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On the stability of the two-dimensional black hole
11 August 1994 PHYSICS LETTERS B
ELSEVIER
Physics Letters B 334 (1994) 29-32
On the stability of the two-dimensional black hole Y.S. Myung Department of Phystcs, lnje Untverstty, Ktmhae 621-749, South Korea Received 12 May 1994 Editor M Dine
Abstract
We reexamine the stability of the static black hole in two-dimensional string theory. The tachyonic mode (t) has a potential barrier near the black hole, while the graviton--dilaton mode (h+ ~) has a potential well near the black hole. ( h + ~) is not a physically propagating mode and nothing but a gauge artefact, although the potential well induces an exponentially growing mode with time. The stability should be based on the physical degrees of freedom. According to the analysis based on the physical mode (tachyonic mode), we find that the d = 2 black hole is stable
Recently there has been a renewed effort to understand the black hole in the simpler context of twodimensional ( d = 2) systems, using the string-inspired models for quantum gravity [ 1,2]. These stringy black hole solutions may play the role of a toy model for the d = 4 real black hole [3]. An easy way of understanding the attributes of a physical system Is to find out how it reacts to external perturbations and, m the first instant, to infinitesimal perturbations. In the case of the black hole, this is the only method available to us because there Is no other way in which an external observer can explore the other side of the horizon. The reaction of an object to an infinitesimal perturbation is determined by the enumeration of the so-called normal modes of oscillation. For the black hole, this enumeration reduces to finding how a black hole reacts to incident waves of different sorts. The solution to this latter problem bears on the stability of a black hole and the determinaUon of the quasi-normal modes [4 ]. From general considerations, one may expect that a fraction of the energy in the incident waves will be irreversibly absorbed by the black hole, while the remaining fraction will be scatElsevier Smence B.V. SSD10370-2693 ( 94 ) 007 75-3
tered (or reflected) back to infinity. In other words, it would appear that it may be possible to vmualize the black hole as presenting an effective potential barrier (or well) to the oncoming waves. In deciding whether the black hole is stable or not, we start with a perturbation which is regular everywhere in space at the mitlal time ~'= 0, and then see whether such a perturbation will grow with time. If there exists an exponentially growing mode, the black hole is unstable. More recently we derived a potential barrier for the tachyon mode (t), and a potential well for the gravitonddaton mode (h + ~). Further it was shown that the d = 2 static black hole is unstable against the (h + ~) external perturbation [ 5 ]. However, th~s conclusion is arrived at from the point of view of d = 2 gravity, not from the point of view of physical d = 4 gravity. In this letter, we will again address this stability problem for the d = 2 stringy black hole. Our reexamination is based on scattering analysis and counting of physical degrees of freedom. In analyzing the twodimensional stringy black hole, we start with one gravlton, one &laton and one tachyon. These in turn give rise to one gravlton-dilaton (h - ~), the other (h + ~),
Y S Myung ~Physics Letters B 334 (1994) 29-32
30
and tachyonic (t) modes. According to scattering analysis [6], the transmission amplitude for the gravltondilaton mode ( h + q~) is a pure phase and thus there is no reflection. This means that this mode propagates freely, - ~ to + oo; as the (h-q~) mode propagates freely. On the other hand the tachyomc mode (t) plays the crucial role m obtaining all information from the black hole. This suggests that only the tachyonlc mode is the physically propagating one in the black hole background. In our theory gauge-fixing is lacking from the beginning, in view of d = 4 gravity theory. The net physical degrees of freedom for the two-dimensional stringy black hole should be given by - l ( g r a v i ton) + 1 (dilaton) + 1 (tachyon) = 1 (tachyon). Consldenng the above, we insist that two grawton-dilaton modes are trivial gauge artefacts. It is emphasized that stabihty should be based on the physical degrees of freedom, not on the gauge degrees of freedom. Hence our prewous conclusion which was based on the unphyslcal mode (h-~p), is not approprmte. Taking together scattering analysis with counting of degrees of freedom, we confirm that the tachyon is only a physical degree of freedom. In this case, we have not found any exponentially growing mode. The o--model actmn of d = 2 critical string theory for the graviton ( g , , ) , the dilaton (q~), and the tachyon (T) (Ix, v = 0 , 1) is given by [5,6,7]
f=l-Mexp(-2Q4,),
Q=~.
