eq. ͑5.9͔͒. Here we show that this negative Ϫ energy is of the order of the Planck mass, i.e., smaller than x0 ϭϪ ln 1Ϫ ϾϪ ln2, O(). This characteristic is present in the RST model as ͩ p ͪ well, where negative energy is carried out by a shock wave ͑the ‘‘thunderpop’’͒ at the endpoint of black-hole evapora- we obtain tion. As shown below, here the analogue endpoint wave is smeared in a Planck time. E͑b͒уϪϪ ln2. We start by considering the particular example of the con- out stant incoming flux given by Eq. ͑3.12͒. Using Eqs. ͑3.13͒, Next, let us estimate the time interval of the negative ͑3.11͒, ͑4.6͒, and ͑5.14͒ one finds the following expressions. energy emission. For simplicity we will consider the case of Subcritical regime ͓a,y(0,1)͔: a constant incoming flux. In Minkowski coordinates Ϯ the ͑ii͒ energy momentum tensor ͑5.9͒ takes the form Eoutϭ͓Ϫa lnyϪa͑1Ϫy͒ϩln͑1Ϫaϩya͔͒. ͑5.18͒
͑b͒ ͑⑀/Ϫ1͒ 1 Intermediate regime ͓a (1,ϱ), y (y ,1)͔: TϪϪϭϪ Ϫ 2 ϩ 2 ϩ Ϫ 2 , 0 ͓ͫ͑⑀/͒e Ϫ1͔ ͓1ϩ͑ px /͒e ͔ ͬ 1
E͑b͒ϭ͓Ϫa lnyϪ͑1ϩbϪ1͒Ϫ1Ϫln͑1ϩb͔͒. ͑5.19͒ Ϫ Ϫ ϩ out ϵ Ϫln͑x1 /͒. ͑5.23͒
Ϫ Supercritical regime ͓a(1,ϱ), y(0,y0)͔: The shifted Minkowski time is such that it starts at 0 when the negative energy emission begins. The second term
͑b͒ Ϫ1 Ϫ1 Ϫ1 Ϫ1 in Eq. ͑5.23͒ is always negligible with respect to the first Eoutϭ͓͑1Ϫa͒ln͑1Ϫa ͒Ϫ͑1ϩb ͒ Ϫln͑1ϩb ͔͒, (b) one. Since ⑀/Ͼ1, TϪϪ is an exponentially decreasing func- ͑5.20͒ Ϫ 1 Ϫ1 tion, with a damping time interval of order ⌬ ϭ 2 , i.e., a ‘‘Planckian’’ interval of time ͓more precisely, ⌬Ϫ 1 Ϫ1 1 Ϫ1 where ϭ 2 (1Ϫ/⑀)Ͻ 2 ͔. 980 D. AMATI AND J. G. RUSSO 56
VI. MORE GENERAL DISTRIBUTIONS This is approximately the same static black hole as in the OF INCOMING MATTER previous case, Eq. ͑5.11͒, except that now p is slightly dif- ϩ ϩ To complete the physical picture, let us also give the ge- ferent ͓since the energy-density flux for x Ͼx1 is subcriti- ϩ ϩ ometry in region ͑b͒ in the case when the incoming energy- cal, it can be easily seen that Pϩ(ϱ)ϪPϩ(x1 )Ͻ/x1 ͔. This ϩ difference produces only a tiny ͑Planck-scale͒ increase in the density flux does not stop at x1 . Let us assume that ϩ ϩ ϩ ϩ final mass m with respect to Eq. ͑5.11͒. Tϩϩ(x ) is a smooth function of x for all x Ͼx0 , and f ϩ ϩ At first sight, the fact that, for xϩϾxϩ , low-energy den- define x1 as the point at which Tϩϩ(x ) becomes less than 1 the critical flux, so that the apparent horizon becomes time- sity matter reflects on the apparent horizon may seem Ϫ strange. However, it must be stressed that this is a quantum like after this point. Continuity along the line x1 requires that effect, since only a subcritical energy-density flux would re- ϩ flect. If, after x1 , supercritical matter is sent in, the apparent ⍀͑b͒͑xϩ,xϪ͒ϭ⍀͑a͒͑xϩ,xϪ͒ϩF͑xϪ͒, ͑6.1͒ horizon will become spacelike and all but a Planckian bit of energy will be eaten by the black hole, increasing its size in Ϫ (a) with F(x1 )ϭ0 and ⍀ as given by Eq. ͑2.5͒. The expres- accordance with the total energy of the additional matter. sion that generalizes Eqs. ͑3.4͒ and ͑5.4͒ is
Ϫ Ϫ2 x ϩ Pϩ͑u͒ VII. OUTLOOK AND DISCUSSION F͑xϪ͒ϭ ln ϩϪ1M͑u͒ϪϪ1M͑xˆ ϩ͒, ͩ xϪϩϪ2P ͑xˆ ϩ͒ͪ ϩ ͑6.2͒ Here we have explored the theory which results from ana- lytically continuing the subcritical regime above the thresh- Ϫ ϩ ϩ ϩ old of black-hole formation. In the corresponding semiclas- with u(x ) given as before by the branch x0 Ͻx Ͻx1 of the solution uϭxϩ(xϪ) to the equation sical theory, quantum effects appear in various ways, but the net result is that only small alterations over a classical pic- 2 ϩ Ϫ Ϫ2 ϩ Ϫ x ͓x ϩ Pϩ͑x ͔͒ϭ, ͑6.3͒ ture appear. Let us summarize the