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Stability of Black Holes in Two-Dimensional Dilaton Gravity (*)

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Stability of Black Holes in Two-Dimensional Dilaton Gravity (*) IL NUOVO CIMENTOVOL. 112 B, N. 12 Dicembre 1997 Stability of black holes in two-dimensional dilaton gravity (*) M. A. AHMED Physics Department, Kuwait University - Kuwait (ricevuto il 14 Aprile 1997; approvato il 3 Giugno 1997) Summary. — We discuss the stability of black-hole solutions in a model of two-dimensional dilaton gravity. We show that the black hole is stable under small nonstatic perturbations. PACS 04.70.Dy – Quantum aspects of black holes, evaporation, thermodynamics. PACS 04.60.Kz – Lower dimensional models; minisuperspace models. 1. – Introduction In recent years there has been a lot of interest in the string-inspired gravity theories in two space-time dimensions [1, 2]. In particular, in ref. [3] Callan et al. (CGHS) included matter fields and discussed the phenomenon of Hawking radiation. A solution in closed form for the quantum corrected CGHS equations had not been found. This led Russo et al. [4] (RST) to present a modification of the CGHS model by adding a term of the form fR to the action, with f being the dilaton field and R the scalar curvature. This form appears in the RST model with a coefficient equal to N/12, where N is the number of matter fields. By performing certain field redefinitions, RST were able to solve the field equations following from the model exactly. In ref. [5] we considered a generalization of the dilaton gravity model by allowing the fR term to enter with an arbitrary coefficient and we were able to solve the resulting field equations exactly. Once one has an exact solution, to investigate its stability becomes an interesting issue. For the model of ref. [1] and [2] this was considered by a number of authors [6-9]. In general to determine whether a black-hole solution is stable or not, a small linear perturbation of the classical equations of motion in the black-hole background is considered [10]. If the perturbation, which is regular everywhere in space to start with, grows with time, then the solution is declared unstable. In ref. [7] it was claimed that a mode growing with time exists in the gravity sector. Later it was conjectured that the mode growing with time should be a gauge artifact [8]. The analysis of ref. [7] was carried out in the Schwarzschild-like gauge [1]. Subsequently, Kim et al. [9], working in the conformal gauge, showed that the mode growing with time can be eliminated by variation of the black-hole position. (*) The author of this paper has agreed to not receive the proofs for correction. G Società Italiana di Fisica 1675 1676 M. A. AHMED In this note we investigate the question of stability for the exact solution of the generalized model considered previously [5]. Working in the Schwarzschild-like gauge we are able to solve the differential equation for the perturbation exactly instead of relying upon asymptotic analysis. In this way we are able to show that perturbations that grow with time are not physically acceptable. In sect. 2 we present a brief review of the dilaton gravity model. Section 3 is devoted to demonstrating the stability of the solution. 2. – The dilaton gravity model The classical action for our two-dimensional dilaton gravity model is 1 k (1) S4 d2 x k2g e 22f [R14(˜f)2 1c]2 fR , 2p m 2 n where f is the dilaton field. The constant k that appears before the Jackiw-Teitelboim term [11] is taken to be arbitrary. From eq. (1) there follow the dilaton equation of motion: k (2) 11 e 2f R24(˜f)2 14˜2 f1c40, g 4 h and the metric equations of motion k k 22f 1 1 2 2 4 (3) e (Rmn 2˜m ˜n f) gmn ˜ f 0. g 4 h 2 The exact solution of the model in the gauge in which the dilaton field is proportional to one of the co-ordinates [1] 1 (4) f4 Qx 2 was found to be [5] 4 2 21 (5) gmn diag [ g(x), g (x) ] , with ae Qx (6) g412 , 11 (kO4) eQx where a is a constant. This solution describes a black hole with an event horizon at 42 1 2 k (7) x0 ln a . Q g 4 h Moreover, for kE0 the solution has a curvature singularity at 84 1 4 (8) x0 ln . Q NkN STABILITY OF BLACK HOLES IN TWO-DIMENSIONAL DILATON GRAVITY 1677 The mass of the black hole is Q k (9) M42 a2 . 