Physics Letters B 312 (1993) 79-82 PHYSICS LETTERS B North-Holland

On the black hole background of two-dimensional string theory

Shyamoli Chaudhur i i and Djordje Minic 2

Theory Group, Department of Physics, University of Texas, Austin, TX 78712, USA

Received 2 February 1993; revised manuscript received 25 May 1993 Editor: M. Dine

The classical black hole background of two-dimensional string theory is examined after including the effect of the tachyon field. Keeping all terms upto O (T 2 ) and making no other approximations, the only consistent classical solution to the resulting dilaton-graviton theory is found to be flat spacetime with a nontrivial dilaton.

The discovery of a nontrivial d i la ton-gravi ton background of critical two-dimensional string theory with the target space geometry of a 2D black hole [1-3] has s t imulated much activity in the quantum mechanics of black holes [4]. Subsequently at tempts have been made to include the effects of the tachyon in the solution of the beta function equations [5,6]. However these treatments have thus far been re- stricted to the asymptot ic regime, since the coupled equations are too difficult to handle in general, and can only be compared to the conformal field theory in this regime [7]. In this paper, we will propose a different approach to including the effect of the tachyon with a rather surprising result. We find that upon el iminating the tachyon through its equation of motion, the two dimensional black hole is no longer a consistent classical solution to the resulting d i la ton-gravi ton theory. It is important to note that while our calculation includes the d i la ton- tachyon and gravi ton- tachyon couplings upto O (T2), higher- order self-interactions and coulings to the massive modes of teh string could, in principle, alter our conclusions.

We follow the sigma model approach of Mandal et al. [2]. As is well known by now, two-dimensional critical string theory can be viewed as a theory of two- dimensional quantum gravity coupled to c = 1 mat- ter [8], or in other words, as a non-critical Polyakov

l E-mail address: sc@utaphy.bitnet. 2 E-mail address: minic@utaphy.bitnet.

string theory in a single embedding dimension x t (~). In conformal gauge, gab = exp [~/(~) ] gab, the confor- mal field theory on the world-sheet is a theory of two scalars, one being the embedding dimension x I (~) and the other the Liouville field r/(~). The correspond- ing sigma-model action is

l/ S = ~ d2~ V/if[ ~gl~ab"'lrUv (X)OaXUObXV

-R(2)CP(X) + T ( x ) ] , (1)

where Gu~, ~ and T denote the low energy excita- tions of a critical two-dimensional bosonic string: the graviton, the dilaton and the tachyon respectively. Also x u = (x I (~), q (~)) . The only propagating mode is the tachyon, which is massless in two dimensions. The graviton and dilaton serve as auxiliary fields that parametr ize the background.

The target space equations of motion for these ex- citations are determined by the condit ions of confor- mal invariance, i.e., the relevant beta-functions are set to zero [9]. As is well-known, these equations can be derived from the target space action

/ .

I = J d 2 x e x p ( - 2 ~ ) v / - G

x [ R - 4 ( V ~ ) 2 + (~7T)2 + V ( T ) + c ] , (2)

where c = - 8 in two dimensions. We will assume that the tachyon potential is quadratic, V ( T ) = - 2 T 2. The corresponding equations of motion are given by

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Volume 312, number 1,2 PHYSICS LETTERS B 5 August 1993

Ru~ - 2 V u V . ~ + VuTV.T = O,

R + 4 ( ~ 7 t P ) 2 -- 4 ( V 2 ~ ) + (~ 'T) 2 + V(T) + c

(3)

= 0, (4)

- 2 V 2 T + 4 V ~ V T + V(T) = 0. (5)

The black hole solution of [2] is obtained by setting the tachyon field to zero. Then eqs. (3) and (4) read

(6) Ru~ - 2 V u V , ~ = 0,

R + 4(~7~) 2 - 4~72~ 4- C = 0. (7)

We will work in the conformal gauge. Then,

ds 2 = exp(~r) d u d v , (8)

and the components of the spacetime metric are Guy = exp(a ) /2 and Guu = G . . . . . 0. The only nonvanishing components of the connection, Fb a, are F f = Oucr, F~l, = Ova. The Ricci tensor is given by Ruv = OuOv~r and the scalar curvature equals R = 4exp(-e)Ou&,tr.

Mandal et al. found a non-trivial solution to eqs. (6) and (7),

(9) e x p ( - 2 q , ) = 2uv + a,

and

1 ( 1 0 ) exp(tr) - 2uv + a '

which implies

ds 2 _ d u d v (11) 2uv + a

For a # 0 this line element represents a black hole in Kruskal coordinates, with horizon at uv = 0 and a curvature singularity at uv = -a/2. The string cou- pling constant g2 = e xp (2~ ) grows infinitely large at the location of the curvature singularity.

