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Static spherically symmetric solutions to Einstein-Maxwell dilaton field equations in D dimensions

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Class. Quantum Grav. 12 (1995) 2799–2816. Printed in the UK

Static spherically symmetric solutions to Einstein–Maxwell dilaton field equations in D dimensions Metin Gürses† and Emre Sermutlu Department of Mathematics, Faculty of Science, Bilkent University, 06533 Ankara, Turkey Received 22 March 1995, in final form 10 August 1995 Abstract. We classify the spherically symmetric solutions of the Einstein–Maxwell dilaton field equations in D dimensions and find some exact solutions of the string theory at all orders of the string tension parameter. We also show the uniqueness of the black-hole solutions of this theory in static axially symmetric spacetimes. PACS numbers: 0470, 1125

1. Introduction Recently we have observed an increase of interest in constructing exact solutions of string theory [1–24]. There are several ways of performing such a construction [1]. One of these is to start with an exact solution of the leading-order field equations (low-energy limit of string theory) and showing that this solution also solves the string equations at all orders of the inverse power of the string tension parameter. Plane-wave spacetimes with appropriate gauge and scalar fields have been shown to be exact solutions of the string theory [2–4]. In these solutions the higher-order terms in the field equations disappear due to the special form of the plane-wave metrics. Levi-Civita–Bertotti–Robinson spacetimes in four dimensions are also an exact solution of the string theory [8, 21]. In this solution the higher-order terms do not disappear but give some algebraic constraints among the constants of solutions [21]. To construct exact solutions in this approach one must first study the low-energy limit of the string theory with some symmetry assumptions like spherical symmetry. Among these solutions one searches for those satisfying the field equations at all orders. Hence we need a classification of the solutions under a symmetry of the low-energy limit of the string theory. In this work, instead of working directly with the low-energy limit of the string theory we consider the Einstein–Maxwell dilaton field theory with arbitrary dilaton coupling constant α. For some fixed values of α this theory reduces to the low-energy limit of the string theory and Kaluza–Klein-type theories. With the assumption of a symmetry, static spherical symmetry for instance, some of the solutions may describe black-hole spacetimes. Non-rotating black-hole solutions of this theory, in spherically symmetric spacetimes, found so far [17–20] carry mass M, electric charge (or magnetic charge) Q and a scalar charge 6. Out of these parameters only two of them are independent. In four dimensions we have the Garfinkle–Horowitz–Strominger (GHS) solution [18]. Gibbons and Maeda (GM) [17] have given black-hole solutions in † E-mail address: [email protected] c 1995 IOP Publishing Ltd 0264-9381/95/112799+18$19.50

2799

2800

M Gürses and E Sermutlu

arbitrary D dimensions. In four dimensions the GM solution describes the region r > r1 of the GHS metric. In this work we find and classify the solutions of the dilatonic Einstein– Maxwell (and hence of the low-energy limit of the string theory) for static and spherically symmetric spacetimes. Recently [22] the most general asymptotically flat spherically symmetric static solution of the low-energy limit of string theory in four dimensions has been found. The uniqueness of the GHS–GM black-hole solutions in four-dimensional static spherically symmetric spacetimes is manifest in this work. Following this work we find the most general Ddimensional spherically symmetric static solution of the Einstein–Maxwell dilaton field theory. The uniqueness of these black-hole solutions with the assumed symmetry (Ddimensional spherical symmetry) is understood, because the solutions of the field equations are completely classified. On the other hand without solving the field equations it is desirable to know the existence of other black-hole solutions under relaxed symmetries. Here we show that the GHS–GM metric in four dimensions is the unique asymptotically flat static axially symmetric black-hole solution of the Einstein–Maxwell dilaton field theory. In this work we assume D-dimensional static spherically symmetric spacetimes. We first write, in section 2, the geometrical quantitites like Riemann and Ricci tensors in terms of the volume form of the sphere S D−2 or in terms of its dual form. With such a representation it is relatively easier to work with the higher-order curvature terms. In the third section we classify the static spherically symmetric solutions of the Einstein–Maxwell dilaton field equations. In the fourth section we prove that the GHS+GM metrics are the unique static axially symmetric black holes of the Einstein–Maxwell dilaton field theory. In the last section we give some theorems for the D-dimensional static spherically symmetric spacetimes which are crucial to finding the exact solutions of the string theory. In this section we give some possible exact solutions of string theory in D dimensions. 2. Static spherically symmetric Riemannian geometry The line element of a static and spherically symmetric spacetime is given by ds 2 = −A2 dt 2 + B 2 dr 2 + C 2 d2D−2

(1)

where A, B and C depend only on r. d2D−2 is the line element on SD−2 . The metric can be rewritten as gij = −A2 ti tj + B 2 ki kj + C 2 hij , where ti = δit , ki = δir , hij = metric on a D − 2 sphere for i, j ≥ 2, h0i = h1i = 0. Components of the Christoffel symbol are given by 0ji m = −AA0 t i (tj km + tm kj ) + AA0 k i tj tm + BB 0 k i kj km − CC 0 k i hj m i +CC 0 (hji km + him kj ) + 0(s)j m

(2)

where 0(s) is the Christoffel symbol on a D − 2 sphere. The Riemann tensor is given by i i Rji ml = 0ji l,m − 0ji m,l + 0nm 0jnl − 0nl 0jnm .