Here the parameter M ( > 0) is proportional to the mass of the black hole. All these solutions approach the linear dilaton vacuum in the asymptotically flat region ( 4 , ~ + ~ ) . The horizon of the black hole occurs at 4hm = ( 1/2Q) In M. Here, for simplicity we choose M = 1 and 4,~n = 0. As a consequence of the fact that 4~H = 0 defines a null surface, the space interior to 4, = 0 is incommunicable to the space outside. To study the stability of the black hole more specifically, let us introduce the small perturbed fields h ~ ~( 4,, ~-), q~(4,, ~-) and t(4,, ~-) as [5]
=ff~[t-h(4,,
r)],
,/,= ,b+ ~(4,, r),
and conformal invarlance requires the following /3function equaUons to be satisfied:
R~,~+ V z V ~ + V~,TV~T=O, R+(Vcrp)z+2~72cl)+(VT)Z-2T2-8=O,
(2)
VET+ VqOVT+2T= 0.
(4)
_ where
(o
-- exp( - ½~) [0+t(4,, z)] .
(9)
First, let us discuss the tachyonic perturbation. Since the region interior to the horizon (4, < 0) is of no relevance to our consideration, let us introduce the new coordinate (4,*)
4,*-4, 1
(10)
Note that 4,* ranges from - oo to + ~, while 4, ranges from the event horizon of the black hole (4,En = 0) to + ~. Assuming t( 4,*, ~-) = L(4,*) exp ( - ik~'),we find the following equaUon:
+k2-V.r t=O,
(11)
where the tachyonlc potential barrier Is given by
Vr= =
T=0,
0 l),
(8)
(3)
Let us begin by considering the static background solutions of the graviton-dilaton sector in the absence of the tachyon, • = 2Q4,,
(7)
T=T+t
+ ~-~ In[1 - M exp( - 2 Q 4 , ) ] .
I f d2x ~ (g~,~Vx~,Vx~+a,R~+2T) ' S~= 8"n'a' (1)
(6)
(5)
2 exp(2~4,*) [1 +exp(2gt24,*)] z
1 2(cosh V~4,*) ~"
(12)
Now let us consider the graviton-dilaton modes. Defining H - h - ~ o and considenng the trial solution of the form
ES. Myung/ PhyswsLettersB 334 (1994)29-32 H(~b*, r ) = I ( ~ b * ) e x p ( - i k ~ ' ) ,
(13)
Here units are used in which G = c = 2/144= 1, so that the horizon is at r * = -oo ( r = 1). r * Is related to the Schwarzschlld radial coordinate r by
(14)
d dr*
we have the free field equation
(O~.2+k2)l(qb*) =0.
The other equation for the gravlton-dilaton mode (J=-h + ~p) is also given by the one-dimensional differential equation. Considering J(~b*, ~-)=K(~b*) exp( - ik¢), we obtain
+(k2-Vj)
K=0,
(15)
- 16 exp(2¢2q~*) [ I + exp(2v/24, *) ]2
-4 = (cosh v ~ b * ) 2"
dr .2
(k 2 - Vz) a/, .~ O.
In these units, the Zerllh potential Vz is given by 2(n + 1) r Vz= .
