Nonabelian solutions in a Melvin magnetic universe
aa r X i v : . [ h e p - t h ] O c t Nonabelian solutions in a Melvin magnetic universe
Burkhard Kleihaus , Jutta Kunz and Eugen Radu ZARM, Universit¨at Bremen, Am Fallturm, D–28359 Bremen, Germany Institut f¨ur Physik, Universit¨at Oldenburg, Postfach 2503 D-26111 Oldenburg, Germany
October 29, 2018
Abstract
We show the existence of D = 4 nonabelian solutions approaching asymptotically a dilatonic Melvinspacetime background. An exact solution generalizing the Chamseddine-Volkov soliton for a nonzeroexternal U(1) magnetic field is also reported. According to the so called ”no-hair” conjecture, an asymptotically flat, stationary black hole is uniquelydescribed in terms of a small set of asymptotically measurable quantities. However, in recent years coun-terexamples to this conjecture were found in several theories, most of them containing nonabelian matterfields. The first nonabelian ”hairy” black hole solutions within the framework of SU(2) Einstein-Yang-Mills(EYM) theory, were presented in [1]. Although the new solutions were static and have vanishing Yang-Mills(YM) charges, they were different from the Schwarzschild black hole and, therefore, not characterized bytheir total mass. Remarkably, in the limit of zero event horizon radius of these configurations, the globallyregular, particle-like solutions originally found in [2] are recovered.It is however worth inquiring what happens with these solutions if we drop the assumption of asymptoticflatness. The asymptotically (anti)-de Sitter solutions which are found for a nonzero cosmological constantenjoyed considerable attention in the last years and present many interesting features [3].Another interesting example of non-asymptotically flat solutions in general relativity is given by theMelvin magnetic universe, describing a bundle of magnetic flux lines in gravitational-magnetostatic equilib-rium [4]. This solution has a number of interesting features, providing the closest approximation in generalrelativity for an uniform magnetic field. The nonsingular nature of this solution (at the cost of losing theasymptotic flatness) motivated Melvin to refer to his solution as a magnetic ”geon”.There exists a fairly extensive literature on the properties of this magnetic universe, starting with astudy by Thorne, which investigates also its dynamical behaviour under arbitrary large radial perturbations[5]. Various generalizations of this type of solution have been proposed (see [6] for a review and relevantreferences), particularly interesting being the Melvin solution in Kaluza-Klein (KK) theory. This configu-ration derives from a flat five dimensional spacetime by performing a U(1) reduction with a twist in theidentifications [7, 8], the four dimensional theory containing an extra dilaton field. An exact solution ofEinstein-Maxwell equations describing a black hole in a background Melvin universe was constructed byErnst [9], and admits also a straightforward generalization in the KK case [7].It is therefore natural to ask whether the well-known hairy black hole solutions admit generalizationswith Melvin-type asymptotics and what new effects emerge due to the presence of a background magneticfield. The main purpose of this paper is to present such solutions in EYM-Higgs-U(1)-dilaton theory, whichapproach asymptotically a Melvin background. 1
General framework
