PHYSICAL REVIEW D VOLUME 5 1, NUMBER 8 15 APRIL 1995

Exact solutions of four-dimensional black holes in string theory

David Gershon* School of Physics and Astronomy, Beverly and Raymond Sackler Faculty of Exact Sciences, Tel-Aviv University,

Tel-Aviv 69978, Israel (Received 8 March 1993; revised manuscript received 5 January 1994)

We construct a class of exact conformal field theories as SL(2, R) x SU(2) x ~ ( 1 ) / ~ ( 1 ) ~ gauged WZW models, which describe a charged black hole in four dimensions. The black hole is axisym- metric and carries both electric and axionic charge. Other cos~~iological backgrounds related to this model are described as well.

PACS number(s): 11.25.Hf, 04 .70.B~

String theory is the first consistent quantum theory that accommodates gravity. However, it is not clear whether it can be established as the quantum theory of gravity.

As a theory of gravity we would expect that string the- orv would exhibit features encountered in general relativ- - ity. Such an important characteristic would, for example, require that string theory could describe objects such as black holes.

Recently, it was shown by Witten [I] that string theory in two-dimensional target space can describe a two-dimensional (2D) black hole. This was obtained as a n SL(2, R ) / U ( l ) Wess-Zumino-Witten (WZW) coset model. Subsequently, a n exact solution of a charged black hole was derived [2] as a n SL(2, R) x U ( l ) / U ( l ) WZW coset model. A related coset model, SL(2, R ) x R I R , can describe a solution of a three-dimensional black string 131. The (ungauged) SL(2, R ) WZW model was re- cently shown to correspond to a three-dimensional black hole [41 which is asymptotically anti-de Sitter type.

action in string theory in four dimensions have a classi- cal solution of a charged black hole. However, a classical solution is not guaranteed to correspond to a conformal field theory.

In this work we construct string theories with a back- ground that corresponds to a charged black hole in four dimensions. Our solution is obtained by gauging the U(1)' subgroup in the SL(2, R)xSU(2)xU(1) WZW model. This coset model describes a n axisymmetric black hole that carries both electric and axionic charges. The model does not describe the case when the electromag- netic field vanishes.

Other backgrounds obtained by related coset mod- els describe a four-dimensional black membrane so- lution [as SL(2, R ) x SU(2) x R/R2] and a cosmologi- cal solution of a universe with charged matter [as SU(1,l) xSU(2) X U ( ~ ) / U ( ~ ) ~ ] .

Closed strings which have gauge fields in their mass- less spectrum can be described in the following way [6,7]: The action contains the fields X p which describe the

~ o h k v e r , toward a more realistic relation between space-time coordinates and compactified free boson fields string theory and nature, we would like to see that string X* which realize the Kac-Moody currents of the gauge theory can describe black holes in four dimensions. I t was group. The a model action is shown in [5] that the equations of motion of the effective

where G,, is the space-time metric, B,, is the antisym- we use is the SL(2, R ) X S U ( ~ ) X U ( ~ ) / U ( ~ ) ~ WZW coset metic tensor, A,(&) is the background space-time gauge model. field, h is the determinant of the world sheet metric, R(') The WZW action [8,9] for a group G is is the curvature of the world sheet, and is the dilaton field. d ' u ~ r ( ~ - ~ a + ~ ~ - ~ d - ~ ) - I? , (2)

In the following we derive exact conformal field theo- - ries (CFT's) which describe a charged black hole in four where is an element of the group and is the Wess- dimecsions, coupled to a n electromagnetic field. Thus zumino (WZ) term; A, (A,) correspond to a U( l ) gauge group. The model

'Electronic address: GERSHON@TAUNIVM.BITNET B is the manifold whose boundary is C. We use

0556-2821/95/5 1(8)/4387(7)/$06.00 - 51 4387 @ 1995 The American Physical Society

4388 DAVID GERSHON - 5 1

Lorentzian metric ds2 = 2du+du- on the world sheet hole solution in Lorentzian signature. We paranletrize C . the group elements of SL(2, R ) and SU(2) by

We start with the group G = SL(2, R ) x S U ( 2 ) x U ( l ) . Since we have a direct product, the action is So(g) =

hl = exp (%US) exp(rul) exp (%u3) , So(h1) + So(h2) +So(X) where h l E SL(2, R), ha ~ S u ( 2 ) and X is a free compactified U ( l ) field (i.e., X - X + 27rR, where R is the radius of compactification). We

(4)

denote the level of the SL(2, R ) WZW by kl and that h2 = exp (i%u3) exp(iOul) exp e o s , of SU(2) by kz. In order that the action is uniquely (!: ) defined the level of a compact group should be integer [g]. where ui are the Pauli matrices. In terms of these coor- Therefore k2 is integer. We shall now derive the black dinates the ungauged action is

