6 March 1997

Physics Letters B 395 ( 1997) IO- 15

PHYSICS LEll-ERS B

Conformal equivalence of 2D dilaton gravity models

Mariano Cadoni * Dipartimento di Scienze Fisiche, Universitir di Cagliari, k+u Ospedale 72, I-09100 Cagliari, Italy

and INFN, Sezione di Cagliari, Italy

Received 12 November 1996 Editor: L. Alvarez-Gaunt?

Abstract

We investigate the behaviour of generic, matter-coupled, 2D dilaton gravity theories under dilaton-dependent Weyl resealings of the metric. We show that physical observables associated with 2D black holes, such as the mass, the temperature and the flux of Hawking radiation are invariant under the action of both Weyl transformations and dilaton reparametrizations. The field theoretical and geometrical meaning of these invariances is discussed.

PACS: 04.50.+h; 04.70.D~; 97.60.Lf Keywords: Two-dimensional gravity models; Black holes

The recent flurry of activity on two-dimensional

(2D) black hole physics [ 1 ] even though it has not succeeded in finding a definite answer to challenging questions such as the ultimate fate of black holes or the loss of quantum coherence in the evaporation pro- cess, has enabled us to gain considerable knowledge on the subject. Among other things, we have got a strong indication that a consistent description of black holes at the semiclassical or even quantum level re- quires us to treat the matter and the gravitational de- grees of freedom on the same footing. Considerable progress has been achieved by considering 2D dilaton gravity models from this purely field theoretical point of view, for example as a non-linear a-model [ 2,3], as a 2D conformal field theory [4] or in the gauge theoretical formulation [ 51.

One serious problem of this kind of approach is the difficulty in giving a geometrical interpretation to

’ E-mail: CADONI@CA.INPN.IT.

some field theoretical concepts. For example, from a purely field theoretical point of view, performing dilaton-dependent Weyl resealings of the metric in the 2D dilaton gravity action should give us equivalent models, since these transformations are nothing but reparametrizations of the field space. The space-time interpretation of this equivalence presents, however, some problems. Though the causal structure of the 2D space-time does not change under Weyl transforma- tions, geometrical objects such us the scalar curvature of the space-time or the equation for the geodesics do change. This discrepancy has generated a lot a confu- sion on the subject. Some authors have assumed ex- plicitly or implicitly this equivalence to hold and used it to simplify the description of the general model [6,7] or even to argue about the existence of Hawk- ing radiation in the context of the Callan-Giddings- Harvey-Strominger model (CGHS) [ 5,8]. Other au- thors, focusing on the space-time interpretation of the gravitational degrees of freedom, have pointed out the

0370-2693/97/%17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00025-7

M. Cadoni/Physics Letters B 395 (1997) lo-15

non-equivalence of 2D dilaton gravity models con- nected by Weyl resealings of the metric [ 8,9].

In this paper we analyze in detail the role of con- formal transformations of the metric in the context of generic, matter-coupled, 2D dilaton gravity theo- ries. We prove a conjecture reported in a previous pa- per [ lo] and based on previous results for the CGHS model [ 111, namely that physical observables for 2D black holes, such us the mass, the temperature and the flux of Hawking radiation, are invariant under dilaton- dependent Weyl resealings of the metric. Moreover, we show that the same observables are also invariant under reparametrization of the dilaton field.

The most general action of 2D dilaton gravity con- formally coupled to a set on N matter scalar fields has the form [ 7,121

Q,4,fl = & J

d2x J-g

(1)

where D, H, V are arbitrary functions of the dilaton (b and A is a constant. Let us consider the following Weyl transformations of the metric:

g,, = e P(b) A &V 7 (2)

for the moment we constrain the form of the func- tion P only by requiring the transformation (2) to be non-singular and invertible in the range of variation of the dilaton. Whereas the matter part of the action ( 1) is invariant under the transformation (2)) the gravi- tational part is not, but it maintains its form, in fact, modulo a total derivative we have

where the new functions B, A, V are related to the old ones through the transformation laws (’ = d/d4)

b=D, A=H+D~P~, V=ePv (3)

Under dilaton reparametrizations 4 = & 4 ) , V and D

behave as scalars, whereas H transforms as

(4)

The transformation laws (2), (3) and (4) enable us to find out how the physical parameters characterizing the solutions of the theory transform under the Weyl transformations (2) and dilaton reparametrizations.