3r 2 + 9r/2n + 9/4n 2 r n ( r + 3/2n) 2 3+
)< ( r - 1) ,
d2qto ~/r,2 + ( k 2 - VRW)qro = 0 , (16)
In the case of the d = 2 black hole, we find a potential barrier for the tachyonic mode, a potential well for the one graviton-&laton mode ( h + ~o) and no potential for the other graviton-dilaton mode (h-~o). It is emphasized that the potential well for J = h + ~ois obviously a new feature of the d = 2 black hole. From (15), we have found an exponentially growing mode with time (k = ia). Naively this means that the two-&mensional stringy black hole is unstable. However, according to scattering analysis [6], the transmission amplitude for the graviton--dilaton mode (h + ~o) is a pure phase and thus there is no reflection. This means that this mode propagates freely, - ~ to + oo, as the (h - ~o) mode propagates freely. On the other hand the tachyonic mode (t) plays the crucial role in obtaining all information from the black hole. This suggests that only the tachyonic mode is the physically propagating one in the black hole background. Now we are in a position to compare this d = 2 black hole with the d = 4 Schwarzschild black hole. In the case of the Schwarzschlld black hole, there exist two kinds of Schrodinger type equation governing a small perturbation of the graviton (gu~). One is the Zerflli equation which arises in even-parity perturbation [4,8]
_a2q,o +
r-ld r dr"
where the parameter n, in terms of the mulfipole Index l~>2 of the perturbation, is n-~ ( l - 1 ) ( l + 2 ) / 2 . The other is the Regge-Wheeler (RW) equation
where the potential (well) is given by
vj=
31
(17)
(18)
which differs only in the details of the potential: VRW =
2(n+ 1)r-3 r4 (r-- 1) .
The RW equation arose m the study of odd-parity perturbations m the same formalism. Chandrasekhar [8] showed exphcitly the connection between the Zerilli and RW equations. The existence of different descnptions of the perturbations of black holes led Chandrasekhar to consider the general question of the relationship between two potentials which are equivalent in the sense of producing the same physical consequences (more specifically, having the same reflection ( 2 ) and transmission ( Y ) coefficients). Further, according to Anderson's formalism of intertwining operators for any potential [ 9], there are equivalent potentials. In fact there exists an infinite number of equivalent potentials. In Schwarzschlld black holes, for example, the Zerilli and RW potentials are only two of an infinite set of possible potentials. Of course all these potentials belong to potential barriers (not potential wells) However, in the case of the two-dimensional black hole, we found a potential well as well as a potential barrier. In order to understand this apparent discrepancy between two-dimensional and four-dimensional black holes, let us consider the counting of degrees of freedom [ 10]. For d-dimensional theory, a symmetric traceless tensor field hg ~ has d(d + 1)/2 - 1 independent
32
Y.S. Myung/ Phystcs LettersB 334 (1994) 29-32
components, d of which are ehminated by the gauge condition that specifies 0 ~h ~ = 0. In addition, ( d - 1 ) are ehminated by our freedom to make further gauge transformations ~h ~ ~= 0 ~ + 0 ~ with O~ ~ = 0 and a 2 ~ , = 0. Hence, the number of degrees of freedom for the gravitauonal field in d dimensions is ~d(d+l)-l-d-(d-1)
= ~d(d-3)
.
For d = 4, we obtain two propagating physical gravitons (Zenlli and R W cases). However, this is - 1 for d = 2. This means that m two dimensions the contribution of the graviton is equal and opposite to that of a spinless particle (dilaton). Thus, from the point of view o f d = 4 theory, general relativity is not much of a theory m two dimensions. In this connection, the Lagranglan ~gR is a total derivative for d = 2, and consequently ( R u ~ - ½ g ~ R ) vamshes identically. This is the reason why one usually chooses either hght-cone gauge or conformal gauge (as in our case) in two dimensions. In analyzing the two-dimensional stringy black hole, we start with one grawton, one dilaton and one tachyon. These m turn gives rise to one graviton-dilaton (h - ~), the other (h + ~) and tachyomc modes. In our theory gauge-fixing is lacking from the beginning, m wew of d = 4 grawty theory. The net physical degrees of freedom for the two-dimensional stringy black hole should be given by - 1 (graviton) + 1 (ddaton) + 1 (tachyon) = 1 (tachyon) . In view of the above, we insist that two graviton-dilaton modes are t n w a l gauge artefacts. It is emphasized
that stability should be based on the physical degrees of freedom, not on the gauge degrees of freedom. Hence, our previous conclusion which was based on the unphysical mode ( h - ~p) [5], is not appropriate. Taking together scattering analysis with counting of degrees of freedom, we confirm that the tachyon is only a physical degree of freedom. In this case we have not found any exponentially growing mode. Therefore, the two-dimensional stringy black hole is stable. This work was supported in part by the Inje Research and Scholarship Foundation and the program of Basic Science Research, Ministry of Education.
