Two-dimensional black hole in the three-dimensional black string

PHYSICAL REVIEW D VOLUME 52, NUMBER 4 15 AUGUST 1995

Two-dimensional black hole in the three-dimensional black string

H. W. Lee and Y. S. Myung Department of Physics, Inje University, Kimhae 621-749, Korea

Jin Young Kim Division of Basic Science, Dongseo University, Pusan 616-010, Korea

(Received 2 1 December 1994)

We discuss the two-dimensional black hole in the (2 + 1)-dimensional black string. When the charge of the black string Q = 0 and thus the torsion H,,, vanishes, a black string with double horizons becomes a simple product of a two-dimensional (2D) black hole with a single horizon and R. It is shown that the two propagating modes are the dilaton and tachyon in a 3D black string background, and L = h - 4 and the tachyon in a 2D black hole@R background. We also calculate the reflection and transmission coefficients for the scattering of string fields off 2D black holes. Using the Bogoliubov transformation we find the Hawking temperature from the reflection coefficient of the tachyon.

PACS number(s): 04.70.Bw, 04.60.Kz, 11.25.Db, 11.25.Pm

I. INTRODUCTION

Lower dimensional gravity has often been used as a toy model for icvestigating various problems that arise in four dimensions, but are not solvable there. Among those that have been extensively studied are quantum gravity in three dimensions and black hole physics in two dimensions. Recently a d = 3 black hole solution has been given by Banados et al. in anti-de Sitter spacetime [I]. Also Horowitz and Welch have shown that this black hole is equivalent under duality to the black string solution [2]. However, it is very important to clarify the fact that the black hole is an asymptotically anti- de Sitter type: whereas the black string is asymptotically flat. A simple extension of Witten's construction for a gauged Wess-Zumino-Witten (WZW) model yields a three-dimensional (3D) charged black string [3]. This solution is characterized by three parameters: M (mass), Q (axion charge per unit length), and k (a constant re- lated to the asymptotic value of the derivative of the dilaton). For 0 < /QI < M, the black string is similar to the Reissner-Nordstrom solution with a charged black hole in four dimensions. In addition to the event (outer) horizon (rEH = M ) , there exists an inner horizon (rIH = Q2/M) . When IQ/ = M , the black string has the usual property of possessing an event horizon, but no singularity. Finally, when /QI > M, the spacetime has neither a horizon nor a curvature singularity. The crucial problem has to do with the inner (Cauchy) horizon of Reissner-Nordstrom black holes. This horizon is believed to be unstable under time-dependent perturbations because it is a surface where infalling radiation is infinitely blueshifted [4]. For the same reason, the inner horizon of a black string is thus an important issue.

Recently, we investigated the propagation and scattering of all string fields (metric g,,, antisyrnmetric tensor B,,, dilaton a, and tachyon T ) off the black string with double (inner and outer) horizons 151. We found that all

string fields except the tachyon and dilaton are nonpropagating in the 3D black string background. This paper concentrates on the study of the propagation of string fields in the (2D black hole)@R background, rather than a 3D black string. Of course this has a single horizon. We wish to find out the distinction between a single horizon and double ones by investigating the propagations of string fields. The organization of this paper is as fol- lows. In Sec. 11, we set up the equations of motion for the black string and linearize this equation around the background solution. In Sec. 111, we review the fact that in the 3D black string background all string fields except the tachyon and dilaton are nonpropagating. Section IV is concerned with the propagation of the string fields under a 2D black hole background. The scattering process of string fields off a 2D black hole is discussed in Sec. V. In Sec. VI we derive the Hawking temperature from the tachyon scattering off a 2D black hole using a Bolgo- liubov transformation. Finally we discuss our results in Sec. VII.