picture. ͑1͒ Collapsing macroscopic matter ͑i.e., with incoming ϩ Ϫ ϩ ϩ and xˆ (x ) given by the upper branch x Ͼx1 .Asinthe energy-momentum tensor far above the threshold for black case of Sec. V, there is no shock-wave discontinuity in going hole formation͒ forms stable black holes with masses of the Ϫ from region ͑a͒ to region ͑b͒, since FЈ(x1 )ϭ0 ͓interestingly, same order as the total imploding energy plus minor emis- Ϫ2FЈ(xϪ)ϭxˆ ϩϪu, i.e., the distance between the two sion. This involves Hawking radiation at early times and a points of the apparent horizon corresponding to a given subsequent burst with tiny energy ͑of order of the Planck xϪ͔. mass͒. The energy-momentum tensor in region ͑a͒ is as in Eq. ͑2͒ If the infalling matter has densities not much larger ͑5.8͓͒since the solution is the same in this region͒, and in than the critical one, the situation looks similar to the con- Ϫ2 region ͑b͒ one finds ͓pϵ Pϩ(ϱ)͔ ventional Hawking picture. This is the intermediate regime where a small black hole is formed and evaporates com- pletely. ⍀ϩ 2ץT͑b͒ ͑xϪ͒ϭϪ ϪϪ Ϫ ͑xϪϩp͒2 ͑3͒ Infalling subcritical matter over an already formed black hole will be reflected from the apparent horizon with a du dxˆϩ 2 2 small accompanying evaporation. ϭ ϪϪ ϪϪ 2 ϩ Ϫ 2 , dx dx xϪ ͑x ϩp͒ ͑4͒ Macroscopic matter falling over a black hole will sim- ply increase its mass and give rise to a limited emission, as in ͑c͒ Ϫ TϪϪ͑x ͒ϭ0. ͑6.4͒ ͑1͒. The bursts have negative energies of order and last a This is essentially the energy-density flux of Eq. ͑5.9͒ plus short Planckian time Ϫ1. A similar feature appears in the an additional ͑positive energy͒ contribution of the form ͑3.8͒ RST model, where the matching with the vacuum is made at ϩ ϩ ϩ representing reflection of the Tϩϩ(x ), x Ͼx1 on the time- the price of a shock-wave discontinuity; this shock wave ͑the like apparent horizon. The total mass of the final black-hole ‘‘thunderpop’’͒ carries out ‘‘Planckian’’ negative energy. In geometry will not vary too much by bombarding it with sub- the present model, the different regions are smoothly critical energy density. Indeed, using Eqs. ͑6.1͒ and ͑6.2͒ we matched. In a sense, the shock wave is smeared-out in a ϩ Ϫ find that the final geometry at x ϱ, x Ϫp is given by Planckian interval of time. → → Why is the final geometry stable? The vanishing of the ͑b͒ ϩ Ϫ 2 ϩ Ϫ 2 ϩ Ϫ ⍀ ͑x ,x ͒ϭϪ x ͑x ϩp͒Ϫ ln͓Ϫ x ͑x ϩp͔͒ energy-momentum tensor at infinity requires—just as when the Boulware vacuum is adopted ͓6͔—a substantial modifi- m f ϩ , cation of the geometry near the line xϪϭϪp. As we have seen in Sec. V B, this is exactly what is happening. In the Ϫ 2 ϩ Ϫϩ Ͼ ϩ allowed space-time region x (x p) the geometry Pϩ͑x2 ͒ is essentially the same as the classical black-hole geometry. m ϭM͑xϩ͒ϩ ln 1Ϫ , xϩϵu͑Ϫp͒. f 2 ͩ 2p ͪ 2 Only at Ϫ2xϩ(xϪϩp)ӶeϪm/ is the geometry signifi- ͑6.5͒ cantly modified, but this lies beyond the boundary. 56 BLACK HOLES BY ANALYTIC CONTINUATION 981
The model agrees with the Hawking theory in the region the fact that the analytic continuation we investigate can only that is not in causal contact with the apparent horizon ͓called be done in the external world; for the infalling observer there region ͑a͒ in Fig. 3͔. Beyond this point ͓region ͑b͔͒, a quan- are no Hawking particles, so no in-out S matrix to be ana- tum theory of gravity is required in order to predict the out- lytically continued. In passing, we would like to stress that going spectrum, since outgoing modes have Planck frequen- no wall prevents a macroscopic infalling object from enter- cies at the moment they arise from the vicinity of the horizon