References [1] E Wltten, Phys Rev D44 (1991)314 [2] G Mandal, A Sengupta and S R. Wadla, Mod Phys Lett A 6 (1991) 1685 [3] S P deAlwlsand J Lykken, Phys Lett B 269 (1991) 464, A Dhar, G Mandal and S R Wadla, Mod Phys Lett. A 7 (1992) 3703, S R. Das, Mod Phys Lett A 8 (1993) 69 [4] T Regge and J A. Wheeler, Phys, Rev 108 (1957) 1403, C V Vlshveshwara,Phys. Rev. D 1 (1970) 2870, FJ Zenlh, Phys Rev Lett 24 (1970) 737 [5] J Y Klm, H W Lee and Y S Myung, to be pubhshed m Phys Lett B, [6] S.K Rama, Phys. Rev Lett 70 (1993)3186 [7] J Y Klm, H.W Lee and Y S. Myung, prepnnt INJE-TH-94-2 (1994) [8] S Chandrasekhar,m Space and Geometry, eds R A Matzner and L C Shepley (Texas U P, Austin, TX, 1982) [ 9] A. Anderson and R H Price, Phys Rev D 43 (1991) 3147 [ 10] S Wemberg, m General relativity, eds S W. Hawking and W. Israel (Cambridge U,P, London, 1979)
ELSEVIER
Physics Letters B 334 (1994) 29-32
On the stability of the two-dimensional black hole Y.S. Myung Department of Phystcs, lnje Untverstty, Ktmhae 621-749, South Korea Received 12 May 1994 Editor M Dine
Abstract
We reexamine the stability of the static black hole in two-dimensional string theory. The tachyonic mode (t) has a potential barrier near the black hole, while the graviton--dilaton mode (h+ ~) has a potential well near the black hole. ( h + ~) is not a physically propagating mode and nothing but a gauge artefact, although the potential well induces an exponentially growing mode with time. The stability should be based on the physical degrees of freedom. According to the analysis based on the physical mode (tachyonic mode), we find that the d = 2 black hole is stable
Recently there has been a renewed effort to understand the black hole in the simpler context of twodimensional ( d = 2) systems, using the string-inspired models for quantum gravity [ 1,2]. These stringy black hole solutions may play the role of a toy model for the d = 4 real black hole [3]. An easy way of understanding the attributes of a physical system Is to find out how it reacts to external perturbations and, m the first instant, to infinitesimal perturbations. In the case of the black hole, this is the only method available to us because there Is no other way in which an external observer can explore the other side of the horizon. The reaction of an object to an infinitesimal perturbation is determined by the enumeration of the so-called normal modes of oscillation. For the black hole, this enumeration reduces to finding how a black hole reacts to incident waves of different sorts. The solution to this latter problem bears on the stability of a black hole and the determinaUon of the quasi-normal modes [4 ]. From general considerations, one may expect that a fraction of the energy in the incident waves will be irreversibly absorbed by the black hole, while the remaining fraction will be scatElsevier Smence B.V. SSD10370-2693 ( 94 ) 007 75-3
tered (or reflected) back to infinity. In other words, it would appear that it may be possible to vmualize the black hole as presenting an effective potential barrier (or well) to the oncoming waves. In deciding whether the black hole is stable or not, we start with a perturbation which is regular everywhere in space at the mitlal time ~'= 0, and then see whether such a perturbation will grow with time. If there exists an exponentially growing mode, the black hole is unstable. More recently we derived a potential barrier for the tachyon mode (t), and a potential well for the gravitonddaton mode (h + ~). Further it was shown that the d = 