We consider the following action in four spacetime dimensions I = 14 π Z d x √− γ h R − ∇ i ψ ∇ i ψ − e aψ f ij f ij − e aψ/ F ′ Iij F ′ Iij − e − aψ/ D i Φ I D i Φ I i , (1)which describes a gravitating system with a scalar triplet Φ I ( I = 1 , , A Ii (with field strength F Iij = ∂ i A Ij − ∂ j A Ii + ǫ IJK A Ji A Kj ), an abelian potential W i ( f ij = ∂ i W j − ∂ j W i beingthe corresponding field strength) and a dilaton field ψ , a being the dilaton coupling constant, and we note F ′ Iij = F Iij + 2Φ I f ij . This expression of the action has a higher dimensional origin and is motivated in thenext Section. The Melvin solution in Einstein-Maxwell-dilaton theory is found for vanishing SU(2) and triplet scalar fields, F ij = Φ I = 0, and reads [7] ds = Λ a ( dr + r dθ − dt ) + Λ − a r sin θdϕ , with Λ = 1 + ( 1 + a B r sin θ, (2)the dilaton and the U (1) potential being e a ( ψ − ψ ) = Λ a a , W i dx i = e − aψ B r sin θ dϕ. (3)The solution is parametrized by ψ , the value of the scalar field on the symmetry axis and B , whichcharacterizes the central strength of the magnetic field. Although not asymptotically flat, the geometry ofthis solution is singularity free and geodesically complete. A curious property of (2)-(3) is that the total fluxΦ m = I ∞ W ϕ = e − aψ π a B (4)is finite and inversely proportional to B . (The total magnetic flux for this cylindrically symmetric solutionis obtained by integrating over the entire physical area perpendicular to the z − axis, with z = r cos θ [4]).However, in the limit B →
0, even if the geometry becomes flat and the field strength goes to zero at thecentre, the total flux diverges.The solution describing a Schwarzschild black hole immersed in the dilatonic Melvin universe is a straight-forward generalization of (2), (3) and has a line element [7] ds = Λ a (cid:18) dr − Mr + r dθ − (1 − Mr ) dt (cid:19) + Λ − a r sin θdϕ , (5)with the same expressions for Λ, dilaton and U(1) potential (the ψ = 0 case was first discussed in [9]).The constant M which enters the line element (5) corresponds to the black hole’s mass. This axiallysymmetric solution contains an event horizon at r = 2 M as in the Schwarzschild vacuum case, but is notasymptotically flat owing to the gravitational effects of the magnetic field. It is evident that standard Kruskalcoordinates may be introduced in order to extend the solution across the event horizon, the only singularityoccurring at r = 0. More details on this solution can be found e.g. in [6],[10],[11]. For a vanishing U(1) and triplet scalar fields, W i = Φ I = 0, one finds a different class of solutions, cor-responding to dilaton generalizations of the SU(2)-EYM hairy black holes [1]. In the simplest, sphericallysymmetric case, these configurations are usually described by using a line element ds = dr N ( r ) + r ( dθ + sin θdϕ ) − σ ( r ) N ( r ) dt , with N ( r ) = 1 − m ( r ) r , (6)2 ( r ) corresponding to the total mass-energy within the radius r , and a SU(2) nonabelian potential A i dx i = w ( r )( τ dθ + τ sin θdϕ ) + τ cos θdϕ, (7) τ i being the Pauli matrices. The metric functions m ( r ), σ ( r ), the gauge potential function w ( r ) and thedilaton function ψ ( r ) are solutions of the equations m ′ = 2 (cid:16) e aψ/ ( w ′ N + ( w − r ) + r N ψ ′ (cid:17) , σ ′ = 2 r (cid:16) e aψ/ w ′ + 12 ψ ′ r (cid:17) , (8) (cid:16) σe aψ/ N w ′ (cid:17) ′ = σe aψ/ w ( w − r , (cid:0) N r σψ ′ (cid:1) ′ = 2 a σe aψ/ (cid:18) w ′ N + ( w − r (cid:19) , (where a prime denotes a derivative with respect to the radial coordinate r ), with suitable boundary condi-tions.Although no exact nonabelian solutions of the above equations are known, Refs. [12, 13] present bothanalytical and numerical arguments for the existence of a discrete family of black hole solutions uniquelycharacterized by the number of nodes p of the function w ( r ), with p ≥