We find that AA0 B 0 CC 0 B 0 Rij ml = AA00 − t[i kj ] t[m kl] + CC 00 − k[i hj ][m kl] B B AA0 CC 0 C2C0 2 + t[m hl][j ti] − hi[m hl]j + C 2 R(s)ij ml 2 B B2

(3)

(4)

Solutions to field equations in D dimensions

2801

where R(s)ij ml = hi[m hl]j are the components of the Riemann tensor on SD−2 . The Riemann tensor in (4) may be rewritten as Rij ml = gj l Sim − gj m Sil + gim Sj l − gil Smj + η2 Hij k...n Hml k...n

(5)

where Sij = η0 Mij + η1 ki kj + 12 η3 gij 1 H 2 gij Mij = Him...n Hjm...n − 2(D − 2) which turns out to be (D − 3)! 1 g . Mij = 2(D−3) hij − ij C 2C 2 Here the tensor Hij...k is the volume form of SD−2 , i.e. √ Hij ...k = − hij ...k

(6) (7)

(8)

(9)

where h denotes det hij . Here the scalars ηi are given by 00 A C 2(D−2) A0 B 0 C 00 B 0C0 (10) − − + 3 η0 = (D − 3)! AB 2 AB 3 CB 2 B C C 00 B 0C0 A0 C 0 − + (11) η1 = AC C BC C0 2 C 2(D−3) A0 C 0 C A00 C 2 A0 B 0 C 2 CC 00 B 0C0C 1− 2 + − + + − (12) η2 = (D − 4)! B AB 2 AB 2 AB 3 B2 B3 A0 C 0 . (13) η3 = − AB 2 C The Ricci tensor, Ricci scalar and Einstein tensor can be computed from Riemann as follows: (D − 3)! 1 (η (D − 4) + η ) + η + η (D − 1) gij Rij = 0 2 1 3 2C 2(D−2) B2 +η1 (D − 2)ki kj + [η0 (D − 2) + η2 ]Mij (14) 2η1 (D − 1) (D − 3)! [η0 (D − 4)(D − 1) + η2 (D − 2)] + + η3 D(D − 1) C 2(D−2) B2 (D − 3)! 1 (η0 (D − 4)(D − 2) + η2 (D − 3)) + 2 η1 (D − 2) Gij = − 2(D−2) 2C B 1 + η3 (D − 1)(D − 2) gij + η1 (D − 2)ki kj 2 +[η0 (D − 2) + η2 ]Mij .

R=

(15)

(16)

The covariant derivatives of Hij...k and ki are given as ∇l Hij...m = −ρ[(D − 2)Hij...m kl + ki Hlj ...m + kj Hil...m + · · · + km Hij ...l ]

(17)

∇i kj = ρ1 gij + ρ2 Mij + ρ3 ki kj

(18)

where

A0 C + AC 0 C0 ρ1 = C 2AB 2 C 2(D−2) 0 A C − AC 0 C (AB)0 ρ . = − ρ2 = 3 (D − 3)! AB 2 C AB

ρ=

(19)

2802


In equation (5) we have given the Riemann tensor of a static, spherically symmetric spacetime in terms of the volume form of the SD−2 (D − 2)-brane field. We may also write the Riemann tensor in the form Rij ml = gj l Sim − gj m Sil + gim Sj l − gil Smj + e2 Fij Fml

(20)

where AB (ti kj − tj ki ) . C2 Other tensors are defined similarly to the previous case, i.e. 1 g Sij = e0 M ij + e1 ki kj + 2 e3 gij m 1 2 g M ij = Fmj Fi − 4 F gij 1 1 g = − g h M ij ij ij . C2 2C 2 Fij =

(21)

(22) (23) (24)

The scalars are given as e0 = C 2 −

C0 2C2 C 3 C 0 A0 + B2 AB 2

e1 = η1

A0 C 0 C A00 C 2 A0 B 0 C 2 CC 00 B 0C0C C0 2 e2 = −C 2 1 − 2 + − + + − B AB 2 AB 2 AB 3 B2 B3 e3 = η3 .

(25)

g Notice that when D = 4, M ij = Mij and e2 = −η2 . The covariant derivatives of Fij , ki and ti are given as 3C 0 AC 0 (tj gil − ti gj l ) Fij kl − C BC 3 g ∇i kj = ρ1 gij + ρ˜2 M ij + ρ3 ki kj 0 A ∇i tj = (ti kj + tj ki ) A ∇l Fij = −

(26) (27) (28)

where ρ˜2 =

(D − 3)! ρ2 . C 2(D−4)

3. Solutions of the Einstein–Maxwell dilaton field equations The field equations of the Einstein–Maxwell dilaton theory can be obtained from the following Lagrangian: √ R 4 1 −αe φ 2 2 − (∇φ) − e F . L = −g (29) 2κ 2 (D − 2)κ 2 4 The field equations are 8 ∂i φ∂j φ − 12 (∇φ)2 gij − κ 2 e−αe φ Fim Fj m − 14 F 2 gij , Gij = (D − 2) ∇i (e−αe φ F ij ) = 0, √ √ (D − 2)κ 2 αe −g −αe φ 2 F = 0, e ∂i ( −gg ij ∂j φ) + 32

(30) (31) (32)