2 static black hole is unstable against the (h + ~) external perturbation [ 5 ]. However, th~s conclusion is arrived at from the point of view of d = 2 gravity, not from the point of view of physical d = 4 gravity. In this letter, we will again address this stability problem for the d = 2 stringy black hole. Our reexamination is based on scattering analysis and counting of physical degrees of freedom. In analyzing the twodimensional stringy black hole, we start with one gravlton, one &laton and one tachyon. These in turn give rise to one gravlton-dilaton (h - ~), the other (h + ~),
Y S Myung ~Physics Letters B 334 (1994) 29-32
30
and tachyonic (t) modes. According to scattering analysis [6], the transmission amplitude for the gravltondilaton mode ( h + q~) is a pure phase and thus there is no reflection. This means that this mode propagates freely, - ~ to + oo; as the (h-q~) mode propagates freely. On the other hand the tachyomc mode (t) plays the crucial role m obtaining all information from the black hole. This suggests that only the tachyonlc mode is the physically propagating one in the black hole background. In our theory gauge-fixing is lacking from the beginning, in view of d = 4 gravity theory. The net physical degrees of freedom for the two-dimensional stringy black hole should be given by - l ( g r a v i ton) + 1 (dilaton) + 1 (tachyon) = 1 (tachyon). Consldenng the above, we insist that two grawton-dilaton modes are trivial gauge artefacts. It is emphasized that stabihty should be based on the physical degrees of freedom, not on the gauge degrees of freedom. Hence our prewous conclusion which was based on the unphyslcal mode (h-~p), is not approprmte. Taking together scattering analysis with counting of degrees of freedom, we confirm that the tachyon is only a physical degree of freedom. In this case, we have not found any exponentially growing mode. The o--model actmn of d = 2 critical string theory for the graviton ( g , , ) , the dilaton (q~), and the tachyon (T) (Ix, v = 0 , 1) is given by [5,6,7]
f=l-Mexp(-2Q4,),
Q=~.
Here the parameter M ( > 0) is proportional to the mass of the black hole. All these solutions approach the linear dilaton vacuum in the asymptotically flat region ( 4 , ~ + ~ ) . The horizon of the black hole occurs at 4hm = ( 1/2Q) In M. Here, for simplicity we choose M = 1 and 4,~n = 0. As a consequence of the fact that 4~H = 0 defines a null surface, the space interior to 4, = 0 is incommunicable to the space outside. To study the stability of the black hole more specifically, let us introduce the small perturbed fields h ~ ~( 4,, ~-), q~(4,, ~-) and t(4,, ~-) as [5]
=ff~[t-h(4,,
r)],
,/,= ,b+ ~(4,, r),
and conformal invarlance requires the following /3function equaUons to be satisfied:
R~,~+ V z V ~ + V~,TV~T=O, R+(Vcrp)z+2~72cl)+(VT)Z-2T2-8=O,
(2)
VET+ VqOVT+2T= 0.
(4)
_ where
(o
-- exp( - ½~) [0+t(4,, z)] .
(9)
First, let us discuss the tachyonic perturbation. Since the region interior to the horizon (4, < 0) is of no relevance to our consideration, let us introduce the new coordinate (4,*)
4,*-4, 1
(10)
Note that 4,* ranges from - oo to + ~, while 4, ranges from the event horizon of the black hole (4,En = 0) to + ~. Assuming t( 4,*, ~-) = L(4,*) exp ( - ik~'),we find the following equaUon:
+k2-V.r t=O,
(11)
where the tachyonlc potential barrier Is given by
Vr= =
T=0,
0 l),
(8)
(3)
Let us begin by considering the static background solutions of the graviton-dilaton sector in the absence of the tachyon, • = 2Q4,,
(7)
T=T+t
+ ~-~ In[1 - M exp( - 2 Q 4 , ) ] .