1. Nontrivial solutions are found forany value of the dilaton coupling constant a , the dilaton field vanishing asymptotically.These solutions approach asymptotically the Minkowski spacetime background ( m ( r ) → M, σ ( r ) → r h . The gauge potential w interpolates between w ( r h ) = w (with | w | <
1) and w ( r → ∞ ) = ±
1, the Schwarzschild solution being recovered for w ( r ) = ± σ = 1, ψ = 0 and m ( r ) = M . In the limit r h →
0, a dilatonic generalization of theBartnik-McKinnon EYM solutions [2] is approached.The thermodynamics of the EYM-dilaton black holes can be discussed in the standard way (see e.g.[14]); it turns out that their entropy is one quarter of the event horizon area S = πr h , while their Hawkingtemperature is T H = σ ( r h ) N ′ ( r h ) / (4 π ). The purpose of this Section is to present a family of solutions which extremizes the action (1), keeping thebasic features of both the Melvin universe (2) and the nonabelian solutions (6), (7).Here we restrict to the case of a dilaton coupling constant a = √
3, in which case the nonabelian solutions(6) can be uplifted to become solutions of the SU(2) EYM equation in five dimensions [15, 16], extremizingthe action I = 14 π Z d x √− g (cid:16) R − F Iµν F Iµν (cid:17) . (9)In a five-dimensional perspective, the solutions of the D = 4 EYMd equations (8) with a = √ x being the extra-direction which is supposed tobe compact and with a unit length) ds = e − aψ (cid:18) dr N + r ( dθ + sin θdϕ ) − σ N dt (cid:19) + e aψ ( dx ) , (10)and the same SU(2) ansatz (7), i.e. a vanishing fifth component of the nonabelian potential, A = 0.The way to introduce a D = 4 magnetic field in a KK setup involves twisting the compactificationdirection. Following [7, 8] one shifts the coordinate ϕ → ϕ + B x (with B an arbitrary real constant), andreidentifies points appropriately. The next step is to consider the KK reduction with respect to the Kilingvector ∂/∂x , according to the generic prescription ds = e − aψ γ ij dx i dx j + e aψ ( dx + 2 W i dx i ) , (11)3 ij dx i dx j being the four dimensional line element and W i the U(1) potential. For the reduction of the YMaction term, a convenient D = 5 SU(2) ansatz is A Iµ dx µ = A Ii dx i + Φ I ( dx + 2 W i dx i ) , (12)where A Ii is a purely four-dimensional YM gauge field potential, while Φ I corresponds after the dimensionalreduction to a triplet Higgs field. It can be verified that the KK reduction of the action (9) with respect tothe x − direction, taken according to (11), (12), yields the four-dimensional action (1).Therefore, upon reduction, the new D = 4 solutions based on the configurations in Section 2.2, have aline element ds = √ Λ (cid:18) dr N + r dθ − σ N dt (cid:19) + r sin θ √ Λ dϕ , with Λ = 1 + e − aψ B r sin θ, (13)the only nonvanishing component of the U(1) potential vector W i being W ϕ = e − aψ B r sin θ . (14)The new D = 4 dilaton field ¯ ψ is ¯ ψ = ψ + 12 a log Λ , (15)while the four dimensional YM field is given by A i dx i = w ( τ dθ + τ sin θdϕ ) + τ cos θdϕ − B ( τ w sin θ + τ cos θ ) W ϕ dϕ . (16)Different from the seed solutions, the new configurations have a nonvanishing Higgs fieldΦ = B ( w sin θτ + cos θτ ) . (17)It can easily be seen that for a vanishing nonabelian matter content ( w = ± B = 0 leads usback to the EYMd seed solution (6), (7).A different type of configuration is found for w ( r ) = 0, describing a magnetic monopole black hole placedin a Melvin universe (note that here the four dimensional