11. 3D BLACK STRING

Let us start with the u-model action of string theory

161

where R(') is the Ricci curvature of the world sheet. The conformal invariance requires the 0-function equations

for gravity,

2214 @ 1995 The American Physical Society

52 - TWO-DIMENSIONAL BLACK HOLE IN THE THREE- . . . 2215

for the dilaton,

for the Kalb-Ramond field, and

for the tachyon, where H,,, = 33~,BVp1 is the torsion corresponding to B,,. The above equations are also de- rived from the requirement that the fields must be an extremum of the low-energy string action [7]:

The static black string solution to the above equations is given by setting the tachyon ( T ) to zero [3] :

nents of the background Christoffel symbols are

To study the scattering problem specifically, we intro- duce small perturbation fields around the background solution as [B]

- 1 ) 0 0 where we choose the metric perturbatmion (h,,) in such a QPV = 0 1 way that the background symmetry should be restored a t

k M -1 0 G ( l - (1 - $)-I the perturbation level. This is a crucial point in studying

(7) a11 black holes [9,10]. In order to obtain the equations governing the perturbations, one has to linearize ( 2 ) - ( 5 )

with N 5 Q 2 / M ( M > N ) . The nonvanishing compo- as

1 - 1 - JR,,(h) - v,V,$J + Jr~,(h)V,(a - -H,pu31~u + q ~ , p o ~ a " h p a = 0,

2 (9)

1 - h @ u ~ , ~ , m - gwJr;,(h)apG + V 2 0 - h"a,Ha,,G + 2guva,G8,~ - -{2&,,?cppu - 3 ~ , , ~ " ~ " h g ) = 0, ( l o )

6

where

111. NONPROPAGATIONS I N A 3D BLACK This is ~ossible because we have no physically propa- STRING gating graviton in 2+1 dimensions. From (11) one can

express-the torsion 31 in terms of and h as

The simplest case with a 3D black string background can be searched with

2216 H. W. LEE, Y. S. MYUNG, AND JIN YOUNG KIM - 52

This means that on shell XTtx is no longer an independent (20)-(24), the equations containing the dilaton are very mode. Also from the diagonal elements of (9), we have complicated in the region of outer and inner horizons.

In order to obtain the simple propagating equation, we ..dTh 8 Q2 - V 2 h - b:(h+ 24) - 3 --- + --(h-24) = 0, consider the asymptotically flat region. For large r we

r k r 2 cannot take the limit r -+ cc naively [3]. First we define

(17) a new variable p as

T T d Q2 - v 2 h - ~ : ( h + 2 4 ) $ 9 -c + - ( h - 2 4 ) = 0. and then take the limit of k + m. Then (20) and (24) r k r

(19) reduce to the free field equations

Adding the above three equations leads to

4 dTh 12 Q2 where V2f = 8; + 82 - df is the flat space-time operator. V2(2h + 4) + -(T - M ) ( T - N) ---- - - -(h - 24)

k r k r 2 And the off-diagonal elements of (21)-(23) take the forms

And the off-diagonal elements of (9) take the form

From (26), we find that h = -24 in the asymptotically atax (5 + d) = 0, (22) flat region. This means that the graviton (h) is not an

independent mode. Substituting this into (25) leads to

ax {(aT - ) ( + 4) - &} = O. (23) the fact that the dilaton is a freely propagating mode.

One remaining equation which describes a physically propagating mode is the tachyon equation (12) discussed

Also from the dilaton equation (10) together with (16), in [s]. we find

4 8, 8 8 Q2 V24 + - ( r - M ) ( r - N ) - (h + 44) - -h + (h - 24) IV. PROPAGATIONS IN (2D BLACK HOLE)@R

k k

= 0. (24) We found that all string fields except the dilaton and

It is not easy to find the solutions which satisfy all Eqs. (20)-(24). Instead, we first check whether the graviton (h) and dilaton (4) are propagating modes in the black string background. We consider the conventional count- - - ing of degrees of freedom. The number of degrees of freedom for the gravitational field (h,,) in d dinlensions is (1/2)d(d - 3). For a d = 4 Schwarzschild black hole, we obtain two degrees of freedom. These correspond to the Regge-Wheeler mode for odd-parity perturbation and Zerilli mode for even-parity perturbation [9]. We have -1 for d = 2. This means that in two dimensions the contri- bution of the graviton is equal and opposite to that of a spinless particle (dilaton). In the 2D dilaton black hole, two graviton-dilaton modes (h - 4, h + 4) are thus trivial gauge artifacts [8] . For d = 3, we have no propagating gravitons. Since the metric is not physically propagating, one can choose this perturbation as (15) for simplicity. In addition, there are couplings of the metric to string fields (4, X,,,, t ) . From (16), XTtx is not an independent mode on shell. The independent fields are two (dilaton and tachyon). Hence we have to obtain the propagating form for the dilaton. The dilaton should be a propagating mode under any kind of conditions. As is shown in

tachyon are nonpropagating in a 3D black string with double horizons. Now let us consider the same problem within a simpler context. For this purpose, we set the graviton modes (h,,) as two degrees of freedom initially:

From ( l l ) , one can express the torsion 31 in terms of 4, h, and hl as

Adding the three diagonal elements of (9) leads to

One can easily check that, when h = hl , (29) is reduced to (20). Further, the off-diagonal components of (9) take the form


Hence all string fields except the torsion seem to be propagating in the (2D black hole)@R background.

V. SCATTERING OF STRING FIELDS OFF (2D BLACK HOLE)@R

In order that 4 and h satisfy (31)-(32) simultaneously, they must have the forms At this stage it is not obvious which one is the most

important mode to extract information from the 2D black hole among h+4 r K, h - 4 L, and t . For this purpose, we need the scattering analysis of these modes off a 2D black hole. Note here that (30) implies h l = hl ( r , x ) . However, this

means a static mode and thus we set

A. Tachyon mode

Let us consider the tachyon mode first. Considering the normal mode solution of the form

for simplicity. From (30) we have the constraint relation for 4 and h:

(42) can be rewritten as

( r - ~ ) r d ; t + (2r - M ) a T t where C ( r , x ) is a residual gauge field and thus we can set C ( r , x ) = 0. Substituting (33) and (34) into (29 ) , we have

Similarly, we find, for the dilaton equation, The exact solution of (44) can be found by defining the new variable z as

With this new variable, z + 0- ( r + M + ) and z + -m ( r -+ +m) correspond, respectively, to the event horizon and asymptotic flat regions. Equation (44) can be rewritten in terms of z as

It is not easy to find the form of h and 4 from (36) and (37) under the constraint (35 ) . However, when Q = 0 [black string reduces to (2D black hole )@R], we obtain the free field equation for the one graviton-dilaton mode ( h + 4 ) from (36 ) ,

and the dilaton equation (37 ) is given by

It is the standard procedure to cast this equation into the form of the hypergeometric equation. Substituting

t ( z ) = z a ( l - ~ ) ~ 9 ~ ,

Substituting (35) into (39) leads to

Calculating (38) - 2 x (40 ) , we obtain the equation for the other mode ( h - 4 ) as

where E = rt, we have

Further, the tachyo11 mode comes from

and the torsion field is obviously nonpropagating; that is,

2218 H. W. LEE, Y. S. MYUNG, AND JIN YOUNG KIM

Comparing this with the general form of the hypergeometric equation,

the general solution is given by

Qt(z) = CIF(a,,O, y; z) + C2z1--'

x F ( a + l - y , P + l - y , 2 - y ; z ) , (49)

where C1, C2 are arbitrary constants and

B. Graviton-dilaton modes

Considering the normal mode solutions for K and L,

K = ~ ( r ) e - ~ ~ ~ , L = ~ ( r ) e - ~ ~ ~ , (50)

we have, from (38) and (41),

We repeat the same procedure as in the previous section. Substituting

into (51) and (52), the general solutions for both equations are

where K1, K Z , L1, and Lz are arbitrary constants and

Here a and b are given by (i) for K mode,

and (ii) for L mode,

VI. HAWKING RADIATION

From the previous scattering analysis, we wish to derive the Hawking radiation. Let us first consider a tachyon solution which in the asymptotic flat region reduces to the sum of two waves (ingoing and outgoing waves), and represents an ingoing wave into the event horizon. This requires that C2 = 0, E = -1. We call this type solution {t+). We expect that part of the waves gets reflected from the event horizon, while the other part will enter the event horizon and will be absorbed by the black hole. Following the procedures in [5,11] for the scattering analysis, we find the on-shell transmission and reflection coefficients for the tachyon mode as

sinh 27rp sinh 2xD =

cosh2 7r(D + p) ' (59)

where

We can repeat the same procedure with the two graviton- dilaton modes. Then we have

~ g - d = 2 sin2 xb + cosh 4xp - cos 2xb '

C O S ~ 47rp - 1

cosh 4xp - cos 2xb '

Each substitution for b = 0 , l for the K mode and b = -1,2 for L give a trivial result R : - ~ = 0 and

~ f - ~ = 1. We see that there is no reflection for the graviton-dilaton modes. Therefore in the (2D black hole physics)@R background, the tachyon is the crucial mode and hereafter we consider only the tachyon mode.