We will now take into account the dynamics of the tachyon mode in this background. (A linearized anal- ysis of eqs. (3) - ( 5 ) for non-vanishing static tachyons, valid in the asymptotic regime, is carried out in [5], and the extension to non-static tachyons is done in

[6].) We begin by manipulating the action. Let us scale the tachyon as follows, t = e x p ( - q ) ) T . Then the tachyonic action, namely

= / d Z x e x p ( - 2 ~ ) x / G [ ( V T ) 2 - 2T2] , (12) Iv

becomes

It = f dZx x/G(GU~VutV~t - ¼Rt2), (13)

where we have used the classical equation of motion for the dilaton field (4), and have dropped terms of O ( T 3) since we are interested in small fluctuations of the tachyon about its quadratic maximum. Let us eliminate t by using its classical equation of motion

(D + ¼R)t = 0. (14)

We choose the boundary condition t ~ 1 at infin- ity. This is motivated by requiring that we match smoothly to the flat metric, linear dilaton background of two-dimensional string theory with non-vanishing cosmological constant, T = exp (~ ) . It would be a worthwhile exercise to explore other possible bound- ary conditions.

Then we can write the solution of the equation of motion:

tel = 1 - ~FR,

where

(15)

p - I _ [] + ¼R. (16)

Note that by using the equation of motion one can substitute in (13) to express the action It as

' / d2xv~R + ~ f d2xv~RFR. (17) II = - a

This procedure is familiar from the work of Fradkin and Vilkovisky on four- dimensional quantum gravity [ 1 o ] .

In conclusion we obtain the following "effective" action for the coupled graviton-dilaton system:

left = f d 2 x e x p ( - 2 ~ ) v / G [ R - 4 (~7 t / ) ) 2 ]

1 -~ f d2xv/-dR + ~ f d2xv~R[] + R/4 - - R .

(18)

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The non-local piece can be evaluated formally as an expansion in powers of [ ] -~R. In the following, we will approximate ( [ ] + R / 4 ) -~ by the leading term in this expansion. The consistency of this approximat ion will be just if ied by the result.

By varying Ieff we obtain the relevant equations of motion for the graviton dilaton system. First we re- call that the Einstein equations are identically satis- fied in two dimensions. Also, the variat ion of W = f d a x R [ ] - l R is well known to be given by (see for example [ 11 ] )

6 W = -v/-GOGu~{2VuV~ ( 0 R ) + V U ( O R ) V ~ ( O R )

- G u~[2R + ½V¢(0R)V ¢ ( 0 R ) ] } , (19)

where 0 = [ ] - l . (In the conformal gauge [] = 4exp(-a)OuO~, so OR = a . )

Hence we are left with the following set of equations:

O, Ova = 20uO~,CI9

+ ½b (-30uOvcr + OuaOva) exp(2~b ). (20)

2 (02q ) - OuaO,~ )

= - ½b [202cr - (0~a) 2 ] exp (2qb), (21)

2 (Offq~ - O,,aO,,~)

= -½b[20,2cr - (O,,o') 2] exp(2q~), (22)

0u0~a + 40~qb0,,q~ - 40~0,,q~ = 2 e x p ( a ) . (23)

Here b = ~. (Of course if we set b = 0 we recover the equations of the gravi ton-d i la ton system with T = 0.)

The goal is now to check whether the black hole metric satisfies the improved equations of motion de- rived from the effective action. We do this in two steps. Note that we have left the dilaton field unspec- ified thus far. We first consider that the two dimen- sional black hole metric is still o f the form (10), while the dilaton is altered. Substituting (10), we get

OuOv q~ - a (2uv + a) 2

b 6a + 4uv 4 (2uv + a) 2 e x p ( 2 q ~ ) , (24)

2'U 2uv +

b 4'U 2 = 4 (2uv + a) 2 exp(2q: ' ) , (25)

2u 2uv +

b 4u z = e x p ( 2 ~ ) , (26)

4 (2uv + a) 2

Ou~O,,~ - GO,,~ - uv + a (27) (2uv + a ) 2

These equations turn out to be consistent only if a = 0. This is seen by eliminating exp (2q~) from equations (25) and (26) which implies

2v + 2uv +

( 2u aOv~ ) (28) "~-'U2 0'2~ "[" 2uv +

and then substituting in eq. (24) which implies

0,0,,q) = V_0zq~ + 10,,q~ ' (29) H U

and it therefore follows that a = 0. We then allow for a small perturbat ion to the form

of the metric (10) by an arbitrary function, f (uv) . Substituting this more general solution we find again that the only consistent solutions have a = 0! We can therefore put a = 0 in eqs. ( 2 4 ) - ( 2 7 ) , and solve for the dilaton. For the original metric, (10), we get

b OuO,, • - exp (2 qb ) ,

4uv

O uq, + LOu _ b u 4u z exp(2q~),

, 2 1 0 ~ , ¢ _ b 0vq ~ + 't) 4V 2 exp(2q~ ) ,

(30)

(31)

(32)

(33) 1

Ou ~ O~, ~ - OuO,, ~ - 4uv "

It is easily seen that the solution of the last equation is given by

( 1 ) e x p ( - ~ ) = Aexp C l l n v ~ + ~ l l n v ~

+ B exp (C2 In v ~ + ~--~2 In v ~ ) . (34)

By subsituting this expression in (30), (31) and (32) we infer that AB = b = -~ and C~ = -C2 = 1.

