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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server A new lo ok at the RST mo del a a a,b Jian-Ge Zhou , F. Zimmerschied , J.{Q. Liang and a H. J. W. Muller{Ki rsten a Department of Physics, University of Kaiserslautern, P. O. Box 3049, D{67653 Kaiserslautern, Germany b Institute of Theoretical Physics, Shanxi University Taiyuan, Shanxi 030006, P. R. China and Institute of Physics, Academia Sinica, Beijing 100080, P. R. China Abstract The RST mo del is augmented by the addition of a scalar eld and a b oundary term so that it is well-p osed and lo cal. Expressing the RST action in terms of the ADM formulation, the constraint structure can b e analysed completely.It is shown that from the view p oint of lo cal eld theories, there exists a hidden dynamical eld in the RST mo del. Thanks to the presence of this hidden 1 dynamical eld, we can reconstruct the closed algebra of the constraints which guarantee the general invariance of the RST action. The resulting stress tensors T are recovered to b e true tensor quantities. Esp ecially, the part of the stress tensors for the hidden dynamical eld gives the precise expression for t . 1 At the quantum level, the cancellation condition for the total central charge is reexamined. Finally, with the help of the hidden dynamical eld , the fact 1 that the semi-classical static solution of the RST mo del has two indep endent parameters (P,M), whereas for the classical CGHS mo del there is only one, can b e explained. With the advent of the mo del prop osed by Callan, Giddings, Harvey and Strominger (CGHS) [1], dilaton gravityintwo dimensions has b een widely recog- nized as an excellent arena in whicha variety of fundamental issues in quantum gravity can b e discussed, esp ecially those concerning quantum prop erties of a black hole. Indeed now a large b o dy of literature on the CGHS mo del and its variants is available, and the notable mo del in the study of the black hole evap- oration problem is the Russo-Susskind-Thorl acio us (RST) mo del which admits physically sensible evap orating black hole solutions [2]. The RST mo del has b een considered a theoretical lab oratory for the study of Hawking radiation [2, 3], black hole entropy [4, 5, 6], critical phenomena [7, 8 ] and so on. However, until now, there are some problems which are still unclear in the RST mo del. For example, the RST action is manifestly invariant under the dif- feomorphism transformation, so the constraints should form the closed algebra at the classical level, i.e., ought to b e rst-class, which guarantees the general covariance or invariance of the theory. Nevertheless, as is well known, the stress tensors T in the earlier semiclassical approach do not transform as tensors but rather as pro jective connections, which means that under a conformal change of co ordinates T pick up an extra term equal to =2 times the schwarzian derivative of the transition function at the classical level [9]. As a result, the Poisson brackets of the constraints have the classical central extension, that is, the closed algebra of the constraints is destroyed [10], which explicitly contra- dicts the fact that the RST action is invariant with resp ect to di eomorphism transformations. Usually, arbitrary functions (more precisely, pro jective connections) t are ; added to T by hand, and under a conformal change of co ordinates t are assumed to pickup =2 times the schwarzian derivative, so that T +t are true tensor quantities. Since t are intro duced by hand, their precise meanings are implicit, so t havevarious physical explanations. For instance, in [11] t are explained as the stress tensors for the ghost sector, and their central charges are equal to 26, whereas in [3, 9] t are considered as the result of the nonlo cality of the Polyakov term, and the corresp onding central charges are 12. So, until now, it is not clear how these con icts could b e reconciled in a consistentway. In the present pap er, the RST mo del is rst dicussed from the viewp ointof the Dirac quantization metho d so as to solve the ab ove mentioned problems. Since there is a nonlo cal term in the RST mo del, the RST Lagrangian must rst b e lo calized so that Dirac quantization can b e p erformed. For this purp ose, the scalar eld and the b