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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.

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

S 4 1 2 p



_d2_xk 2g

m

e22 f_{[R 14(˜f)}2 1 c] 2 k 2 fR

n

, (1)

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:

g

1 1 k 4 e 2 f

h

R 24(˜f)2 1 4 ˜2f 1c40 , (2)

and the metric equations of motion

g

e22 f 1 k 4

h

(Rmn1 2 ˜m˜nf) 2 k 2 gmn˜ 2 f 40 . (3)

The exact solution of the model in the gauge in which the dilaton field is proportional to one of the co-ordinates [1]

f 4 1 2 Qx (4) was found to be [5] gmn4 diag [2g(x), g21(x) ] , (5) with g 412 ae Qx 1 1 (kO4) eQx , (6)

where a is a constant. This solution describes a black hole with an event horizon at

x04 2 1 Q ln

g

a 2 k 4

h

. (7)

Moreover, for k E0 the solution has a curvature singularity at

x 804 1 Q ln 4 NkN . (8)

The mass of the black hole is M 42Q 2

g

a 2 k 4

h

. (9)

Requiring M to be positive we see that a2kO4 must be positive for QE0 and vice versa. Using the fact that Rab4 ( 1 O2 ) gabR 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:

4(˜f)2_{2 2}

g

_{1 1} k 4 e

2 f

h

˜2_{f 2c40 .} (10)

Taking, for definiteness, a 2k/4 D0 and QE0 we obtain for xDx0 the Kruskal

coordinates:

.

`

/

`

´

u–_{4 2}

k

_{1 2}

g

_{a 2} k 4

h

e Qx

l

1 O2_e Q(a 2kO4)(t2x) 2 a , v–₄

k

_{1 2}

g

_{a 2} k 4

h

e Qx

l

1 O2_e2 Q(a 2kO4)(t1x) 2 a . (11)

The metric in terms of these coordinates reads

ds2_{4 2}

y

2 a Q(a 2kO4)

z

2 _e Q(a 2kO4) x a 1 1 (kO4) eQxdu – dv–, (12)

in which the coordinate does not appear singularly at x 4x0any 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 gmnand f

of eqs. (4)-(6): gmn4 g – mn1 hmn, (13) f 4f–1 df . (14)

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:

(15) d(˜m˜nf) 42 1 2g –ar_(˜– nhrm1 ˜ – mhrn2 ˜ – rhmn) ˜ – af – 1 ˜–m˜ – ndf , (16) d(˜2_{f) 42g}–mag–nbhab˜–m˜nf – 2 1 2g –mn g–ab(˜–nhrm1 ˜ – mhrn2 ˜ – rhmn) ˜ – af – 1 ˜–2df , (17) dRmn4 2 1 2g –lr_(˜– n˜ – mhlr2 ˜ – l˜ – nhrm2 ˜ – l˜ – mhrn1 ˜ – l˜ – rhmn) .

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 22 df e22 f – R–mn2 1 2

g

e 22 f– 1 k 4

h

g –lr_(˜– n˜ – mhlr2 ˜ – l˜ – nhrm2 ˜ – l˜ – mhrn1 ˜ – l˜ – rhmn) 2 (18) 2 4 dfe22 f – ˜ – m˜ – nf – 2

g

e22 f– 1 k 4

h

g –ar_(˜– nhrm1 ˜ – mhrn2 ˜ – rhmn) ˜ – af – 1 1 2

g

e22 f– 1 k 4

h

˜ – m˜ – ndf 2 k 2 hmn˜ –₂ f – 1 k 2 g – mng–lag–tbhab˜ – l˜ – tf – 2 2k 2 g – mn˜ –₂ df 1 k 4g – mng–ltg–ar(˜ – thrl1 ˜ – lhrt2 ˜ – rhtl) ˜ – af – 4 0 , 2 4 g–amg–bn_hmn ˜ – af – ˜ – bf – 1 8 g–ab˜–adf˜ – bf – 2 kdfe2 f – ˜ –₂ f – 2 2

g

1 1 k 4 e 2 f–

h

_Q (19) Q

k

2g–mag–nb_h ab˜ – m˜ – nf – 2 1 2g –mn_g–ar_(˜– nhrm1 ˜ – mhrn2 ˜ – rhmn) ˜ – af – 1 ˜–2df

l

4 0 . In the following analysis we shall take Q E0, aD0 and a2kO4 D0 [5]. Moreover we also take [6, 7]

hmn4 h(x , t) g–mn.

(20)

Using eq. (20) in eq. (18) we obtain, from the sum of the xx and tt components of the resulting equation, the following equation:

eQx [2Qg2 ¯2xh 12(¯2t1 g2¯2x) df] 1 k 4 g 2

g

21 2Q¯xh 12¯ 2 xdf

h

4 0 , (21)

while the xt component of the equation yields

2Q

2 ¯th 1 (2¯x2 g

21_{g 8) ¯}

th 40 .

(22)

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:

2Q2gh 1Qg 8

g

1 1 k 4e Qx

h

h 2 k Q 2 e Qx g 8 df14Qg¯xdf 2 (23) 22

g

1 1 k 4e Qx

h

(g¯2xdf 2g21d 2 tdf 1g 8 ¯xdf) 40 .