The standard way to gauge HL x HR subgroup of GL x GR in WZW action [lo] is by replacing derivatives by covariant derivatives. The gauged action is

where the symmetry transformation is 6g = vg - gu, s i n p s i n p y bAi = -Diu, 6Bi = -D;v. However the WZ term r ( g ) T A = -iu sin p cos ,By has a gauge-invariant extension only if one restricts to iv cos p a n "anomaly-free" subgroup of GL x GR. Denote the - - -

generators H L and HR by TE and Tfl. The condition for a gauged subgroup to be anomaly-free is [ll]

trTzT: = t r T ~ T k for a , b = 1 , . . . , d imH . ( 7 )

( t r is the trace on the GL x GR Lie algebra. When G is a product of groups G, with levels ki this reads t r= C kitr; where tri is the trace in the representations of the Lie algebra of the group G;.) In the case a t hand we can write the group elements g = diag(hl , h2 , eix) as a 5 x 5 matrix. Now we shall gauge a (axial) U(1)' subgroup of SL(2, R ) x SU(2) x U ( l ) , generated by the matrices1

sin .J, sin a

Ti = i u sin .J, cos a y iv cos .J,

cos p y T; =

where u2 = kl/kz, v = m, and a , P,$, p are arbi- trary. (Notice that since in the SL(2, R) WZW model we have replaced the sign of kl in order to have a signature (- + +), the anomaly free condition is -kl t r S L ( 2 , ~ ) + k2 trsv(2) + tru(').) To gauge the above symmetry we introduce two Abelian gauge fields Al , A2. The explicit infinitestimal symmetry transformation is

u3 u3 Sh1 = (€1 sir1 .J, sin cu + €2 cos a ) h l + hl - (el sin p s i n p + e2 COS P) ,

2 2

0 3 '-73 . 6h2 = iu(el sin .J, cos a - c2 sin a ) -ha + h2 -zu(-cl s i n p cos P + t 2 s inp ) , 2 2

where t l , t2 are infinitesimal. Now we pick two gauge conditions. We take

'A similar procedure of gauging anomaly-free subgroup was used by Horava [12] and by Nappi and Witten [13].

51 - EXACT SOLUTIONS OF FOUR-DIMENSIONAL BLACK HOLES . . . 4389

In the above we concentrate on a solution of a black hole, so we take p = 0. Notice that once sinp = 0 we cannot take also sin$ = 0, since the gauge fixing (10) will not be valid. As we see later, this is the reason that this model cannot describe uncharged black hole.

The full gauged action is

kl s(A,, A ~ , g ) = So(g) + J d2a(1 - cosh 2r)[(sin$ sin a A l + + cos aA2+)a-t - A 2 cosfla+t]

+a 1 d 2 a ( l - cos 28) [(sin $ cos a A l + - sin aA2+)8-4 - A 2 sin pa++] 4rr

+' J d20[(sin$ sin a c o s P A l + A 2 + cos a cos4A2+A2-) cosh 2r 4rr

+(sin$ cos a sin/3A1+A2- - sin a sin/3A2+A2-) cos 281

+h J d 2 0 [ ~ l + ~ l ( l + COS$) + A2+A2-] . 47l

After we integrate out the gauge fields Al , A2, we obtain the action

where

and

I = J D [ ~ , t, 0, *]eS"" det - [;::I

L

kl sinh2 r[l + cos(a + 0 ) + 2 sin a s i n 0 sin2 Q] -- a+ ta- t k2 A sin2 Q[2 cos a cosp cosh2 r - cos(a - 0) + 11 + --

A a+4a-4 + a+ea- Q

+ 2 E s i n h 2 ;sinz Q (cos p sin aa+td-* - cos a s inpa- t8+4)

a tan (g ) sinh2 r [2 sin sin2 Q + (sin a - sin 0)] --

A a+ ta- X

k2

- tan (') sin2 Q[2 cos /3 cosh2 r + (cos a - cos P) ] A

d+ 48- X 1

Following Buscher [14], we can calculate explicitly the determinant to one-loop order 1151. We obtain

where a is a n arbitrary constant. This term introduces the dilaton field to the action. By solving the one-loop equations of motion we obtain the same dilaton term.