ing into the black hole. As a macroscopic object is falling in, ͑i.e., about one Planck proper distance from the horizon; see the apparent horizon expands and the object always remains Refs. ͓1,7–9͔͒. Lacking a microscopic theory, some extra inside the black hole. It is only the small emitted radiation phenomenological input is needed, and several possibilities that is effectively described as if there was a boundary at the have been discussed ͓8,10–12͔. In the context of this two- apparent horizon. dimensional model, this is naturally realized in two sce- The boundary condition at the apparent horizon is not narios. The first one, described in RST, is based on a quan- equivalent to the ’t Hooft S-matrix approach ͓1,11͔ or the 3 tum field theory with a boundary at ⍀ϭ⍀cr ͑the singularity͒; stretched horizon of Susskind et al. ͓15,8͔, where an effec- the other, described here, follows from analytic continuation tive dynamical boundary is proposed with the task of ‘‘trans- -ϩ⍀ϭ0 ͑the apparent horizon͒. fering’’ the quantum mechanical information of the incomץ and implies a boundary at For the former, the physics above the threshold reproduces ing matter to the outgoing modes. In those approaches, the the Hawking model of gravitational collapse, and thus leads discrepancy in the descriptions of distant and freely infalling to information loss. But, as we have seen, this physics is not observers has to be explained in terms of a rather strong analytically connected to the subcritical regime: an S matrix notion of complementarity, where quantum gravity effects constructed on the basis of the subcritical theory would not must be such that they ‘‘destroy’’ the infalling object, extract 2 describe this conventional approach. its quantum mechanical information, and transfer it to the In Refs. ͓1,11͔ the unitarity property of the S matrix was outgoing modes ͑which must happen even before the object used to constrain the number of fundamental degrees of free- reaches the apparent horizon͒. This is not the case in the dom in quantum gravity within a given volume ͑see also present approach: both ingoing and outside observers agree ͓14,12͔͒. But unitarity is not the only implication of having that the bulk of matter and its information remain in the hole an S matrix: an S matrix also requires that the physics above ͑i.e., beyond the event horizon͒. There is no need of dupli- the thresholds is described by the same ͑analytically contin- cation of the information and no conflict with causality. Here ued͒ formulas that govern the physics below the thresholds. we only make use of ’t Hooft’s arguments insofar as the Surprisingly, the consequence is that black holes stop evapo- emission of trans-Planckian modes must be affected by rating. It is important to understand the physics that may give quantum gravity effects. We are just exploiting the fact that rise to the different picture. The difference stems from the the quantum theoretical description which is appropriate to quanta that a distant observer in our model is absent, while an outside observer does not need to be extrapolated up to the observer of the conventional approach would be inter- the singularity; analyticity indeed dictates that simple preted as being originated from the small trapped region ͑of Dirichlet- or Neumann-type boundary conditions must be Planckian proper length͒ which is in the causal past of null imposed to the quantum fields at the apparent horizon ͑which infinity, i.e., in between the receding apparent horizon and amounts to exclude the region where the contours rϭe Ϫ the null line x ϭϪp. Those quanta would have