I f d2x ~ (g~,~Vx~,Vx~+a,R~+2T) ' S~= 8"n'a' (1)
(6)
(5)
2 exp(2~4,*) [1 +exp(2gt24,*)] z
1 2(cosh V~4,*) ~"
(12)
Now let us consider the graviton-dilaton modes. Defining H - h - ~ o and considenng the trial solution of the form
ES. Myung/ PhyswsLettersB 334 (1994)29-32 H(~b*, r ) = I ( ~ b * ) e x p ( - i k ~ ' ) ,
(13)
Here units are used in which G = c = 2/144= 1, so that the horizon is at r * = -oo ( r = 1). r * Is related to the Schwarzschlld radial coordinate r by
(14)
d dr*
we have the free field equation
(O~.2+k2)l(qb*) =0.
The other equation for the gravlton-dilaton mode (J=-h + ~p) is also given by the one-dimensional differential equation. Considering J(~b*, ~-)=K(~b*) exp( - ik¢), we obtain
+(k2-Vj)
K=0,
(15)
- 16 exp(2¢2q~*) [ I + exp(2v/24, *) ]2
-4 = (cosh v ~ b * ) 2"
dr .2
(k 2 - Vz) a/, .~ O.
In these units, the Zerllh potential Vz is given by 2(n + 1) r Vz= .
3r 2 + 9r/2n + 9/4n 2 r n ( r + 3/2n) 2 3+
)< ( r - 1) ,
d2qto ~/r,2 + ( k 2 - VRW)qro = 0 , (16)
In the case of the d = 2 black hole, we find a potential barrier for the tachyonic mode, a potential well for the one graviton-&laton mode ( h + ~o) and no potential for the other graviton-dilaton mode (h-~o). It is emphasized that the potential well for J = h + ~ois obviously a new feature of the d = 2 black hole. From (15), we have found an exponentially growing mode with time (k = ia). Naively this means that the two-&mensional stringy black hole is unstable. However, according to scattering analysis [6], the transmission amplitude for the graviton--dilaton mode (h + ~o) is a pure phase and thus there is no reflection. This means that this mode propagates freely, - ~ to + oo, as the (h - ~o) mode propagates freely. On the other hand the tachyonic mode (t) plays the crucial role in obtaining all information from the black hole. This suggests that only the tachyonic mode is the physically propagating one in the black hole background. Now we are in a position to compare this d = 2 black hole with the d = 4 Schwarzschild black hole. In the case of the Schwarzschlld black hole, there exist two kinds of Schrodinger type equation governing a small perturbation of the graviton (gu~). One is the Zerflli equation which arises in even-parity perturbation [4,8]
_a2q,o +
r-ld r dr"
where the parameter n, in terms of the mulfipole Index l~>2 of the perturbation, is n-~ ( l - 1 ) ( l + 2 ) / 2 . The other is the Regge-Wheeler (RW) equation
where the potential (well) is given by
vj=
31
(17)
(18)
which differs only in the details of the potential: VRW =
2(n+ 1)r-3 r4 (r-- 1) .