geometry has a closed form expression [7], for adifferent parametrization instead of (6), however).For the generic case, one can see that the causal structure of the seed EYMd solution is not changedby the twisting procedure. Supposing one starts with an initial EYMd hairy black hole solution, one findsthat the Melvin-type metric (13) describes, in terms of the usual definitions, a black hole, with an eventhorizon and trapped surfaces. It has a horizon located at r = r h (where N ( r h ) = 0), which is independentof the value of the magnetic field strength. A globally regular configuration (which differs from the Melvinsolution) is found in the limit of zero event horizon radius. For r → ∞ , the line element (13) approaches theMelvin background (2) with a = √ S = e iπτ / e iθτ / e iϕτ / . Their new expression,written in terms of a general ansatz used before in the literature on axially symmetric nonabelian solutions(see e.g. [19, 20]) is A µ dx µ = (cid:20) H r dr + (1 − H ) dθ (cid:21) τ ϕ − sin θ [ H τ r + (1 − H ) τ θ ] dϕ , Φ = ( φ r τ r + φ θ τ θ ) , (18)where H = 0, H = w , H = − B W ϕ cot θ , H = w (1 − W ϕ B ) and φ r = B cos θ , φ θ = − wB sin θ .As usual, the symbols τ r , τ θ and τ φ in the above relation denote the dot products of the cartesian vector4f Pauli matrices, ~τ = ( τ , τ , τ ), with the spatial unit vectors ~e r = (sin θ cos φ, sin θ sin φ, cos θ ) , ~e θ =(cos θ cos φ, cos θ sin φ, − sin θ ) , ~e φ = ( − sin φ, cos φ, , respectively.The matter fields of the new solution possess a nontrivial dependence on the polar coordinate θ . Themodulus of the Higgs field | Φ | = √ Φ I Φ I approaches a constant value at infinity and vanishes on p circles inthe xy -plane ( θ = π/ w ( r ). (The occurranceof asymptotically flat vortex ring solutions in a pure EYMH theory has been noticed in [17] for a setof monopole-antimonopole solutions). Given the non asymptotically flat character of the spacetime, theinterpretation of the matter field configurations in this solution is not obvious. However, since the modulusof the Higgs field is constant at infinity, as in the asymptotically flat case, we suggest that the ’t Hooftelectromagnetic field strength tensor F µν = ε IJK ˆΦ I ∂ µ ˆΦ J ∂ ν ˆΦ K + ∂ µ ( ˆΦ I A Iν ) − ∂ ν ( ˆΦ I A Iµ ) , (19)(where ˆΦ I is the normalized Higgs field) might be used to analyze the solutions. Then, following [18],one would evaluate the total nonabelian magnetic charge of the configurations, by integrating the ’t Hooftelectromagnetic field strength tensor, F θϕ = ((1 − B W ϕ ) p w sin θ + cos θ ) ,θ . (20)Thus the magnetic charge inside a closed surface S would be expressed as m = V ( S ) R S F µν dx µ dx ν , whichturns out to vanish for the new solution (although locally the magnetic charge density would be nonzero).To interpret the new solution we now suggest to consider the asymptotic expansion of the function w ( r ) = ± (cid:0) − cr . . . (cid:1) in the ’t Hooft field strength tensor (20), yielding F θϕ = ((1 − B W ϕ )(1 − c sin θr + O ( r ) )) ,θ and compare with the gauge potential of a magnetic dipole with dipole moment µ , ˜ W ϕ = µ sin θr [17, 18]. Theanalogous functional dependence then hints at the possibility to interpret the new solution as a magneticdipole with dipole moment µ = − c , immersed in a Melvin background. In the asymptotically flat casediscussed in [17], the vortex ring solutions (where the Higgs field vanishes on one or more rings) analogouslycorrespond to magnetic dipoles.A computation of the thermodynamic properties of the solution (13)-(17) can be done by applying thesame