2803

where Fij is the Maxwell and φ is the dilaton field. Here i, j = 1, 2, . . . , D ≥ 4. In static spherically symmetric spacetimes, gravitational field equations first lead to η0 (D − 4)(D − 2)! η2 (D − 3)(D − 3)! η1 (D − 2) η3 (D − 1)(D − 2) + + + 2C 2(D−2) 2C 2(D−2) B2 2 4φ 0 2 =0 − (D − 2)B 2 8φ 0 2 =0 η1 (D − 2) − (D − 2) κ 2 Q2 eαe φ = 0. η0 (D − 2)! + η2 (D − 3)! − A2D−2 The dilaton equation is A D−2 0 0 αe ABκ 2 Q2 eαe φ 8 φ − = 0. C D−2 B 2C D−2 A2D−2 From equations (33) and (34) we obtain " 0 #0 AC d C = d 2 ABC d−1 B

(33) (34) (35)

(36)

(37)

where d = D − 3. Using the freedom in choosing the r coordinate we can let ABC d−1 = r d−1 .

(38)

Using equations (37) and (38) we obtain A2 C 2d = r 2d − 2b1 r d + b2

(39)

where b1 and b2 are integration constants. A combination of the dilaton (32) and gravitational field equations (30) give dT 2 −

d −1 8 φ0 2 = 0 , T +T0 + r (d + 1)2

(40)

where T is defined as T =

(r d + c1 )r d−1 16φ 0 . − (r 2d − 2b1 r d + b2 ) (d + 1)2 αe

(41)

Here c1 is an integration constant. Now defining a new function ψ(ρ) as follows: k1 dr d−1 ψ(ρ) − 2b1 r d + b2 )

(42)

r 2d − 2b1 r d + b2 dψ dρ = (ψ + µ)2 + ν 2 . dr d−1 dρ dr

(43)

αe φ 0 =

(r 2d

equation (40) becomes

The constants are given by (d + 1)2 αe2 µ = −(c1 + b1 ) 32d Now, if we solve the auxiliary equation a=

r 2d − 2b1 r d + b2 dρ =ρ dr d−1 dr

ν 2 = aµ2 − 1(a + 1) .

(44)

(45)

2804


and insert ρ above in (43), we obtain dψ = (ψ + µ)2 + ν 2 . ρ dρ Note that ρ depends on the sign of 1 = b12 − b2 . Hence φ can be found from ψ by Z ψ(ρ) k1 dρ + φ0 φ= αe ρ where k1 = as

2a a+1

(46)

(47)

and φ0 is an arbitrary constant. The metric function C is connected to φ

(r d + c1 )r d−1 16φ 0 C0 = 2d − . C (r − 2b1 r d + b2 ) (d + 1)2 αe This gives us

ln

C c0

Z

d =

u + b1 + c1 αe du − φ 2 u −1 2a

(48)

(49)

where u = r d − b1 and c0 is an arbitrary integration constant. Metric functions A and B can be found from C through (38) and (39). We have three different cases according to the sign of 1: d r − r1d 1 ln d case 1: 1 > 0 ln ρ(r) = d r1 − r2d r − r2d 1 case 2: 1 = 0 ln ρ(r) = − d (50) r − r3d d r − b1 π 1 arctan √ − case 3: 1 < 0 ln ρ(r) = √ 2 −1 −1 √ √ d d d where r1 = b1 + 1, r2 = b1 − 1, r3 = b1 . The integration constants c0 and φ0 are determined by taking the asymptotic behaviour of the functions C(r) and φ(r) to be lim φ = φ0 = 0

r→∞

(51)

C(r) = c0 = 1 . (52) r To determine the remaining integration constants, we restrict the asymptotic behaviour of the metric, scalar field and the tensor field Fij as follows: lim

r→∞

2κ 2 M lim r d (A2 − 1) = − r→∞ Ad+1 (d + 1) √ κ d +1 (53) 6 lim r d+1 φ 0 = − r→∞ 2Ad+1 Q lim r d+1 Ftr = r→∞ Ad+1 where M, 6 and Q are the mass, dilaton and electric charges, respectively [17]. From the definition of these constants it is clear that we are interested in the asymptotically flat solutions of the Einstein–Maxwell dilaton field equations: Case 1 1>0

(54)


2805

which means there are two roots to (39). According to the sign of ν 2 we have three distinct solutions: Type 1 ν2 < 0

λ2 = −ν 2

ψ=

λ − µ + c2 (λ + µ)ρ 2λ . 1 − c2 ρ 2λ

The metric functions become (r d − r1d )(r d − r2d ) r 2d−2 C 2 2 A2 = B = C 2d (r d − r1d )(r d − r2d ) k 1 − c2 ρ 2λ 2 k3 ρ . C d = (r d − r2d ) 1 − c2 The dilaton field is given as k (1 − c2 )ρ λ−µ 1 αe φ . e = 1 − c2 ρ 2λ The constants are given by (r d − r2d ) (ν + aµ) 1 , k3 = 1 − . (a + 1) 2 (a + 1) Undetermined integration constants are r1 , r2 , c1 and c2 . From the boundary condition we find that √ c2 1 + c2 1 + c2 2M = λ + aµ e1 6= λ − µ e2 Q=λ e3 . 1 − c2 1 − c2 1 − c2 The constants ei are given by s 2Ad+1 (d + 1) Ad+1 αe (d + 1)3/2 2Ad+1 (d + 1)d e1 = e3 = . e2 = (a + 1)κ 2 8(a + 1)κ κ (a + 1) k2 =