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Therefore the improved classical equations of mo- tion yield flat spacetime with the dilaton field given by

q~ = - I n A v / - ~ + 5 ' (35)

while the tachyon asymptotically (t ~ 1 ) approaches

1 T = exp(05) = (36)

A v / ~ + ( b / A ) ( 1 / x / - ~ ) '

and we recall that b = ~. By formally letting b ~ 0, and comparing to the

T = 0 solution we determine A = x/2. Our solution again has the property that the string coupling con- stant g2 = exp(2q~ ) grows infinitely large at uv = _ b / A 2 = 1 .

What is the implicat ion of our result? To O (T 2) in the equations of motion, it appears that the tar- get space black hole background of two-dimensional string theory is unstable when the dynamics of the only propagating field in the theory is taken into account, or in other words this background does not represent a true vacuum for the gravi ton-di la ton system once the nonlinear d i la ton- tachyon and gravi ton- tachyon cou- plings are considered. (This result was known to Sasha Polyakov [12].) Note that these couplings are not probed in the l inearized tachyon equation solved in [5,6]. Our calculation neglects higher order tachyon self-interactions as well as couplings to the massive modes of the string. It is possible that these are es- sential to the stability of the black hole solution, as suggested by the conformal field theory analysis [7].

What we find particularly intriguing is the ap- pearance of a non-local term proport ional to f d2xv/-'GR (F1 + R / 4 ) - I R in the spacetime effective action for the gravi ton-di la ton system. This term is of a similar form to the one proposed years ago by Fradkin and Vilkovisky which when added to the usual Eins te in-Hilber t term would represent the appropria te quantum action for four-dimensional quantum gravity (with manifest off-shell conformal invariance) and would yield identical on-shell prop- erties as the Einstein theory! (For more details con- sult [ 10 ]. ) Of course, from the point of view of string theory, this term is a logical possibility in the low- energy effective action that should account for the infrared properties of the gravitational field. In view

of recent activity on the problem of black hole evap- oration [13] it is perhaps worthwhile reconsidering the role of such non-local terms in the gravitational effective action.

We thank W. Fischler, J. Lykken and J. Polchin- ski for discussions. Our interest in this problem was considerably influenced by lectures delivered by L. Susskind on black hole evaporat ion at the Universi ty of Texas. D.M. would like to acknowledge many il- luminating conversations with Sasha Polyakov about the black hole problem in general. This work is sup- ported in part by the Robert A. Welch Foundat ion, NSF grant PHY 9009850, and the Texas Advanced Research Program.

References

[ 1 ] S. Elitzur, A. Forge and E. Rabinovici, Nucl. Phys. B 359 (1991) 581.

[2] G. Mandal, A.M. Sengupta and S.R. Wadia, Mod. Phys. Lett. A 6 (1991) 1685.

[3] E. Witten, Phys. Rev. D 44 (1991) 314. [4] For a review and further references see J.A. Harvey

and A. Strominger, Lectures at TASI'92. [5] S. DeAlwis and J. Lykken, Phys. Lett. B 269 (1991)

264. [6] G.A. Diamandis, B.C. Georgalas, X. Maintas and N.E.

Mavromatos, Phys. Lett. B 297 (1992) 67; J.G. Russo, University of Texas preprint, UTTG-31- 92 (December 1992).

[7] S. Chaudhuri and J. Lykken, Nucl. Phys. B 396 (1993) 270.

[8] A. Dhar, T. Jayaraman, K.S. Narain and S.R. Wadia, Mod. Phys. Lett. A 5 (1990) 863; S.R. Das, S. Naik and S.R. Wadia, Mod. Phys. Lett. A 4 (1989) 1033; S.R. Das, A. Dhar and S.R. Wadia, Mod. Phys. Lett. A 5 (1990) 799; J. Polchinski, Nucl. Phys. B 234 (1989) 123; T. Banks and J. Lykken, Nucl. Phys. B 331 (1990) 173.

[9] C. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B 262 (1985) 593.

[10] E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 73 (1978) 209; Phys. Lett. B 77 (1978) 262.

[1 I]G.A. Vilkovisky, in: Quantum theory of gravity ed. S. Christensen (Adam Hilger, Bristol, 1984) pp. 169- 209.

[12]A.M. Polyakov, Lectures at Les Houches Summer School (July 1992).

[13] J.G. Russo, L. Susskind and U Thorlacius, University of Texas and Stanford University preprint, UTTG-19- 92 (SU-ITP-92-24) (August 1992).

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