Let us begin with the mass of the solutions. Mann has shown that for the generic theory defined by the action ( 1) , one can define the conserved quantity [ 121

,=z[jdDVexp( - J&z)

- (VD)2exp (- JdrE)]. where Fa is a constant. M is constant whenever the equations of motion are satisfied and, in this case, it can be interpreted as the mass of the solution. Us- ing Eqs. (2)-(4), one can easily demonstrate that the mass M given by the expression (5) is invariant under both Weyl transformation and dilaton reparametriza- tions.

The Hawking temperature associated with a generic black hole solution can be defined as the inverse of the periodicity of the Euclidean time necessary to remove the conical singularity at the event horizon. The generic static solutions in the conformal frame in which Z? = 0, have already been found in Ref. [ 61,

+a-2 (j- z)-‘dr’, D(4) +, (6)

where d.f/db = V, a is an arbitrary integration con- stant and M is the mass of the solution given by Eq. (5). The static solutions in the generic conformal frame can easily be obtained from these solutions us-

ing Eq. (2) with P = - J’~T [H(T)/D+-)],

(7)

A straightforward calculation gives for the Hawking temperature associated with an event horizon of the solution (7), located at (f, = 40,

12 M. Cadoni/Physics Letters B 395 (1997) lo-15

T= $V($bo) exp (8)

The temperature is invariant both under Weyl trans- formations and dilaton reparametrizations. This can easily be checked using Eqs. (3) and (4) in E!q. (8) and taking into account that the transformations (2) do not change the position of the event horizon, since by assumption they are everywhere non-singular and invertible.

In Eq. (6) and in the expressions (5)) (8) for the mass and the temperature appear two arbitrary con- stants a and Fo. To fix the values of these parame- ters we need a Weyl-invariant notion of asymptotic re- gion (spatial infinity) for our space-time. As already noted in a previous paper [ lo], the dilaton C$ gives a coordinate-independent notion of location and there- fore it can be used to define the asymptotic region, the singularities and the event horizon of our 2D space- time. Moreover, the natural coupling constant of the theory is D-‘f2 so that we have a natural division of our space-time in a strong-coupling region (D = 0) and a weak-coupling region (D + co). These con- siderations limit the range of variation of the function D to 0 5 D < co and enable us to identify the weak- coupling region D --+ 00 with the asymptoticregion of our space-time [ lo]. This notion of location is Weyl- invariant because D is invariant under the action of the transformations (2).

In the conformal gauge

ds2 = -e2Pdxfdx-, (9)

using a Weyl transformation (2), one can always put the solution (7) into the form

e’P=o’(l-g$),

K= IdDvexp (-J&z). (10)

This form of the solution can be used to fix the values of the parameters a and Fo. Using arguments similar to those of Ref. [ lo], one can show that a black hole interpretation of the solution (10) requires K -+ 00 for D -+ co. The condition that the metric (10) has asymptotically a Minkowskian form fixes now a = 1. The constant FO can be fixed to FO = l/h by requir- ing that in the conformal frame where the metric has

the form (lo), the norm of the Killing vector of the solution approaches, for D + 00, the value - 1.

We study the black hole solutions in the particular conformal frame in which the metric is asymptotically Minkowskian and has, therefore, the form ( IO). In this conformal frame, the ground state solution M = 0 coincides with Minkowsky space. Furthermore, we assume that the black holes exist in any conformally related frame. In the conformal frame defined by Eq. ( IO), the scalar curvature of the black hole space-time is

R = 2MAK& In K.

We require that the M # 0 solutions behave asymp- totically as the ground state solution, i.e., R + 0 for D -+ cm. This singles out three main classes of 2D dilaton gravity models, according to the asymptotical, D + co, behaviour of the function K:

K-D”, O<a<2,

KNylnD, O<y<oo,

K - epD, o<p<cQ. (11)

The first class of models has already been found and discussed in Ref. [ lo]. Our discussion, including the Hawking effect, holds also for models with LY = 2. In this case the solutions describe space-times that are asymptotically anti-de Sitter.