(55) (56)

(57)

(58) (53) (59)

(60)

Note that, we have four integration constants (c1 , c2 , r1d and r2d ) but there exist only three equations to determine them. Also note that c1 does not appear in the solution directly, so we have a freedom in c1 (c2 6= 1). In order to complete the solution,we need to determine the integration constants in terms of the physical parameters M, 6 and Q. Let us define some auxiliary variables to solve the set of algebraic equations (59): 1 a6 2M 2M 1 6 T1 = + − T2 = a + 1 e2 e1 a + 1 e1 e2 s (61) 2 4Q T3 = 1 − 2 2 . e3 T1 Then the integration constants are λ = T3 T1

µ = T2

c2 =

1−T3 . 1 + T3

(62)

Also aµ2 + λ2 . a+1 The reality of T3 imposes 1=

(63)

M + g6 ≥ s|Q|

(64)

2806


where g and s are given by

r 1 (d + 1)(a + 1) αe (d + 1)3/2 , s= . (65) g= 4dκ κ d Such an inequality has been found by Gibbons and Wells for D = 4. When 1 > 0 we have two roots to (39). In general these roots are the singular points of the spacetime. If the integration constants satisfy some additional constraints one of these roots becomes regular. An invariant of the spacetime is the scalar curvature which is given by z2 +2 2c2 λρ 2λ 2 1−c2 ρ z1 −µ + λ + 1−c ρ z1 +2λ 1−c 2λ 2λ 2ρ 2ρ + A2 (66) R = A1 1 1 [(r d − r1d )(r d − r2d )]1+ d (1 − c2 ρ 2λ )z2 [(r d − r1d )(r d − r2d )]1+ d

where 2 2 (aµ + λ) z2 = d(a + 1) d(a + 1) 8 k12 d 2 κ 2 (D − 4) . A2 = A1 = 2 d + 1 αe 2 (D − 2)

z1 =

(67)

As r → r1 we have a singularity unless we choose µ = λ. This choice, by the use of (44), r d −r d gives µ = λ = 1 2 2 . Inserting this in (66), we find that as r → r1 , R = A˜1 (r d − r1d ) + A˜2

(68)

where A˜1 and A˜2 are constants, so the horizon is regular. The choice µ = λ gives T2 = T1 T3 from (62), which means 8d6 8(a − 1) 2κ Q2 = M + 6 . (69) αe (d + 1)3/2 αe (d + 1)2 The connection between physical parameters and integration constants is r1d − r2d 1 + c2 + a e1 2M = 2 1 − c2 r d − r2d 2c2 6= 1 e2 2 1 − c2 √ c2 r d − r2d Q= 1 e3 . 2 1 − c2 The metric is (r d − r1d )(r d − r2d ) 2 r 2d−2 C 2 dt + dr 2 + C 2 d2D−2 ds 2 = − C 2d (r d − r1d )(r d − r2d ) where

6/e2 1+ d C = (r − r − r2d −k1 6/e2 eαe φ = 1 + d r − r2d Q Ftr = Ad+1 r d+1 d

and

r1d −r2d 2

= T2 .

d

r2d )

(70)

(71)

k2 (72) (73) (74)


2807

If we choose the integration constant r2 as zero, these black-hole metrics are identical to the metrics given by Gibbons and Maeda [17]. Although the extreme limit (1 = 0), in general, is going to be studied in case 2 we obtain the extreme case of the above solution by setting c2 = (r2 /r1 )d and r2 = r1 . Then we obtain 6 = r1d e2

2M = r1d e1

2Q = r1d e3 .

(75)

Type 2 ψ = ν tan(c2 + ν ln ρ) − µ ν2 > 0 Z ψ(ρ) dρ = − ln [cos(c2 + ν ln ρ)] − µ ln ρ + c3 . ρ

(76) (77)

After similar steps to the previous type, we arrive at the solution A2 =

(r d − r1d )(r d − r2d ) C 2d

C d = (r d − r2d )ρ

r1d −r2d 2

−

µk1 2

r 2d−2 C 2 (r d − r1d )(r d − r2d ) cos(c2 + ν ln ρ) k2 . cos c2 B2 =

The scalar field is given as k1 cos c2 eαe φ = . ρ µ cos(c2 + ν ln ρ)

(78) (79)

(80)

Physical parameters are found using (53): 2M = (aµ + ν tan c2 )e1 µ = T2

sin c2 =

6 = (−µ + ν tan c2 )e2

e3 T1 Q

Q=

ν e3 cos c2

ν tan c2 = T1 .

(81) (82)

The condition |sinc2 | < 1 imposes M + g6 < s|Q| where g and s are defined in (65). We also have to check the sign of 1: 2 a6 4M 2 Q2 1 + − 2 . (a + 1)1 = 2 2 a + 1 e2 e1 e3

(83)

(84)

Here the sign of 1 puts a constraint on the physical variables. Type 3 1 ψ = −µ − ν2 = 0 ln ρ + c2 Z ψ(ρ) dρ = − ln(ln ρ + c2 ) − µ ln ρ + c3 ρ k1 c2 eαe φ = ρ µ (c2 + ln ρ) 1/2 −µk1 c2 + ln ρ k2 d d C = ρ 2 . r − r1d r d − r2d c2

(85) (86) (87) (88)

2808


Physical parameters are found using (53) 1 1 2M = aµ − 6 = −µ − e1 e2 c2 c2

Q=

e3 . 2c2

(89)

The solution is µ = −T2

−

1 = T1 c2

(90)

which gives 2Q = T1 . e3 This is the equality case of the inequality (64).