Frist, let us now choose a = /3 and make a coordinate transformation i; = cosh2 r. All we have to do now is to

identify this action (14) with the string action (1). We denote

Q = t a n a (16)

absorb the constant in t and absorb m t a n ( $ / 2 ) in X. The last rescaling is allowed since, as we mentioned, we should consider only sin$ # 0; otherwise our gauge fixing is not valid. Then we readily see that the gauged action describes (to one-loop order) the background (we omitted the hat from r and have an

4390 DAVID GERSHON 5 1 -

overall factor of kz) G x x = gives rise to an additional scalar field ( that has only zero modes in our case) associated with the ver-

( r - l ) ( l + ~ ~ s i n ~ 8 ) dS2 = - k1 dr2 tex operator [16]

r + Q2 sin2 8 dt + k 2 r ( r - 1)

the antisymmetric tensor (the L'axion field") which has only the t , $h component,

the electromagnetic vector potential

and the dilaton field is

Notice that A, in (1) vanishes in our case. In fact we could get also (differen) A, = A, with the same 4D space-time metric if we used the following gauge: taking q!~ = cp and gauging X also with A2 in (8), i.e., putting 1 instead of 0 in Tz,R.

Now let us examine the space time we obtained: The space-time metric (17) describes a static axisymmetric black hole. The sphere r = 1 is the event horizon and r + Q2 sin2 8 = 0 is the singularity. A similar type of singularity occurs in the Kerr solution, in which the sin- gularity is a t r2 + a cos2 8 = 0, with a being the total angular momentum divided by the mass. I t can be seen there that the singularity is a ring ("ring singularity") with a topology S' x R (a circle "cross" time) [17]. In our solution the singularity has a topology S2 x R (a two sphere "cross" time) as can be seen by extending r up to -Q2. Q is associated with the axion charge as we show later. We can define a new coodinate

which is like the Boyer-Lindquist coordinates for the Kerr solution [la]. Then i: = 0 is the singularity and the event horizon is a t P = Q 2 sin8 + 1. The above observations are obtained by calculating all the scalar curvatures of the metric (Riemann curvature, Ricci curvature, scalar curvature) which are all singular only a t the point i: = r + Q2 sin2 8 = 0. The expressions are rather long; thus, we bring here only the scalar curvature

Q 2 ( r - 1) (+ sin2 28 - cos 26 + Q2 sin2 8 - Q 2 ( r - 1) cos 28

A;: +

A2P

where

It is easy to see that RcurVatur, blows up when r^ -+ 0. The metric is not asymptotically flat. In Cartesian coordinates, a t r -+ m the metric approaches (for kl = k2)

In other words, it describes a distribution of matter all over the space-time. In terms of the new coordinate P = r + Q 2 sin2 8 the metric is

51 - EXACT SOLUTIONS OF FOUR-DIMENSIONAL BLACK HOLES. . . 4391

d s 2 = - (i - Q2 sin2 6 - 1)(1 + Q2 sin')

dt r

:di2 - $Q2 sin 26did6 + [i2 - i (1 + Q2 sin2 6) + Q2(Q2 + 1) sin2 6]d02 (i - Q2 sin2 6) sin2 6 + + (i - Q2 sin2 6 ) ( i - Q2 sin2 0 - 1)

dqh2 . (24) r

The metric G,, in (17) was read directly from the u-model action (14). The Einstein metric is obtained by conformal rescaling of the m-model metric involving the dilaton field. In our notation G:,, = e'G,,, where 9 is given in (19).

The transformation of the curvature tensors are well known. For example, under the transformation G,, = RG,,, the scalar curvature transforms as [17]

where d is the number of the dimensions. It can easily be seen that in our model, the Einstein metric describes a four-dimensional axisymmetric black hole. The singularity remains at r + Q2 sin2 6 = 0 only, and the event horizon at r = 1. For Q = 0 and kl = k2 the Einstein metric, after making coordinate transformation r + r 2 , is

and the black hole has a magnetic field only. The charged black hole is described by a Conformal

Field Theory (CFT) with a central charge c = 3kl/(k1 - 2)+3kz/(kz +2)-1 [as the gauged U ( l ) subgrops decrease the central charge by 1 each]. Now, if we want this model to describe the complete space-time we need to have ei- ther c = 26 in the bosonic strings or c = 15 in the su- persymmetric strings. For any integer k2 we can find the appropriate kl. On the other hand we can describe our space-time as M4 x K where M is the four-dimensional Lorentzian space-time and K accounts for another con- formal field theory [19] so that the total central charge is 26 (or 15 in the supersymmetric case). Then we should take kl, k2 -+ ca. In this case our model is regarded as a Kaluza-Klein model, with one compactified dimension that is part of K.