There are various ways to analyze the Hawking ef- fect. Here, we will use the relationship between Hawk- ing radiation and quantum anomalies [ 13,8,14,10]. It is well-known that in quantizing the scalar mat- ter fields f in a fixed background geometry the Weyl resealing and/or part of the diffeomorphism invari- ance of the classical action for the matter fields has to be explicitly broken. The quantization procedure has two sources of ambiguity. First, one can decide to preserve at the semiclassical level either the diffeo- morphism or the Weyl resealing invariance 115,161 (for sake of simplicity we do not consider here the case in which both symmetries are broken). Second, if one decides to preserve diffeomorphism invariance, one has still the freedom of adding local, covariant, dilaton-dependent counterterms to the semiclassical action [ 17,3,4]. The nature of these ambiguities is particularly clear in the path integral formulation. By choosing the diffeomorphism-invariant measure [ 18 I

M. Cadoni / Physics Letters B 395 (1997) I O-l 5 13

(12)

one breaks explicitly the Weyl invariance of the clas- sical matter action, and introduces an ambiguity re- lated to the choice of the metric to be used in the measure. One is allowed to use in Pq. ( 12) the metric gp, or a Weyl-resealed metric &. The corresponding semiclassical actions differ one from the other for the presence of local, covariant, dilaton-dependent coun- terterms. On the other hand, by choosing the Weyl- invariant measure [ 15,161

(13)

one breaks part of the diffeomorphism invariance of the classical action, but there is no ambiguity asso- ciated with the choice of the metric to be used in the measure. Let us first consider a measure defined by Eq. ( 12). The semiclassical effective action is diffeomorphism-invariant and in the conformal frame where the metric is asymptotically Minkowskian, it is given by

SC = SC1 - 4p 3 (14)

where $1 is the classical action ( 1) and Stp is the usual non-local Liouville-Polyakov action

sip = & J d2xJ-gR[g]~-2R[g],

where the notation g has been used in order to avoid confusion with the metric in the generic conformal frame.

The semiclassical action has its “minimal” Liouville-Polyakov form, with no dilaton-dependent counterterms present, exactly in the conformal frame where the solutions are asymptotically Minkowskian. This fact follows from very simple physical require- ments. Dilaton-dependent counterterms are forbid- den if one requires the expectation value of the stress-energy tensor to vanish when evaluated for Minkowsky space (the A4 = 0 ground state solution of our models). Under a Weyl transformation (2) the Liouville-Polyakov action acquires local, dilaton- dependent terms that have the same form as those already present in the action ( 1) . These terms depend on the form of the function P( 4) in Eq. (2), so that - as expected - the trace anomaly depends on the

particular conformal frame chosen. Using the equa- tion gpy = g,,exp (JdT[H(T)/D’(7)] - 1nK) in the expression ( 14), one easily finds the form of the semiclassical action in the generic conformal frame:

-& Jd’x J-g [2( htK-~dr~)R[gl

- ($ - ;)2(V&2]. (15)

The black hole radiation can now be studied along the lines of Ref. [lo], working in the conformal gauge (9) and considering a black hole formed by collapse of a f-shock-wave, travelling in the x+ direction and described by a classical stress-energy tensor T++ = M&x+ - x0’). The classical solution describing the collapse of the shock-wave, for x+ < x0+, is given by

, 4J J dr

-2,x+-x-,, K(7) 2

and, for xf 2 xt, it is given by

(16)

e2p=exp (-1dTs) (K- y)F’(x-1,

6

J dr

K(T) - y =; [x+-x; 4(x-)], (17)

F/(x-) = g = (18)

The next step in our semiclassical calculation is to use the effective action ( 15) to derive the expression for the quantum contributions of the matter to the stress- energy tensor. The flux of Hawking radiation across spatial infinity is given by (T__) evaluated on the asymptotical D + co region. For the class of mod- els in Eq. ( 11) a straightforward calculation, which follows closely that of Ref. [ lo], leads to

(T-h.5 = ;&{Ex-}, (19)

14 M. Cadoni/Physics Letters B 395 (1997) lo-15

where {E x- } denotes the Schwarzian derivative of the function F( X- ) . This is a Weyl resealing and dila- ton reparametrization invariant result for the Hawking flux. In fact the function F( x- ) is defined entirely in terms of the function K( 4) (see Eq. ( 18) ), which in turn is invariant under both transformations (see Eq. ( 10). When the horizon 40 is approached, the Hawk- ing flux reaches the thermal value

which is the result already found in Ref. [lo], written in a manifest Weyl resealing and dilaton reparametrization invariant form.