(91)

Case 2 1 = 0.

(92)

Then there is one root to (39). Denote it by

r1d ,

AC d = (r d − r1d )

(93)

(r d − r1d )2 A2 = C 2d C d = (r d − r1d )ρ −

µk1 2

r 2d−2 C 2 B2 = d (r − r1d )2 cos(c2 + ν ln ρ) k2 . cos(c2 )

The scalar field is given as k1 cos c2 αe φ e = . ρ µ cos(c2 + ν ln ρ)

(94) (95)

(96)

Case 3 1 < 0. This case is similar to the previous one: k1 cos c2 eαe φ = ρ µ cos(c2 + ν ln ρ) (r 2d − 2b1 r d + b2 ) r 2d−2 C 2 2 B = A2 = C 2d (r 2d − 2b1 r d + b2 ) µk cos(c2 + ν ln ρ) k2 d 2d d 1/2 − 21 C = (r − 2b1 r + b2 ) ρ . cos(c2 )

(97)

(98) (99) (100)

In cases 2 and 3, the relation of physical parameters to integration constants are exactly the same as case 1, type 2. In addition we have (84) with the sign of 1 chosen according to the case. It is perhaps noticed that we solve field equations exactly without assuming any special ansatz†. In that respect we have all the possible integration constants in the solutions, hence we have the following assertion: in D dimensions the metric given in (71) and (72) is the only asymptotically flat metric corresponding to a spherically symmetric static black hole carrying mass M, dilaton charge 6 and electric charge Q and covered by a regular horizon. † After this work was completed we became aware of a recent paper [23], which similarly solves the field equations (30)–(32) without any special ansatz, but in a completely different way to those used here.


2809

In D dimensions the Gibbons–Maeda metric is diffeomorphic to the region r > r1 where r1 is the location of the outer horizon. Relaxing the condition of asymptotical flatness there is a possibility of having a black-hole (with a regular horizon and singularity located at the origin) solution of the Einstein–Maxwell dilaton field equations [24]. Expressing the boundary conditions of the black-hole solution (72) (the behaviour of the metric as r approaches the outer horizon r1 and to infinity) as the black-hole boundary conditions in static axially symmetric four dimensions one can prove that the GHS or Gibbons–Maeda solution also describes the unique asymptotically flat static black-hole solution in Einstein–Maxwell dilaton gravity. In the next section we give a proof of this statement. 4. Uniqueness of black-hole solutions In this section we are interested in the black-hole solutions of the Einstein–Maxwell dilaton theory in the static axially symmetric spacetimes in four dimensions. Using a different approach the uniqueness of a static charged dilaton black hole (GHS+GM black hole) has been shown recently in [27]. In this proof the dilaton coupling constant was taken as fixed (a = 2) which corresponds to the low-energy limit of the string theory. Here we shall show that the static black holes of the Einstein–Maxwell dilaton theory are unique for arbitrary values of the dilaton coupling constant a. We are not going to solve the corresponding field equations but first formulate these field equations as a sigma model in two dimensions and use this formulation in the proof of the uniqueness of the solutions under the same boundary conditions. Uniqueness of the stationary black-hole solutions of the Einstein theory is now a very well established concept [27–34]. This proof is based on the sigma model formulation of the stationary axially symmetric Einstein–Maxwell field equations [30–37]. Here we shall follow the approach given by [30, 34]. The line element of a static axially symmetric four-dimensional spacetime is given by ds 2 = e2ψ [e2γ (dρ 2 + dz 2 ) + ρ 2 dφ 2 ] − e−2ψ dt 2 .

(101)

The field equations of the Einstein–Maxwell dilaton field theory with the above metric and Aµ = (A, 0, 0, 0) are ∇ 2 ψ + 12 κ 2 e2ψ−aφ (∇A)2 = 0

(102)

∇ φ−

(103)

2

a 18 κ 2 e2ψ−aφ (∇A)2

=0

∇ A + ∇(2ψ − aφ)∇A = 0 1 γ,z = 2ψ,ρ ψ,z + 4φρ φ,z − κ 2 e2ψ−aφ Aρ A,z ρ 2 2 γ,ρ = 2ψ2,ρ − 2ψ2,z + 4φ,ρ − 4φ,z2 − κ 2 e2ψ−aφ (A2,ρ − A2,z ) . ρ 2

Let E = ψ − a2 φ and B =

κ0 √ A 2

(104) (105) (106)

we then find

∇ 2 E + e2E (∇B)2 = 0

(107)

∇ 2 B + 2∇E∇B = 0

(108)

where

r 0

κ =κ 1+

a2 . 8

(109)

2810


We wish to write (107) and (108) as a single complex equation by introducing a complex potential. In order to achieve this we introduce pseudopotential ω by use of (108), ωρ = ρe−2E Bz ,

ωz = −ρe−2E Bρ

(110)

then the resulting equations can be written as the following single complex equation (the Ernst equation) for ε = ρeE + iω: Re(ε)∇ 2 ε = ∇ε∇ε .