approach as for the B = 0 case. The computation of the mass and total Euclidean action is donewith respect to the Melvin background (2) (with a = √ τ = it in (13). Similar to the pure Einstein-Maxwell-dilaton case[11], it follows that the thermodynamic properties of these black holes are not affected by the background U(1)magnetic field. In particular we find the same entropy and mass as for the asymptotically flat configurations;the value of the Hawking temperature is also unchanged. A similar behaviour has been noticed in [11] forthe Ernst solution (5). Therefore, this seems to be a generic property of static black hole solutions in abackground U (1) magnetic field extending to infinity. The procedure above may be applied to other nonabelian solutions with a higher dimensional origin. Aparticularly interesting case is given by the Chamseddine-Volkov solution [21, 22] of the N = 4 , D = 4Freedman-Schwarz gauged supergravity model [23]. This exact solution is globally regular, preserves 1 / S [22]. As conjectured by Maldacena andNu˜nez, this solution provides a holographic description for N = 1 , D = 4 super-Yang-Mills theory [24].The four dimensional Chamseddine-Volkov solution in [21, 22] can be uplifted to D = 5 [25] to becomea solution of a consistent truncation of the N = 4 Romans’ model [26] with an action I = 14 π Z d x √− g (cid:16) R − ∂ µ φ ∂ µ φ −
14 e √ / φ F Iµν F Iµν + 18 e − √ / φ (cid:17) , (21)5ith F Iµν the SU(2) YM field strength. The uplifted Chamseddine-Volkov solution reads [25] ds = r e ν (cid:0) − dt + dr + Y ( dθ + sin θdϕ ) + ( dx ) (cid:1) , with Y = 2 r coth r − r sinh r − , e ν = sinh rY , (22) r being an integration constant, a dilaton field φ = φ + r ν (23)and a SU(2) field given by (7), with w = r/ sinh r. This configuration is neither asymptotically AdS norasymptotically flat, a common situation in the presence of a Liouville dilaton potential [27, 28].To generate a nontrivial D = 4 Melvin-type solution, one twists again the five dimensional configuration ϕ → ϕ + B x , and considers the KK reduction along the x − direction. Thus we find that (21) leads to thefour dimensional action, which different from (1), contains two dilatons with a nontrivial potential I = 14 π Z d x √− γ h R − ∇ i ψ ∇ i ψ − ∇ i φ ∇ i φ − e √ ψ f ij f ij − e ψ/ √ √ / φ F ′ Iij F ′ Iij (24) − e − ψ/ √ √ / φ D i Φ I D i Φ I + 18 e − √ / φ − ψ/ √ i , (one can see that (24) differrs also from the bosonic truncation of the N = 4 , D = 4 Freedman-Schwarzgauged supergravity model used in [21, 22]).The four-dimensional line element reads ds = r e ν √ Λ (cid:18) − dt + dr + Y ( dθ + sin θdϕ Λ ) (cid:19) , with Λ = 1 + B Y sin θ, (25)while the expression of the new dilaton ψ and the nonvanishing U(1) potential is e aψ = r e ν √ Λ , W i dx i = B Y sin θ dϕ. (26)The four-dimensional YM and Higgs fields are still given by (16), (17), with w = r/ sinh r .One can easily see that for B = 0 the Chamseddine-Volkov solution is recovered, since the scalars φ, ψ are not independent in this case. Asymptotically, the geometry (25) approaches the Melvin-type solution in N = 4 , D = 4 gauged supergravity found in [29]. Therefore we interpret the solution (25)-(26) as describinga nonabelian soliton in a magnetic universe. The same procedure can be applied to the more general globallyregular and black hole solutions in [30]. The main purpose of this paper was to propose a generalization of the known D = 4 spherically symmetricnonabelian solutions by including the effects of a background U(1) magnetic field. In this case, the resultingconfigurations