On the conformal equivalence between 2D black holes and Rindler spacetime

更新时间：2023-07-17 22:17:01 阅读量：实用文档文档下载

说明：文章内容仅供预览，部分内容可能不全。下载后的文档，内容与下面显示的完全一致。下载之前请确认下面内容是否您想要的，是否完整无缺。

on是开还是关推荐度：
相关推荐

We study a two-dimensional dilaton gravity model related by a conformal transformation of the metric to the Callan-Giddings-Harvey-Strominger model. We find that most of the features and problems of the latter can be simply understood in terms of the class

a r

:g r

v 1

INFNCA-TH9516May 1995ON THE CONFORMAL EQUIV ALENCE BETWEEN 2D BLACK HOLES AND RINDLER SPACETIME Mariano Cadoni and Salvatore Mignemi Dipartimento di Scienze Fisiche,Universit`a di Cagliari,Via Ospedale 72,I-09100Cagliari,Italy.and INFN,Sezione di Cagliari.ABSTRACT We study a two-dimensional dilaton gravity model related by a conformal transformation of the metric to the Callan-Giddings-Harvey-Strominger model.We ?nd that most of the features and problems of the latter can be simply understood in terms of the classical and

semiclassical dynamics of accelerated observers in two-dimensional Minkowski space.PACS:04.70.Dy,11.25.-W,97.60.Lf

In the past few years,a great deal of work has been dedicated to the study of a two-dimensional model of dilaton gravity?rst proposed by Callan et al.(CGHS)[1].The reason of this interest is due to the fact that such model provides a good approximation of low-energy scattering from a nearly extremal black hole of four-dimensional string gravity. Such approximation is much more tractable than the original model and in fact permits the investigation of the process of formation and evaporation of a black hole in a semiclassical approximation.

The model is described by the two-dimensional action:

S= d2x 2 (??f i)2 ,(1)

whereφis the dilaton and f i are a set of scalar?elds.

At the classical level,this model admits an exact solution describing the formation of a black hole caused by a shock wave of incident matter.It is then possible to discuss the Hawking radiation of the black hole by means of the standard semiclassical calculation performed by quantizing the matter?elds f i in the background constituted by the classical solution.As is well known,in this approximation the contribution of the scalars to the energy-momentum tensor is proportional to the one-loop anomaly,which in the conformal gauge is in turn proportional to the curvature of the background metric.

The?nal result of the calculation is that a constant?ux of radiation is emitted,which is independent of the mass of the black hole[1].This surprising result can be improved by making a better approximation,taking into account the backreaction of the gravitational ?eld to the radiation.This topic has been widely investigated[2].

A further problem arises when one considers conformal transformations of the original metric.As is well known,in fact,a conformal transformation in two dimensions consists essentially in a rede?nition of the?elds and therefore the physical content of the theory should not depend on it.In particular,if one de?nes a rescaled metric

gµν=e?2φ?gµν,(2) the action(1)becomes

S= d2x√2 (?f i)2 .(3)

However,the action(3)admits only?at solutions(in Rindler coordinates)with non-trivial dilaton.It has therefore been argued that the theory de?ned by(1)should be trivial,since it is equivalent to?at space under conformal transformations and in particular does not admit Hawking radiation,since the conformal anomaly is of course zero for?at space[3], at variance with the results of[1].In this letter,we wish to clarify this point,by observing that in order to solve the puzzle,one has to take into due account the role of the dilaton ?eld,which should be considered on the same footing as the metric in the discussion of the structure of the spacetime.In particular,the requirement of reality of the dilaton ?eld implies that even?at solutions have a non-trivial structure,since one is forced to

impose a boundary to the spacetime on the curve where e?2φchanges its sign(from a four-dimensional point of view the line e?2φ=0corresponds to the origin of the radial

coordinate r).This fact prevents the possibility of performing a change of coordinates to put the Rindler metric in a Minkowski form.But it is well-known that to Rindler coordinates is associated a constant?ux of Hawking radiation,which is in this case of purely topological origin,and whose value turns out to coincide with the one calculated from(1).This mechanism is very similar to that discussed in[4]for two-dimensional anti-de Sitter spacetime,in the context of the Jackiw-Teitelboim model.