oundary term are intro duced in order that the reformulated RST mo del is well-p osed and lo cal [6, 15 ]. Expressing the RST action in terms of the ADM formulation [12, 13 , 14 ], the constraint structure can b e easily analysed. It is found that there are four rst-class constraints in the RST mo del, and twoof these generate the well-known Virasoro algebra without classical central charge. At the quantum level, the cancellation condition for the total central charge is reexamined. Three typ es of measures are discussed and the corresp onding 2 results are obtained. By the Hamiltonian constraint analysis, it is shown that except for the N scalar matter degrees of freedom, the true physical degrees of freedom for gravity, the dilaton and the new eld , are nonzero. From the viewp oint of the lo cal eld theories, there is a hidden dynamical eld in the RST mo del, whichwas omitted in the usual semiclassical approach. Exploiting the equations of motion, the stress tensors T can b e derived from our original constraints H . In comparison with the known results, it is found that just the stress tensors of the hidden dynamical eld give the precise expression for 1 t . Thus we conclude that the previous semiclassical approachisintrinsically inconsistent due to the omission of this hidden dynamical eld , which results 1 in the ab ove mentioned con icts. Finally, with the help of the hidden dynamical eld , the fact that the semi-classical static solution for the RST mo del has 1 two indep endent parameters (P,M), whereas for the classical CGHS mo del it has only one, can b e well elucidated. Wenow consider the RST mo del with the action [2] " # Z N X p 1 1 1 2 2 2 2 2 S = g e (R +4(r) +4 ) (rf ) (R R +2R) d x i 2 2 2 4 r M i=1 (1) where g is the metric on the 2D manifold M, R is its curvature scalar, is the i dilaton eld, and the f , i = 1,...,N, are N scalar matter elds. The nonlo cal term 1 R R comes from the familiar conformal anomaly. The lo cal and covariant term 2 r 2R is to preserve the simple form of the current j = @ ( ), with @ j =0. The co ecient has to b e p ositive, since in the case of b eing negative, there is no singularity in gravitational collapse [2]. Obviously, Eq. (1) is invariant with resp ect to the di eomorphism transformation. According to Ref. [15], one can intro duce an indep endent scalar eld to lo calize the conformal anomaly term, and add a b oundary term to de ne the variational problem prop erly. Then Eq. (1) b ecomes [15 ] Z p 1 2 2 2 2 S = d g fR~ + 4[(r) g x + ]e @ @ 2 4 Z N p X 1 1 2 (rf ) g d hK~ (2) i 2 i=1 2 where ~ = e (), h is the induced metric on the b oundary of M (assumed 2 spacelike), and K is the mean extrinsic curvature of @ M. As in (3+1)-dimensional gravity, the b oundary term serves to eliminate second time derivatives of the metric from the action which are contained in R. Following the ADM formulation, the metric can b e parametrized as follows [12, 13 , 14]: 3 2 g = e g^ (3) ! 2 2 + g^ = (4) 1 where (x) and (x) are lapse and shift functions resp ectively, and we factor out 2 the conformal factor e . In terms of this parametrization, the action (2) can b e written as Z q n 1 2 2 ^ d x g^ R ~ +2^g @ @~ 2^g @ @ e S = 2 N o X 1 2 2() +4 e g^ @ @ g^ @ f @ f i i 4 2 i=1 Z p d hK ~ (5) 1 ^ where R is the curvature scalar forg ^ , and for simplicity, the factor in front of action (2) has b een omitted. If weintro duce momenta , , resp ectively for the elds ; ; , the Hamiltonian would b ecome so complicated that we cannot quantize the theory. Thus we need a eld rede nition to diagonalize the kinetic term of action (5), which is rst given by p p 1 2 p = e + 0 2 p p = + 1 2 p 1 2 p = e + (6) 2 2 Here we should p oint out that the physical value of is restricted, i.e., it is a 2 non-negative quantity. If this restriction is ignored, the semi- classical solution of the mo del is unstable [16]. The black holes radiate forever at a xed rate and the Bondi mass tends to negative in nity. This feature will not b e changed by the addition of the auxilliary eld to the mo del [17]. Then wehave p p Z n 0 0 0 0 0 2 _ _ ( ) + ( )( ) S = d x 0 1 0 1 N o X 2 1 1 p ( ) 0 2 2 + g^ @ @ +2 e @ f @ f (7) g^ i i 2 4 i=1 4 00 2 01 10 2 where ; =0;1;2 with =(1;1;1),g ^ = ,^g =^g = , 11 2 2 2 g^ =( ) .

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