2 g 4 h Requiring M to be positive we see that a2kO4 must be positive for QE0 and vice versa. 4 O Using the fact that Rab (1 2) gab R identically in two dimensions in eq. (3), taking the trace of the resulting equation and substituting for R from eq. (2) we obtain an equation involving the dilaton field only: k (10) 4(˜f)2 22 11 e 2f ˜2 f2c40. g 4 h 2 D E D Taking, for definiteness, a k/4 0 and Q 0 we obtain for x x0 the Kruskal coordinates: Q(a2kO4)(t2x) 1O2 . – k 2a u42 12 a2 eQx e , ` k g 4 h l (11) Q(a2kO4)(t1x) / 1O2 2 – k 2a ` v4 12 a2 eQx e . ´ k g 4 h l The metric in terms of these coordinates reads Q(a2kO4) x a 2a 2 e – – (12) ds 2 42 du dv , yQ(a2kO4) z 11(kO4) eQx 4 in which the coordinate does not appear singularly at x x0 any longer. Further analysis of the model described by eq. (1) can be found in ref. [5]. 3. – Stability of the black hole In order to discuss the stability of the black-hole solution one introduces small nonstatic perturbations hmn (x, t) and df(x, t) about the background solutions gmn and f of eqs. (4)-(6): 4 – 1 (13) gmn gmn hmn , – (14) f4f1df . We substitute eqs. (13) and (14) in eqs. (3) and (10) retaining only terms of first order in perturbations. For this purpose we need the following relations: 1 – – – – – – 42 –ar 1 2 –1 (15) d(˜m ˜n f) g (˜n hrm ˜m hrn ˜r hmn ) ˜a f ˜m ˜n df , 2 – 1 – – – – – 2 42–ma –nb –2 –mn –ab 1 2 –1 2 (16) d(˜ f) g g hab ˜m ˜n f g g (˜n hrm ˜m hrn ˜r hmn ) ˜a f ˜ df , 2 1 – – – – – – – – 42 –lr 2 2 1 (17) dRmn g (˜n ˜m hlr ˜l ˜n hrm ˜l ˜m hrn ˜l ˜r hmn ). 2 1678 M. A. AHMED – – In eqs. (15)-(17) ˜m denotes the covariant derivative with respect to the metric g. The linearized field equations for the perturbations that follow, respectively, from eqs. (3) and (10) are – – 1 – k – – – – – – – – 2 22f 2 22f 1 –lr 2 2 1 2 (18) 2df e Rmn e g (˜n ˜m hlr ˜l ˜n hrm ˜l ˜m hrn ˜l ˜r hmn ) 2 g 4 h – – – – k – – – – 2 22f –2 22f 1 –ar 1 2 –1 4dfe ˜m ˜n f e g (˜n hrm ˜m hrn ˜r hmn )˜a f g 4 h – k – – k – k – – 1 22f 1 2 2 –1 – –la –tb –2 2 e ˜m ˜n df hmn ˜ f gmn g g hab ˜l ˜t f g 4 h 2 2 k – k – – – – 2 – 2 1 – –lt –ar 1 2 –4 gmn ˜ df gmn g g (˜t hrl ˜l hrt ˜r htl ) ˜a f 0, 2 4 – – – – – – k – 2 –am –bn – –1 –ab –2 2f 2 –2 1 2f (19) 4g g hmn ˜a f˜b f 8g ˜a df˜b f kdfe ˜ f 2 1 e Q g 4 h – – 1 – – – – – 2–ma –nb –2 –mn –ar 1 2 –1 2 4 Q g g hab ˜m ˜n f g g (˜n hrm ˜m hrn ˜r hmn ) ˜a f ˜ df 0. k 2 l In the following analysis we shall take QE0, aD0 and a2kO4 D0 [5]. Moreover we also take [6, 7] 4 – (20) hmn h(x, t) gmn . Using eq. (20) in eq. (18) we obtain, from the sum of the xx and tt components of the resulting equation, the following equation: k 1 Qx 2 2 2 1 2 1 2 2 1 2 2 1 2 4 (21) e [ Qg ¯x h 2(¯t g ¯x ) df] g Q¯x h 2¯x df 0, 4 g 2 h while the xt component of the equation yields Q 2 1 2 21 8 4 (22) ¯t h (2¯x g g ) ¯t h 0. 2 In eqs. (21) and (22) we dropped the bar over g for ease of writing, with g of course being given by eq. (6), and the prime denotes differentiation with respect to x. On the other hand eq. (19) gives rise to the following equation: k kQ 2 2 1 8 1 Qx 2 Qx 8 1 2 (23) Q gh Qg 1 e h e g df 4Qg¯x df g 4 h 2 k 2 1 Qx 2 2 21 2 1 8 4 2 1 e (g¯x df g d t df g ¯x df) 0. g 4 h STABILITY OF BLACK HOLES IN TWO-DIMENSIONAL DILATON GRAVITY 1679 Integrating eq. (22) we obtain Q 2 1 2 21 8 4 (24) h (2¯x g g ) df F(x), 2 where F(x) is an arbitrary function of x only. Below we shall at first set F(x) 40 and return to the more general case of F(x) c0 later on. Then using eq. (24) to eliminate h from eq.
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