Integrating eq. (22) we obtain

2Q

2 h 1 (2¯x2 g

21_{g 8) df4F(x) ,}

(24)

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) c 0 later on. Then using eq. (24) to eliminate h from eq. (23) we obtain an equation involving df only:

g2 ¯2xdf 2gg 8 ¯xdf 2¯2tdf 1

y

g 821 Q

(

(kO4) eQx 2 1

)

1 1 (kO4) eQx gg 8

z

df 40 . (25)

The solution of eq. (25) is best discussed in terms of the variables (x * , t), where x * is defined by x * 4x2 a Q(a 2kO4) ln

k

1 2

g

a 2 k 4

h

e Qx

l

_, (26)

for x Dx0. Then x KQ corresponds to x *KQ and the event horizon x0is located at

x * 42Q. Equation (25) then becomes

(¯2x *2 2 g 8 ¯x *2 ¯2t1 V) df 4 0 , (27) where V is given by V 4g 82 1 Q

(

(kO4) e Qx 2 1

)

1 1 (kO4) eQx gg 8 . (28)

We seek solutions of eq. (27) in the form

df 4e2ikt_{n ,} (29) and obtain

g

d2 dx *2 2 2 g 8 d dx * 1 k 2 1 V

h

n 40 . (30)

We can transform eq. (30) to a form in which the first derivative no longer appears. Let us define

b(x * ) f 22g 8

(

x(x * )

)

, (31)

i.e. b(x * ) is the function obtained from the coefficient of the first-derivative term in eq.

(30) by expressing x in terms of x*. We then consider the transformation

n 4ve21 O2



b(x * ) dx *_,

(32)

and deduce the following equation for the function v: d2_v

dx *2 1 k

2

v 40 .

Note that we have from eqs. (31) and (32) that

n 4gv ,

(34)

up to an arbitrary multiplicative constant.

For real frequencies the solutions of eq. (33) are

v 4A6e6ikx *.

(35)

These solutions correspond to outgoing and incoming waves. Next we consider solutions with purely imaginary k, which can give rise to unstable perturbations, i.e. perturbations that grow exponentially with time. We set k 4ia, where a is real and positive. Then the time dependence of the perturbation df becomes exp [at], and the solutions of eq. (33) are

v 4A6e Zax *_.

(36)

We require that the perturbation falls off to zero for large values of x and this leads us to select the solution

v 4A1e2ax *.

(37)

Correspondingly, we have

df 4A1ge2a(x * 2 t).

(38)

Consequently, from eq. (24) we obtain

h 422 A1

Q ( 2 a 1g 8) e

2a(x * 2 t)_.

(39)

Expressed in terms of the regular Kruskal coordinates this reads

h 422 A1

Q ( 2 a 1g 8)(2u

–₎2 aaOQ(a2kO4)_.

(40)

In deciding the stability of a black hole, one starts with a perturbation that is regular everywhere in space at t 40 and then finds out whether it grows with time. It is clear that at t 40, by choosing u small, h can be made arbitrarily large, i.e. the perturbation diverges as u K0, whereas the background metric remains finite. This contradicts the assumption that the perturbation is small in comparison with the background. Hence perturbations with k 4ia, that grow exponentially with time, are physically unacceptable and thus ruled out.

We now go back and analyse the case in which F(x) c 0 in eq. (24). Equation (27) is then replaced by (¯2 x *2 2 g 8 ¯x *2 ¯t2_{1 V) df 4} Qg 1 1 (kO4) eQxF(x) . (41)

We next carry out the analogue to the transformation of eq. (32) by writing

df(x * , t) 4v(x *, t) e21 O2



b(x * ) dx *

4 gv(x * , t) , (42)

and eq. (41) simplifies to

(¯2x *2 ¯2t) v 4U ,

(43)

where U stands for the r.h.s. of eq. (41). Next we substitute for h from eq. (24) into eq. (21) and employ eq. (41) in the resulting equation, thereby arriving at

k 8 e Qx (2g 8 ¯x *1 V) df 2 g2

g

1 1 k 8e Qx

h

F 8(x)1 Qg 1 1 (kO4) eQxF(x) 40 . (44)

For bounded df at spatial infinity one can use this equation to compute the asymptotic behaviour of F. We find for large x that

F AeQx

k

1 2

g

a 2 k 4

h

e Qx

h

2((a 2kO4)O(a2kO8))

k

1 1 k 8e Qx

l

2kO( 8 a 2 k)_. (45)

We now solve eq. (43) using Laplace transform methods. We write

v–_{(x * , s) 4}



0 Q

e2st_{v(x * , t) dt ,}

(46)

where we assume that the integral exists for some large enough s. We then find that ¯2x *v – (x * , s) 2s2_v– (x * , s) 4 1 sU 2¯tv(x * , 0 ) 2sv(x *, 0) . (47)

A particular solution v–p of eq. (47) can be obtained using the method of variation of

parameters. We find that

v–p(x * , s) 4esx *



x * e2sj 2 s r(j , s) dj 2e 2sx *



x * esj 2 s r(j , s) dj , (48) where r 4 1 s U 2¯tv(j , 0 ) 2sv(j, 0) . (49)

The function vp(x * , t) can be obtained by computing the inverse Laplace transform

and we readily derive that

vp4



x *

(x * 2j) U dj . (50)

Note that the r.h.s. of eq. (50) does not involve the values of the functions v and ¯tv at t 40. Given the asymptotic behaviour of F we observe that vpsatisfies our requirement

of vanishing as x KQ. In any case the solution in eq. (50) is time-independent and thus does not play a role in deciding the issue of stability. Stability is then completely decided by the solution of the homogeneous equation, eq. (27), which corresponds to the case F 40. It is thus settled in the manner we described before.

R E F E R E N C E S

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