(111)

Hence the above complex equation represents (107) and (108) if we let ε = ρeE + iω. The above Ernst equation (111) defines a sigma model on SU (2)/U (1) with the equation of motion E E P −1 ∇P =0 (112) ∇ where P =

1 ρeE

−B ρ 2 e2E + B 2

1 −B

.

(113)

In the following we assume enough differentiability for the components of the matrix P in V ∪ ∂V . Here V is a region in M with boundary ∂V . In our case V is the region r > 0 (see section 2, type I solutions) and hence ∂V has two disconnected components. We also assume that P is positive definite. Let P1 and P2 be two different solutions of (112). The difference of their equations satisfy E E P1−1 (∇Q)P ∇ (114) 2 =0 where Q = P1 P2−1 . Multiplying both sides by Q† (Hermitian conjugation) and taking the trace we obtain E E E tr Q† P1−1 (∇Q)P E † )P1−1 (∇Q)P ∇ = tr (∇Q (115) 2 2 . The left-hand side of the above equation can be simplified further and we obtain E † )P1−1 (∇Q)P E ∇ 2 q = tr (∇Q 2

(116)

where q = tr(Q). Using the Hermiticity and positive definiteness properties of the matrices P1 and P2 we may let †

Pi = Ai Ai ,

(i = 1, 2)

where A1 and A2 are non-singular n × n matrices given by 1 1 0 . Ai = √ Ei −Bi ρeEi ρe 2

(117)

(118)

With the aid of (117), equation (116) reduces to ∇ 2 q = tr(JE† JE)

(119)

E JE = A−1 1 (∇Q)A2 .

(120)

where

Equation (119) is a crucial step towards the proof of the uniqueness theorems. It is clear that the right-hand side is positive definite at all point of V . Before going on let us give the scalar function q: 1 (121) q = 2 + 2 E +E ρ 2 (eE1 − eE2 )2 + (B1 − B2 )2 . 1 2 ρ e


2811

It is clear that q = 2 and its first derivatives vanish on the boundary ∂V of V . Let M be a two-dimensional manifold with local coordinates (ρ, z). Let V be a region in M with boundary ∂V . Let P be a Hermitian positive-definite 2 × 2 matrix with unit determinant and let P1 and P2 be two such matrices satisfying (112) in V with the same boundary conditions on ∂V then we have P1 = P2 at all points in region V . The proof is as follows. Integrating (119) in V we obtain Z Z E dE (122) ∇q σ = tr JE† JE dV ∂V

V

and using the boundary condition q = 2 on ∂V we get Z tr JE† JE dV = 0 .

(123)

V

Then the integrand in (123) vanishes at all points in V . This implies the vanishing of JE which implies that Q = Q0 = a constant matrix in V . Since Q is the identity matrix I on ∂V then Q = I in V . Hence P1 = P2 at all points in V . Another way to obtain this result is to use (119) directly. The vanishing of the integrand in (123) implies that q is a harmonic function in V . Since q = 2 on the boundary ∂V of V then it must be equal to the same constant in V as well. This implies that P1 = P2 in V . In four dimensions for the Einstein–Maxwell dilaton field theory a static black hole should carry mass M, electric charge Q and dilaton charge 6. Such a black-hole solution exists and was found by Gibbons and Maeda for an arbitrary dilaton coupling parameter a. Here the above proof implies that all those solutions with the same black-hole boundary conditions (asymptotically flat and regular horizons) as the GM solution are the same everywhere in spacetime. 5. Exact solutions of string theory In this section we introduce some theorems for static spherically symmetric D-dimensional spacetimes which will be utilized to obtain exact solutions of string theory. This section is a natural extension of the work in [21] to arbitrary dimensions. We shall not give the proofs of the theorems. The necessary tools for the proofs are given in the second section. The covariant derivatives of Hij ...k and ki given in (17) and (18) are expressed only in terms of themselves and the metric tensor. Hence we have the following theorem: Theorem 1. Covariant derivatives of the Riemann tensor Rij kl , the tensor Hij ...k and the vector ki at any order are expressible only in terms of Hij ...k , gij , ki . Since contraction of k i with Hij ...k vanishes, the only symmetric tensors constructable out of Hij ...k , gij and ki are Mij , the metric tensor gij and ki kj . Then the following theorem holds: Theorem 2. Any second rank symmetric tensor constructed out of the Riemann tensor, antisymmetric tensor Hij ...k , dilaton field φ = φ(r) and their covariant derivatives is a linear combination of Mij , gij and ki kj . Let this symmetric tensor be Eij0 . Then we have Eij0 = σ1 Mij + σ2 gij + σ3 ki kj

(124)

where σ1 , σ2 and σ3 are scalars which are functions of the metric functions, invariants constructed out of the curvature tensor Rij kl , Hij ...k and the dilaton field φ(r) and their covariant derivatives.