Next we construct another solution {t-) by imposing different boundary conditions. We demand a solution which, close to the event horizon, behaves as the sum of an ingoing wave with an outgoing wave (Hawking radiation), and reduces to the outgoing wave for the asymptotically flat region. The appropriate choice for the constants is E = -1 and the ratio of C1 and C2 should be

The states {t+} and {t-) form two different bases of which any state can be expanded. Consequently there


are two distinct Fock spaces corresponding to two different vacua. It is a standard procedure to show that the expectation value of the occupation number operator N+ for the {t+) basis in the vacuum of the {t-) basis is

One can define the Hawking temperature by rewriting it in the form

where E is the eigenvalue of the timelike vector (at). From the reflection coefficient for the tachyon, one finds the Hawking temperature

When the tachyon energy is large (E + m), we find

which corresponds to the statistical temperature and Hawking temperature of the 3D black string in the classical limit [5]. In this range, the classical thermodynamical equilibrium is realized. The well-defined thermodynamical picture is obtained in the limit where the energy is larger than the height of the potential barrier. Here we can understand the problem according to the classical picture.

V I I . DISCUSSION

Here we have investigated the propagation and scattering of all string fields (metric h,,, torsion %,,,, dilaton 4, and tachyon t ) off the black string with double (inner and outer) horizons. On shell, XrtX is no longer an independent mode. According to the counting of degrees of freedom in 3D, the independent fields are two (dilaton and tachyon). Hence we have to obtain the propagating form for the dilaton. The dilaton should be a propagating mode under any kind of conditions. As shown in

(20)-(24), the equations containing the dilaton are very complicated in the region of outer and inner horizons. It is not easy to obtain forms of the dilaton which satisfy all Eqs. (20)-(24) simultaneously. Instead, we consider the asymptotically flat region to obtain the simple propagating equation. In this region we found that the dilaton is freely propagating.

Further we consider the string fields in the (2D black hole)@R background, rather than the 3D black string. This corresponds to Q = 0 and has a single horizon. Fortunately, it is a simple matter to obtain the modes within this background. In this case we must also have two physical degrees of freedom. All string fields ( h - 4, h + 4, t ) except Xrtx are apparently propagating in the (2D black hole)@R background. It is well known that the black hole can be visualized as presenting a potential barrier (well) to the true oncoming waves [&lo]. From (38), one graviton-dilaton mode (K = h + 4) is just the free field in the whole region and thus does not feel the presence the black hole. From (41) and (42), L = h - 4 and t propagate under the potentials. Thus we consider L = h - 4 and t as the physical mode, while K = h + 4 is a trivial gauge artifact [8]. Further it is not obvious which one is an important mode to extract information from the black hole among h + 4, h - 4, and t. For this purpose, we need the scattering analysis of these modes off a 2D black hole. We calculate the on-shell reflection and transmission coefficients for the scattering of all fields. The tachyon (t) scattering provides a lot of information about the 2D black hole. For example, we found the Hawking temperature from the tachyon scattering off the (2D black hole)@R background using the Bogoliubov transformation. However, the scatterings of two graviton-dilaton modes are trivial, since R+ = 0 and T+ = 1 for these modes.

We conclude that the two propagating modes are the dilaton and tachyon in the 3D black string background, and L = h - 4 and the tachyon in the 2D black hole background.

ACKNOWLEDGMENTS

This work was supported in part by the Basic Sci- ence Research Institute Program, Ministry of Education, Project No. BSRI-95-2413, and by the Nondirected Re- search Fund, Korea Research Foundation, 1994.

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Two-dimensional black hole in the three-dimensional black string