It is clear that the model we described will have higher- order corrections in l /k i in the space-time metric, the antisymmetric tensor, the gauges fields, and in the dila- ton. Only when kl, k2 m this model can be regarded as the exact to all order solution. In order to find the exact solution for finite levels in our case [20] we can apply either the Hamiltonian method as in [21] or the Lagrangian method suggested in [22]. This is the advan- tage of having the action derived by gauging the WZW model rather than using the O(d,d) symmetry trans- formations [23]. (In fact, we found [24] a similar class of solutions by applying 0(3,3) transformations on the [SL(P,R)/U(l)] x [SU(2)/U(l)] xU(1) WZW model.)

Let us now analyze some properties of the black hole. The temperature of the black hole associated with the metric in(17) can be obtained by calculating the surface gravity [17]. The metric is static and the Killing vector d:, which we denote xu, defines a quantity K that is con- stant on the horizon. This quantity is associated with the temperature of the black hole. The surface gravity is defined as

where "limn stands for the limit as one approaches the horizon. In our case we obtain

(For comparison, the temperature of the two-dimensional black hole is a [25].)

From the explicit expression for the electromagnetic vector field we obtain the electromagnetic tensor F,,, defined by

F,, = V,A, - V,A, .

Therefore the electric (Fo,,) and the magnetic (Fi,,) fields are

Now if we observe the action (14) we see that vector potential is multiplied by tan($/2), which we have ab- sorbed in X. Only when tan($/2) = 0 the black hole has no electromagnetic field. Since sin$ = 0 invalidates our gauge fixing ( lo) , our model cannot describe properly uncharged black hole.

K2 = lim{-(xbVbxc) ( X ~ V ~ X ~ ) / X ~ X ~ } , (27) The effective action [7] of this theory is obtained in

4392 DAVID GERSHON

the Kaluza-Klein fashion [26] as a dimensional reduction from the five-dimensional effective action. Denote the scalar field which was introduced as G x x = e'. Then the effective action can be written in the following way (we drop the volume element due to the integration over the fifth dimension):

where g is the four-dimensional Lorentzian metric: F = dA, H = dB, and V p = 0 in our case. From the above we see that the total electric charge is

where the integral is over a two-sphere at infinity ( * F is F-dual) . Thus

The magnetic charge vanishes since

Two axionic charges are associated with the action (33). The first one. which vanishes in our solution is

(although in our solution locally FAF # 0) and the other axionic charge is

Thus, this black hole carries both electric and axionic charges but has no magnetic charge. On the other hand, it is well known that the equations of motion are invari- ant under the duality transformation F -t * F [5 ] while keeping the metric fixed. Hence, magnetically charged black hole solutions may also be obtained as equivalent string theories.

Let us now turn to describe very briefly other back- grounds that are related with the coset model we have constructed in (14). In this action both the fields X and 4, of which the background is independent of, are compactified. Let us now consider the model with 4 as- sociated with the Abelian background gauge field rather than X and take the coordinate X in (5) to be noncom- pact and be associated with a space coordinate. After transforming cosh2 r -+ r and z = X/2 we obtain the background

+ ( r - 1) sin2 B + 2 ~ r r + Q2 sin2 8 dtdz (39)

the antisymmetric tensor (the "axion field") which has only the t , z component,

the electromagnetic vector potentials A and 2,

and the dilaton field

In this background the coordinates r , 8 , z are "cylindrical"-like coordinates. The event horizon is on the "cylinder" r = 1 (it is easy to see that the area of the event horizon is infinite). Thus this background describes a four-dimensional black membrane.

Finally, let us consider the analytical continuation of this model. We take i ( T + ;) = r , where T is now the time coordinate. This background is therefore described as SU(1,l) x SU(2) ~ U ( l ) / u ( l ) ~ coset. We denote t by pl and q5 by p2. We obtain

the antisymmetric tensor

the electromagnetic vector potential

and the dilaton field is

51 - EXACT SOLUTIONS OF FOUR-DIMENSIONAL BLACK HOLES . . . 4393

This model describes a closed expanding a n d recollapsing A C K N O W L E D G M E N T S universe with charged mat te r . It c a n b e considerd a s t h e "charged" version of t h e closed expanding universe derived by Nappi a n d W i t t e n [13]. A t T = 0 t h e Universe I would like t o t h a n k S. Yankielowicz for encour-

s t a r t s wi th a collapsed s ta te , t h e "big bang," a n d at T = agement a n d interesting discussions a n d N. Marcus, B. .rr i t recollapses. T h e factor ka which is a prefactor of Reznik, a n d A. Casher for helpful comments. This work t h e action, multiplied by k l / k z sets t h e scale of the t ime was supported i n p a r t by t h e U.S.-Israel Binat ional Sci- parameter T t o b e k l . ence Foundat ion a n d t h e Israel Academy of Science.

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