2812


Theorem 3. Any vector constructed out of the Riemannian tensor Rij kl , Hij ...k the dilaton field φ = φ(r) and their covariant derivatives is proportional to ki . Let this vector be Ei0 . Hence Ei0 = σ ki

(125)

where σ is a scalar like σ1 , σ2 , σ3 . The covariant derivatives of Fij , ki and ti are expressed in terms of themselves, a metric tensor. So we have a similar theorem: Theorem 4. Covariant derivatives of the Riemann tensor Rij kl , the antisymmetric tensor Fij , the vectors ki and ti at any order are expressible only in terms of Fij , ki , ti and gij . As an example to an exact solution of the low-energy limit of the string theory in D dimensions with constant dilaton field, we propose the LCBR (Levi-Civita Bertotti–Robinson) metric which is given by q2 (−ti tj + c02 ki kj − r 2 hij ) (126) r2 where hij is the metric on SD−2 , ti = δit , ki = δir , q and c0 are constants. Then the antisymmetric tensor Fij defined in (21) becomes c0 Fij = 2 (ti kj − tj ki ) (127) r m 1 2 g (128) M ij = Fmj Fi − 4 F gij gij =

2 g with e1 = e3 = 0 ⇒ Sij = e0 M ij , e0 = q

q 2 (c02 − 1) g g g g Rij kl = q 2 [gj l M Fij Fml ik − gj k Mil + gik Mj l − gil Mkj ] + c02 1 1 g 1 + (D − 3) − gij Rij = q 2 (D − 3) + 2 M ij 2q 2 c0 c02 1 1 1 g 2 2 Gij = q (D − 3) + 2 Mij + 2 2 − (D − 3) gij . 2q c0 c0

(129) (130) (131)

It is easily seen that ∇l Fij = 0,

∇m Rij kl = 0 .

(132)

Here the origin of the antisymmetric tensor Fij is geometrical. It is the volume form of the two-dimensional part ds 2 = −A2 dt 2 + B 2 dr 2 of the D-dimensional spacetime line element (1). We can define the electromagnetic tensor field, Fijel , simply as Fijel = QFij . Here Q is a constant (in asymptotically flat cases Q plays the role of electric charge). Equation (131) becomes q2 1 1 1 el 2 (133) Gij = 2 (D − 3) + 2 Mij + 2 2 − (D − 3) gij Q 2q c0 c0 g where Mijel (= Q2 M ij ) is the energy–momentum tensor of the electromagnetic field tensor el Fij . In D dimensions the LCBR metric is the solution of the Einstein–Maxwell equations with a cosmological constant. This constant vanishes when (D − 3) = c12 . In this case 0 spacetime is conformally flat only when D = 4. For this metric, the field equations of a


2813

most general Lagrangian (for instance, the Lagrangian of the low-energy limit of the string theory at all orders in string tension parameter) yields Eij = σ2 gij + σ1 Mij = Gij ,

(134)

Ei = σ ki ,

(135)

0

E = σ,

(136) 0

where σ1 , σ2 , σ and σ are constants. This means that, Einstein’s equations reduce to algebraic equations. The Lagrangian of the most general theory is a scalar containing the curvature tensor, metric tensor, matter fields and their derivatives, contractions and multiple products of all orders. According to the theorem 2 given above all second-rank symmetric tensors constructed out of these are expressible in terms of gij , Mij and ki kj . So whatever the theory is we have three equations (under our assumptions all the terms containing ki drop out) for five constants. Two constants (c0 , q) come from the definition of the D-dimensional LCBR metric (126). One constant Q comes from the definition of the electromagnetic field tensor Fijel . The other two constants are the constant dilaton field φ0 and the cosmological constant λ. In some cases λ is set to zero. In these cases as well, the number of equations are less than the number of unknown constants. In the general case, under our assumptions the field equations reduce to the following algebraic equations for q, c0 , Q, φ0 and λ: 1 q2 (137) D − 3 + = σ2 (q, c0 , Q, φ0 , λ) Q2 c02 1 1 2 + (138) −(D − 3) = σ4 (q, c0 , Q, φ0 , λ) 2q 2 c02 (139) 0 = σ (q, c0 , Q, φ0 , λ) 0 = σ 0 (q, c0 , Q, φ0 , λ)

(140)

where σ1 , σ2 , σ and σ 0 are defined in (134)–(136). Choosing φ (= φ0 ) and the metric function C (= c0 ) as constants the function σ given above vanishes identically. Hence we have three algebraic equations for q, c0 , Q, φ0 and λ. These algebraic equations may or may not have solutions. For instance, when we consider the Lovelock theory [26] without a cosmological contant these equations have solutions when D = 4 and D > 6. With a cosmological constant these algebraic equations have solutions for D ≥ 4. In the case of string theory the LCBR metric is a solution of the low-energy limit of the string theory. LCBR spacetime in four dimensions is also known to be an exact solution of the string theory [8, 21]. Here we have the generalization of this result for an arbitrary value of D > 2. If there exists a solution of the above-mentioned algebraic equations then Theorem 5. The

LCBR

metric is an exact solution of the string theory at all orders.

As in the case of the LCBR spacetime [8] the symmetric spaces, ∇i Rj klm = 0 and recurrent spaces, ∇i Rj klm = li Rj klm where li is a (gradient) vector, play an effective role in constructing exact solutions of the string theory. These spaces are, in general, product spaces [38]. In the case of static spherically symmetric spacetimes the only possibility to have such spaces is to choose C = c0 . Then D-dimensional spacetime becomes M2 × S D−2 , where M2 is a two-dimensional geometry with metric ds 2 = −A2 dt 2 + B 2 dr 2 . If we take C = c0 in the metric (1), then 0 0 c02(D−2) 1 A η0 = η1 = 0 (141) (D − 3)! AB B η2 = ξ − (D − 3)η0

η3 = 0

2814 where ξ =

M Gürses and E Sermutlu c02(D−3) : (D−4)!