From our discussion will also emerge that the CGHS vacuum is semiclassically un-stable,unless one imposes a priori a cosmic censorship hypothesis(which can however be justi?ed from a four-dimensional point of view).In doing that,we clarify some results obtained in[5]by means of a moving mirror model.Our arguments will also permit us to consistently de?ne a mass for the dilaton Rindler spacetimes,which coincides with that of the conformally related CGHS solution.

The?eld equations stemming from(3)are:

R=0,?2f i=0,

(gµν?2??µ?ν)e?2φ=2λ2+T(f)µν,(4) where T(f)

µνis the energy-momentum tensor for the?elds f i.The general static solutions of these equations in the Schwarzschild gauge are given,for vanishing f i,by a locally?at metric and a non-trivial dilaton,namely:

ds2=?(2λr?c)dt2+(2λr?c)?1dr2,

(5)

e?2φ=2λr+d.

Without loss of generality,one can take d=0.It is important to notice that the

static solutions arise naturally in Rindler coordinates[6].A change of coordinatesσ=√2λr?c sinhλt brings the metric to the Minkowski form λ?1

ds2=?dτ2+dσ2,but the dilaton becomes time-dependent,e?2φ=c+λ2(σ2?τ2).

The central point of this paper is however the observation that,if one wishes to have a real dilaton,one is forced to cut the spacetime at the curve where e?2φchanges its sign (r=0in the coordinates(5)),so that one cannot obtain the full Minkowski space even by changing coordinates.One can interpret the curve where e?2φvanishes as a singularity of spacetime.Indeed this curve corresponds to a true curvature singularity in the CGHS model,as one can easily verify performing the rescaling(2).For c>0,this singularity is shielded by a horizon at r=c/2λand is spacelike,while for c≤0it is naked and timelike. In the case c=0it coincides with the coordinate singularity and is lightlike.

It is also possible to assign a mass to the solutions(5)by means of the ADM procedure. In fact,one can de?ne a conserved mass function[7]:

Of course,the nonvanishing value of the mass is due to the contribution of the dilaton, which cannot be separated from that of the metric.

The temperature at the horizon of the metric(5)can be easily obtained and is inde-pendent of c.Its value is given by

ln(2λr?c).

2λ

Notice that the boundary at r=0is not visible in these coordinates if c>0,since they cover only the region of the black hole spacetime outside the horizon at r=c/2λ.

Once the interpretation of solution(5)as a black hole has been established,at the semiclassical level one would naively expect this black hole to evaporate.The emergence of the Hawking radiation in our simple two-dimensional gravity model is however a point which deserves a careful study.Indeed conformal anomaly arguments have been used to argue that,being the space everywhere?at,there is no Hawking radiation in this model [3].In the following we will demonstrate,using standard quantization techniques,that the semiclassical dynamics of the black holes naturally gives rise to particle creation with

thermal spectrum.Even though this radiation cannot be interpreted in the usual way as the Hawking radiation seen by an observer in an asymptotically?at region,its existence sheds some light on the properties of the CGHS model and is crucial to understand the physical equivalence of the models related by the rescaling(2).

Let us assume that the black hole is formed by the collapse of a f-wave.For example one can easily construct,using light-cone coordinates,solutions describing the collapse of a f-shock wave at y+=y+0:for y+≤y+0the solution is given by the vacuum solution (c=M=0),whereas for y+≥y+0a black hole is formed.The key point is that for the

vacuum solution the boundary at r=0is lightlike so that one can introduce coordinates

x±=±

e2πω/λ?1.(11) This is precisely the Planck spectrum for radiation at temperature given by(6).When integrated it gives the total?ux of f-particle energy:

λ2

which is in agreement with the CGHS result.Our results simply state that the semiclassical behavior of our black holes is described by a quantum?eld theory of accelerated observers in Minkowski space.The physical interpretation of the associated thermal radiation deserves however careful analysis.It is in fact evident that it cannot be interpreted in the usual way, as the Hawking radiation seen by an observer in an asymptotically?at spacetime region. This prevents the usual interpretation of the Hawking e?ect as the distortion,due to the black hole geometry,of incoming quantum modes de?ned in a?at region into outcoming ones de?ned in another?at region.The point is that in our model the black hole geometry is truly equivalent to that seen by accelerated observers in Minkowski space.The event horizon of the black hole has to be considered as an acceleration horizon.This discussion clari?es a point which appears rather puzzling in the CGHS model.The CGHS black holes have temperature and Hawking?ux,given by(6)and(12)respectively,independent of the mass of the black hole;there is no explanation of this fact in CGHS theory.In our model a natural explanation of this fact is at hand:the thermal properties are independent of the mass because the semiclassical dynamics of the black hole is equivalent to that of accelerated observers,they depend only on the parameterλwhich de?nes the proper acceleration of these observers.Moreover our results restore some of the physical intuition about the equivalence of the models under the rescaling of the metric(2).What we have found is that the models(1)and(3)have an equivalent classical and semiclassical dynamics.Indeed the two metrics,being related by a Weyl rescaling,have the same causal structure(the same Penrose diagram)and at the semiclassical level have the same thermal properties.

At?rst glance our result seems to contradict the well-known relationship between conformal anomaly and Hawking radiation.Being the black hole spacetime everywhere ?at the conformal anomaly vanishes and there should be no Hawking radiation.However the presence of thermal radiation in our model is related to a topological e?ect which is independent of the presence of a conformal anomaly.On the other hand the possibility of having Hawking radiation in globally?at spaces is related to the anomalous transformation law of the quantum energy-momentum tensor T??under coordinates change.

To conclude this letter let us comment about the question of the stability of the ground state in our model and in the related CGHS model.Soon after the discovery of the CGHS model it was realised that the semiclassical ground state of this model is unstable and plagued by the presence of naked singularities[2].The meaning and the origin of this instability has been further clari?ed in Ref.[5],where a dynamical moving mirror in?at spacetime was used to model the semiclassical evolution of a CGHS black hole. It was found that vacuum solution do exist which describe a forever accelerating mirror resulting in an unphysical process of a forever radiating black hole.These features have a natural explanation in the context of our model.Di?erently from the CGHS case,for which the classical vacuum(the so called linear dilaton vacuum)is a perfectly regular and geodesically complete spacetime,in our model one can de?ne consistently a classical ground state only using a cosmic censorship conjecture to rule out the states with negative mass. Moreover this vacuum does not describe a complete spacetime but Minkowsky space with a light-like boundary.The presence of this boundary is a potential source of instability of the vacuum.This can be easily understood if one allows in the spectrum the states

with negative mass(thus giving up the cosmic censorship conjecture).The dynamics can be now described in terms of the evolution of a timelike boundary in Minkowski space, it is therefore equivalent to the dynamical moving mirrors discussed in ref.[5].There the authors considered a boundary on the curve where e?2φ=C,where C is an arbitrary positive paring with the expression(10)one easily sees that this is equivalent to consider our solutions with negative mass given by M=?Cλwith a boundary on the curve where e?2φ=0.The runaway solutions describing a forever accelerating moving mirror found in[5]are therefore related to the in?nite tower of black hole states with negative mass,i.e to spacetimes with a timelike boundary at e?2φ=0.Thus,unless one uses a cosmic censorship hypothesis to discard these states as unphysical,one has to deal with processes in which the black hole mass becomes arbitrarily negative.A similar point of view,stating the role of spacetime singularities acting as regulators forbidding the appearance of unphysical states in the spectrum,has been put forward recently by Horowitz and Myers[10].

References.

[1]C.G.Callan,S.B.Giddings,J.A.Harvey and A.Strominger,Phys.Rev.D45,

1005(1992);

[2]J.G.Russo,L.Susskind and L.Thorlacius Phys.Rev.D46,3445(1992);Phys.

Rev.D47,533(1993).

[3]D.Cangemi and R.Jackiw,Phys.Rev.D50,3913(1994);T.Fujiwara,Y.Igarashi

and J.Kubo,Phys.Lett.B316,66(1993).

[4]M.Cadoni and S.Mignemi,Phys.Rev.D(in press),hep-th9410041.

[5]T.D.Chung and H.Verlinde,Nucl.Phys.B418,305(1994).

[6]W.Rindler,Essential Relativity(Springer-Verlag,1969).

[7]R.B.Mann,Phys.Rev.D47,4438(1993);S.Mignemi,preprint INFNCA-TH-

94-28.