have axial symmetry and approach asymptotically a dilatonic Melvin background. In ourapproach, we have used a twisting procedure applied to a set of five-dimensional configurations in EYMtheory. It would be interesting to construct this type of solutions for a simpler version of the action than(1), without making use of the twisting procedure; however, this would require to solve a complicated set ofpartial differential equations with suitable boundary conditions.More complicated solutions with Melvin-type asymptotics in EYM-Higgs-U(1)-dilaton theory are foundby starting with other static EYM configurations instead of (6), (7). The general procedure works asfollows: one starts with an axially symmetric EYMd ( a = √
3) solution ( γ ij , A (0) Ii , ψ ), where γ ij dx i dx j =6 ℓ + γ ϕϕ dϕ , and uplifts it to D = 5 according to (11). After twisting and reducing back to four dimensions,one generates in this way a new configuration with ds = γ ij dx i dx j = √ Λ( dℓ + γ ϕϕ Λ dϕ ) , with Λ = 1 + e − aψ B γ ϕϕ , e aψ = e aψ Λ , (27) W i dx i = e − aψ B γ ϕϕ dϕ, Φ I = B A (0) Iϕ , A Ii dx i = A (0) Ii dx i − B A (0) Iϕ W i dx i . For example, the D = 4 EYM(-dilaton) theory possesses also static axially symmetric black hole solutions[19, 31], with (these configurations are not known in closed form) dℓ = − f dt + mf dr + mr f dθ , γ ϕϕ = lr sin θf , (28)where the metric functions f , m and l are functions of the coordinates r and θ , only. After a suitable gaugetransformation, the SU(2) matter fields of these solutions are written in terms of four potentials H i ( r, θ ) as A (0) i dx i = n sin θ ( H τ + (1 − H ) τ ) dϕ − (( H /r ) dr + (1 − H ) dθ ) τ + τ dθ + nτ cos θdϕ − nτ sin θdϕ. (29)These asymptotically flat solutions are characterized by their horizon radius and three positive integers( k, n, p ), where k is related to the polar angle, n to the azimuthal angle and p to the node number of somegauge functions (the spherically symmetric solutions have k = n = 1). As r h →
0, a nontrivial globallyregular solution is approached [32]. By using this type of seed solutions one can construct more generalaxially symmetric configurations describing vortex ring solutions in a background U(1) magnetic field, wherethese vortex ring solutions need not only be located in the xy -plane, but might also come in pairs locatedsymmetrically above and below the xy -plane (similar to the asymptotically flat vortex ring solutions [17]).Similar to the asymptotically flat case, one expects all these configurations to be unstable.Fluxbrane solutions with with nonabelian fields in 4 + N spacetime dimensions can be generated in asimilar way, by starting again with solutions of the Eqs. (8) (the dilaton coupling constant there woulddepend on N ). Also, a similar construction to that presented in this paper can be done starting with a morecomplicated higher dimensional action instead of (9), a particularly interesting case being the D = 10 lowenergy heterotic string theory action, which contains nonabelian fields in the bulk.The Einstein-Maxwell-dilaton theory has also a solution describing a pair of oppositely charged blackholes in an external gauge field [33, 34]. Its euclideanised version describes the analogue of the Schwinger pairproduction of charged particles in a uniform electromagnetic field [34]. It would be interesting to constructthe nonabelian counterparts of these configurations. Acknowledgement
BK gratefully acknowledges support by the German Aerospace Center. The work of ER was supported bya fellowship from the Alexander von Humboldt Foundation.