experienced ϭconst are spacelike͒. a tremendous red shift from trans-Planckian frequencies of The dynamics could be quite different in the order econst M/, so their inhibition seems in line with the (3ϩ1)-dimensional physics, where it is possible to have ultraviolet softening expected from quantum gravity. The classical scattering without black hole formation ͑e.g., in boundary has thus the same effect as imposing a cutoff at a terms of the impact parameter͒.In1ϩ1 dimensions there is Planck-scale frequency, which leads to a termination of the no classical scattering without black holes; the threshold is a Hawking process ͓7͔. pure quantum effect ͑a model in 3ϩ1 dimensions is inves- As pointed out by ’t Hooft ͓1͔, the inertial infalling ob- tigated in ͓17͔͒. The results of this two-dimensional model, server cannot be used to argue that outgoing radiation at provide, however, a simple and concrete example of how the sub-Planckian distances from the horizon is not affected by requirement of analytic continuation from a subcritical quantum gravity effects, since this observer sees no Hawking regime—inevitable in an S-matrix approach—may shed light radiation at all. There is no contradiction, since it is not on black-hole behavior in a theory where there is no loss of possible for the ingoing observer to communicate the result quantum coherence. of any physical measurement to the outside world. The de- scription of the physics in the infalling frame is different. To an inertial infalling observer, strong quantum gravity effects ACKNOWLEDGMENTS only occur near the singularity, so no substantial change with respect to the classical Einstein physics is expected in the This work was partially supported by EC Contract No. horizon region. In the present context, this is consistent with ERBFMRXCT960090.
2We stress that we have continued the expressions for the energy- 3The apparent horizon always lies inside the stretched horizon and momentum tensor, since a satisfactory S-matrix formalism in 1 it coincides with it after the incoming flux terminates ͓16͔—in the ϩ1 dimensions is, unfortunately, still lacking ͑despite some inter- present model the stretched horizon is just given by Ϫ2xϩ͓xϪ Ϫ2 esting attempts ͓13͔͒. ϩ Pϩ(ϱ)͔ϭ. 982 D. AMATI AND J. G. RUSSO 56
͓1͔ G. ’t Hooft, Nucl. Phys. B256, 727 ͑1985͒; B335, 138 ͑1990͒. ͓11͔ C. Stephens, G. ’t Hooft, and B. F. Whiting, Class. Quantum ͓2͔ D. Amati, M. Ciafaloni, and G. Veneziano, Int. J. Mod. Phys. Grav. 11, 621 ͑1994͒. A 3, 1615 ͑1988͒; Nucl. Phys. B403, 707 ͑1993͒. ͓12͔ L. Susskind, J. Math. Phys. ͑N.Y.͒ 36, 6377 ͑1995͒. ͓3͔ S. Hawking, Commun. Math. Phys. 43, 199 ͑1975͒. ͓13͔ H. Verlinde and E. Verlinde, Nucl. Phys. B406,43͑1993͒;K. ͓4͔ J. G. Russo, L. Susskind, and L. Thorlacius, Phys. Rev. D 46, Schoutens, H. Verlinde, and E. Verlinde, Phys. Rev. D 48, 2670 ͑1993͒. 3444 ͑1992͒; 47, 533 ͑1993͒. ͓14͔ G. ’t Hooft, Phys. Scr. T36, 247 ͑1991͒;inSalamfestschrift, ͓5͔ S. Giddings, Phys. Rev. D 46, 1347 ͑1992͒. Proceedings of the Conference, Trieste, Italy, 1993, edited by ͓6͔ D. G. Boulware, Phys. Rev. D 11, 1404 ͑1975͒. A. Ali et al. ͑World Scientific, Singapore, 1993͒, gr-qc/ ͓7͔ T. Jacobson, Phys. Rev. D 44, 1731 ͑1991͒. 9310026. ͓8͔ L. Susskind, L. Thorlacius, and J. Uglum, Phys. Rev. D 48, ͓15͔ K. Thorne, R. Price, and D. MacDonald, Black Holes: The 3743 ͑1993͒. Membrane Paradigm ͑Yale University Press, New Haven, CT, ͓9͔ Y. Kiem, H. Verlinde, and E. Verlinde, Phys. Rev. D 52, 7053 1986͒. ͑1995͒. ͓16͔ J. G. Russo, Phys. Lett. B 359,69͑1995͒. ͓10͔ L. Susskind, Phys. Rev. Lett. 71, 2367 ͑1993͒. ͓17͔ J. G. Russo, Phys. Rev. D 55, 871 ͑1997͒.