Rij =

(D − 3)!

[−η0 + ξ ]gij + [η0 + ξ ]Mij 2c02(D−2) (D − 3)! R = 2(D−2) [−2η0 + (D − 2)ξ ] c0 (D − 3)! Gij = − 2(D−2) [−η0 + (D − 3)ξ ]gij + [η0 + ξ ]Mij 2c0

(142)

∇l Hij ...m = 0 ∇i kj =

c02(D−2) A0 A0 (AB)0 g + M − ki kj . ij ij 2AB 2 (D − 3)! AB 2 AB

If we let η0 be a constant, then we have to solve the equation B2 =

1 A0 0 AB B

A0 2 αA2 + β

(143) = α. The solution is (144)

where α and β are constants. Hence the metric (1) can be written as ds 2 = −A2 dt 2 +

dA2 + c02 d2D−2 . αA2 + β

(145)

The case β = 0 is identical with the LCBR metric. As in the case of the LCBR metric the metric constants α, c0 and the magnetic charge satisfy three coupled (due to the theorems given before) algebraic equations. If these algebraic equations have solutions then the metric given above with nonvanishing α and β is a candidate as an exact solution of the string theory. Above we have given the Riemann tensor in terms of the volume form of SD−2 (D − 2)brane field. We may also write the Riemann tensor in the form Rij ml = gj l Sim − gj m Sil + gim Sj l − gil Smj + e2 Fij Fml .

(146)

In this case e0 = c02

e1 = 0

e2 = −c02 + c02 ζ 0 0 c2 where ζ = AB0 AB : Rij = R=

e3 = 0

D−3−ζ g gij + (D − 3 + ζ )c02 M ij 2c02

D 2 − 5D + 6 − 2ζ c02

Gij =

(147)

(148)

ζ − (D − 3)2 g gij + (D − 3 + ζ )c02 M ij 2c02

∇l Fij = 0 0 (149) A0 (AB)0 4 A g ki kj . g + c − M ij ij 0 2 2 2AB AB AB The solution given in (144) is not valid when A is set to a constant. In such a case B can be chosen as unity. Therefore letting A = B = 1 and C = c0 . We find that e0 = −e2 = c02

∇i kj =


2815

∇l Fij = 0,

(150)

and ∇m Rij kl = 0

∇i kj = 0,

∇i tj = 0 eij + 1 gij Rij = (D − 3)c02 M 2c02 (D − 3)(D − 2) R= c02 1 2 e Gij = (D − 3)c0 Mij + 2 (3 − D)gij . 2c0

(151) (152) (153) (154)

Since the metric contains only one arbitrary constant c0 it may not be possible to eliminate the coefficient of gij (the cosmological constant) in the field equations. Hence the metric given by ds 2 = −dt 2 + dr 2 + c02 d2D−2

(155)

is a candidate of an exact solution of the string equations with a cosmological constant. 6. Conclusion We have found all possible asymptotically flat solutions of the Einstein–Maxwell dilaton field theory in D-dimensional static spherically symmetric spacetimes with arbitrary dilaton coupling constant. We proved that the asymptotically flat black-hole solutions carrying mass M, electric charge Q and dilaton charge 6 in four dimensions are unique not only in static spherically symmetric spacetimes but also in static axially symmetric spacetimes. This proof may be extended to arbitrary dimensions and also to spacetimes which are not asymptotically flat [24]. We searched for all possible static spherically spacetimes which may solve the string equations exactly (to all orders in string tension parameter). Acknowledgments This work is partially supported by the Scientific and Technical Research Council of Turkey (TUBITAK) and Turkish Academy of Sciences (TUBA). We would like to thank the referees for their several suggestions, in particular for informing us of the recent work of [23, 24]. References [1] Tseytlin A A 1994 Exact string solutions and duality Proc. 2nd Journée Cosmologie, Observatoire de Paris Imperial/TP/93-94/46, hep-th/9407099 (Singapore: World Scientific) to appear [2] Güven R 1987 Phys. Lett. 191B 275 [3] Amati D and Klimcik C 1989 Phys. Lett. 219B 443 [4] Horowitz G and Steif A 1990 Phys. Rev. Lett. 64 260 [5] Horowitz G and Tseytlin A A 1994 Phys. Rev. Lett. 73 3351 [6] Tseytlin A A 1994 Black holes and exact solutions in string theory Proc. Int. School of Astrophysics ‘D Chalonge’ 3rd Course: Current Topics in Astrofundamental Physics (Erice) ed N Sanchez (Singapore: World Scientific) to appear [7] Witten T and Nappi 1993 Phys. Rev. Lett. 71 3751 [8] Lowe D and Strominger A 1994 Phys. Rev. Lett. 73 1468 [9] Kallosh R and Ortin T 1994 Phys. Rev. D 50 R7123 [10] Horowitz G and Tseytlin A A 1995 Phys. Rev D 51 2896 [11] Zwiebach B 1985 Phys. Lett. 156B 315

2816 [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]


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