References [1] M. S. Volkov and D. V. Galtsov, JETP Lett. (1989) 346;H. P. Kuenzle and A. K. Masood- ul- Alam, J. Math. Phys. (1990) 928;P. Bizon, Phys. Rev. Lett. (1990) 2844.[2] R. Bartnik and J. McKinnon, Phys. Rev. Lett. (1988) 141.[3] E. Winstanley, Class. Quant. Grav. (1999) 1963 [gr-qc/9812064];J. Bjoraker and Y. Hosotani, Phys. Rev. D (2000) 043513 [hep-th/0002098];T. Torii, K. Maeda and T. Tachizawa, Phys. Rev. D (1995) 4272 [gr-qc/9506018].[4] M. A. Melvin, Phys. Lett. (1964) 65. 75] K. S. Thorne, Phys. Rev. (1965) B244.[6] V. Karas, Z. Bud´ınov´a, Physica Scripta (2000) 253;J. Biˇc´ak, V. Karas, in Proc. of 5th M. Grossmann Meeting in General Relativity , eds. D. Blair et al(World Scientific, Singapore) 1989, p. 1199;[7] F. Dowker, J. P. Gauntlett, S. B. Giddings and G. T. Horowitz, Phys. Rev. D (1994) 2662[arXiv:hep-th/9312172].[8] F. Dowker, J. P. Gauntlett, G. W. Gibbons and G. T. Horowitz, Phys. Rev. D (1995) 6929[arXiv:hep-th/9507143].[9] F. Ernst, J. Math. Phys. (1976) 54.[10] W. A. Hiscock, J. Math. Phys. (1981) 1828;W. J. Wild, R. M. Kerns Phys .Rev. D21 , 332, (1980);R. A. Konoplya, Phys. Lett. B (2007) 219 [arXiv:gr-qc/0608066].[11] E. Radu, Mod. Phys. Lett. A (2002) 2277 [arXiv:gr-qc/0211035].[12] G. V. Lavrelashvili and D. Maison, Nucl. Phys. B (1993) 407.[13] E. E. Donets and D. V. Galtsov, Phys. Lett. B (1993) 411 [arXiv:hep-th/9212153].[14] M. Visser, Phys. Rev. D (1993) 583 [arXiv:hep-th/9303029].[15] M. S. Volkov, Phys. Lett. B (2002) 369 [arXiv:hep-th/0103038].[16] Y. Brihaye, B. Hartmann and E. Radu, Phys. Rev. D (2005) 085002 [arXiv:hep-th/0502131].[17] B. Kleihaus, J. Kunz and Y. Shnir, Phys. Rev. D (2003) 101701 [arXiv:hep-th/0307215].[18] B. Kleihaus and J. Kunz, Phys. Rev. D (2000) 025003 [arXiv:hep-th/9909037].[19] B. Kleihaus and J. Kunz, Phys. Rev. D (1998) 6138 [arXiv:gr-qc/9712086].[20] B. Hartmann, B. Kleihaus and J. Kunz, Phys. Rev. D (2002) 024027 [arXiv:hep-th/0108129].[21] A. H. Chamseddine and M. S. Volkov, Phys. Rev. Lett. (1997) 3343 [arXiv:hep-th/9707176].[22] A. H. Chamseddine and M. S. Volkov, Phys. Rev. D (1998) 6242 [arXiv:hep-th/9711181].[23] D. Z. Freedman and J. H. Schwarz, Nucl. Phys. B (1978) 333.[24] J. M. Maldacena and C. Nu˜nez, Phys. Rev. Lett. (2001) 588.[25] A. H. Chamseddine and M. S. Volkov, JHEP (2001) 023 [arXiv:hep-th/0101202].[26] L. J. Romans, Nucl. Phys. B (1986) 433.[27] K. C. Chan, J. H. Horne and R. B. Mann, Nucl. Phys. B (1995) 441 [arXiv:gr-qc/9502042].[28] R. G. Cai, J. Y. Ji and K. S. Soh, Phys. Rev. D (1998) 6547 [arXiv:gr-qc/9708063].[29] E. Radu and R. J. Slagter, Class. Quant. Grav. (2004) 2379 [arXiv:gr-qc/0311075].[30] S. S. Gubser, A. A. Tseytlin and M. S. Volkov, JHEP (2001) 017 [arXiv:hep-th/0108205].[31] R. Ibadov, B. Kleihaus, J. Kunz and M. Wirschins, Phys. Lett. B (2005) 180 [arXiv:gr-qc/0507110].[32] R. Ibadov, B. Kleihaus, J. Kunz and Y. Shnir, Phys. Lett. B (2005) 150 [arXiv:gr-qc/0410091].[33] F. Ernst, J. Math. Phys. (1976) 515.[34] F. Dowker, J. P. Gauntlett, D. A. Kastor and J. H. Traschen, Phys. Rev. D49