The RW equation arose m the study of odd-parity perturbations m the same formalism. Chandrasekhar [8] showed exphcitly the connection between the Zerilli and RW equations. The existence of different descnptions of the perturbations of black holes led Chandrasekhar to consider the general question of the relationship between two potentials which are equivalent in the sense of producing the same physical consequences (more specifically, having the same reflection ( 2 ) and transmission ( Y ) coefficients). Further, according to Anderson's formalism of intertwining operators for any potential [ 9], there are equivalent potentials. In fact there exists an infinite number of equivalent potentials. In Schwarzschlld black holes, for example, the Zerilli and RW potentials are only two of an infinite set of possible potentials. Of course all these potentials belong to potential barriers (not potential wells) However, in the case of the two-dimensional black hole, we found a potential well as well as a potential barrier. In order to understand this apparent discrepancy between two-dimensional and four-dimensional black holes, let us consider the counting of degrees of freedom [ 10]. For d-dimensional theory, a symmetric traceless tensor field hg ~ has d(d + 1)/2 - 1 independent
32
Y.S. Myung/ Phystcs LettersB 334 (1994) 29-32
components, d of which are ehminated by the gauge condition that specifies 0 ~h ~ = 0. In addition, ( d - 1 ) are ehminated by our freedom to make further gauge transformations ~h ~ ~= 0 ~ + 0 ~ with O~ ~ = 0 and a 2 ~ , = 0. Hence, the number of degrees of freedom for the gravitauonal field in d dimensions is ~d(d+l)-l-d-(d-1)
= ~d(d-3)
.
For d = 4, we obtain two propagating physical gravitons (Zenlli and R W cases). However, this is - 1 for d = 2. This means that m two dimensions the contribution of the graviton is equal and opposite to that of a spinless particle (dilaton). Thus, from the point of view o f d = 4 theory, general relativity is not much of a theory m two dimensions. In this connection, the Lagranglan ~gR is a total derivative for d = 2, and consequently ( R u ~ - ½ g ~ R ) vamshes identically. This is the reason why one usually chooses either hght-cone gauge or conformal gauge (as in our case) in two dimensions. In analyzing the two-dimensional stringy black hole, we start with one grawton, one dilaton and one tachyon. These m turn gives rise to one graviton-dilaton (h - ~), the other (h + ~) and tachyomc modes. In our theory gauge-fixing is lacking from the beginning, m wew of d = 4 grawty theory. The net physical degrees of freedom for the two-dimensional stringy black hole should be given by - 1 (graviton) + 1 (ddaton) + 1 (tachyon) = 1 (tachyon) . In view of the above, we insist that two graviton-dilaton modes are t n w a l gauge artefacts. It is emphasized
that stability should be based on the physical degrees of freedom, not on the gauge degrees of freedom. Hence, our previous conclusion which was based on the unphysical mode ( h - ~p) [5], is not appropriate. Taking together scattering analysis with counting of degrees of freedom, we confirm that the tachyon is only a physical degree of freedom. In this case we have not found any exponentially growing mode. Therefore, the two-dimensional stringy black hole is stable. This work was supported in part by the Inje Research and Scholarship Foundation and the program of Basic Science Research, Ministry of Education.
References [1] E Wltten, Phys Rev D44 (1991)314 [2] G Mandal, A Sengupta and S R. Wadla, Mod Phys Lett A 6 (1991) 1685 [3] S P deAlwlsand J Lykken, Phys Lett B 269 (1991) 464, A Dhar, G Mandal and S R Wadla, Mod Phys Lett. A 7 (1992) 3703, S R. Das, Mod Phys Lett A 8 (1993) 69 [4] T Regge and J A. Wheeler, Phys, Rev 108 (1957) 1403, C V Vlshveshwara,Phys. Rev. D 1 (1970) 2870, FJ Zenlh, Phys Rev Lett 24 (1970) 737 [5] J Y Klm, H W Lee and Y S Myung, to be pubhshed m Phys Lett B, [6] S.K Rama, Phys. Rev Lett 70 (1993)3186 [7] J Y Klm, H.W Lee and Y S. Myung, prepnnt INJE-TH-94-2 (1994) [8] S Chandrasekhar,m Space and Geometry, eds R A Matzner and L C Shepley (Texas U P, Austin, TX, 1982) [ 9] A. Anderson and R H Price, Phys Rev D 43 (1991) 3147 [ 10] S Wemberg, m General relativity, eds S W. Hawking and W. Israel (Cambridge U,P, London, 1979)