Let us now consider a Weyl-invariant measure de- fined by Eq. ( 13). The Hawking effect is now a con- sequence of the explicit breaking of diffeomorphism invariance. T,, does not transform as a tensor under general coordinate transformations yfi = yp (xp) , but it transforms as follows [ 141:

Applied to the shock-wave solution ( 16), ( 17) this transformation law reproduces our previous result ( 19). It is important to note that our result for the Hawking effect has been derived ignoring back- reaction effects of the radiation on the geometry. Al- though for the CGHS model the back-reaction effects do not change the Hawking flux [ 31, in the general case these effects could spoil the Weyl Invariance of our result. Moreover, when back-reaction effects are taken into account our assumptions used to derive ( 19) could become too restrictive. For example, they would exclude all those models for which there is a non-trivial one-loop correction to the classical ground state.

Let us now come to a central question: What is the physical meaning of the equivalence we have found? Does it imply the complete physical equivalence (at least at the semiclassical level and ignoring back- reaction effects) of 2D dilaton gravity models con- nected by Weyl transformations? In general the struc- ture of the space-time singularities is not preserved by Weyl transformations, since the scalar curvature of the

space-time changes under such transformations. Also the notion of geodesic motion depends on the choice of the conformal frame. In Fact, the geodesic equation is not invariant under the transformation (2), but ac- quires terms depending on the derivatives of the func- tion P. As a consequence, a space-time that is geodesi- tally complete in the range 0 5 D( 4) < cc can be mapped by the transformation (2) into a space-time that is not geodesically complete in the same range of variation for the dilaton. A nice example of this be- haviour is given by the model discussed in Ref. [ 111.

The answer to the previous questions depends on the features of the model we are interested in. Af- ter all, 2D dilaton gravity models are just toy mod- els for studying 4D gravity in a simplified context. Differently from the 4D case, where geometrical ob- jects such as the metric or the curvature have a di- rect physical meaning, in the 2D case these objects acquire a physical significance only through their re- lation with the 4D problem. There are examples in which 2D space-times with asymptotical anti-de Sit- ter behaviour can be used to model asymptotically flat 4D black holes near extremality [ 191. If geometrical features of the 2D model such as the curvature or the geodesic completeness of the space-time are crucial for our problem, we will consider models related by Weyl transformations as non-equivalent. On the other hand, if these geometrical features are irrelevant be- cause we want to treat the gravitational degrees of freedom on the same footing as the matter degrees of freedom or because our problem is focused on phys- ical observables such as masses or temperatures, we can regard the former models as equivalent.

In this discussion the form of the function P (4) in Q. (2) plays a crucial role. Until now we have assumed, generically, that P is such that the transfor- mation (2) is non-singular and invertible in the range of variation of the field 4. However, one can con- sider a function P subjected to stronger (or weaker) conditions. For example, one can consider as confor- mally equivalent models connected by a transforma- tion (2) such that only those functions P that preserve the structure of the singularities of the model are al- lowed. Conversely, one can also allow for functions P leading to transformations that become singular at some points of the dilaton field space. As an example, let us consider the models investigated in Ref. [ 201, they are defined by the action

M. Cadoni/Physics Letters B 395 (1997) IO-15 15

S= &/d2x J-g [e-‘m (R, ;(v,,i)

+ 4A2e-24 . 1 (21) One can easily show that the models with different values of the parameter n are conformally equivalent. In fact, by defining

D = e&, gpp = D”g,,, (22)

the action (2 1) becomes, modulo a total derivative,

S= & s

d2x J-g (DR[g] +4A2), (23)

which is the model studied in Ref. [ 111. This fact explains why the black hole solutions of the models (2 1) have the same values of mass, temperature and Hawking flux as the solution of the CGHS model (the model in IQ. (2 1) with n = 1) . However, the existence and the position of curvature singularities depend cru- cially on the parameter n. In fact, the transformation (22) implies the following on-shell relation:

R[g] = D”R[j] + 2nMADne2.

For 0 5 n 5 2 the transformation (22) does not change the position of the singularities. The black hole solutions of the models (21) with these values of n have a curvature singularity for D = 0 (the black hole solution of the model (23) have no curvature singularities but we treat the region of strong coupling D = 0 as a singularity). Conversely, for n > 2 the position of the curvature singularity is changed by the transformation (22) from D = 0 to D = cm.

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