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7 Solutions with a Gauge Field

7.1 Introduction

Black holes with nontrivial gauge fields play an important role in string theory, not least because the types of charge that they carry help identify their microscopic constituents (e.g., branes). In this section we shall review briefly such solutions for d ≥ 5. We shall concentrate exclusively on solutions of maximal supergravity theories arising as consistent toroidal reductions of d = 10,11 supergravity.

The bosonic sector of maximal supergravity theories contains massless scalars taking values in some coset space G ∕H where G is noncompact and H is compact. Since the scalars are massless, their asymptotic values (for an asymptotically-flat solution) are arbitrary, i.e., they are moduli. G is a global symmetry of the supergravity theory (broken to a discrete U-duality symmetry in string/M-theory). By acting with G we can choose the moduli to be anything convenient. Given a particular choice for the moduli, the global symmetry group is broken to H. See, e.g., [210] for a review of this.

The only type of conserved gauge charge that an asymptotically flat solution can carry is electric charge with respect to a 2-form field strength (or magnetic charge with respect to a d − 2 form field strength, but this can always be dualized into a 2-form). The group H acts nontrivially on the charges of a solution. Hence, the strategy in looking for black-hole solutions is to identify a “seed” solution with a small number of parameters, from which a solution with the most general charge assignments can be obtained by acting with H. For example, dimensional reduction of d = 11 supergravity on 6 T gives the maximal 𝒩 = 4, d = 5 supergravity, which has 27 Abelian vectors and 42 scalars parameterizing the coset E6 (6)∕U Sp (8 ). It turns out that the a solution with 27 independent charges can be obtained by acting with U Sp (8) on a seed solution with just 3 charges [58Jump To The Next Citation Point].

7.2 Topologically-spherical black holes

7.2.1 Nonextremal solutions

The construction of stationary, charged, topologically-spherical, black-hole solutions (“charged Myers–Perry”) in maximal supergravity theories was described in [58]. It turns out that the seed solutions are the same as for toroidal compactifications of heterotic string theory. The latter seed solutions were constructed in [63Jump To The Next Citation Point] for d = 5 and [64184] for 6 ≤ d ≤ 9. In addition to their mass and angular momenta, they are parameterized by three electric charges in d = 5 and two electric charges for 6 ≤ d ≤ 9. For d = 10 (type IIA theory), the solution describing a rotating black hole charged with respect to the Ramond-Ramond 2-form field strength (i.e., D0-brane charge) can be obtained from the general rotating brane solution given in [55Jump To The Next Citation Point].

In the limit of vanishing angular momenta, all of these solutions reduce to d > 4 generalizations of the Reissner–Nordström solution. This static case is the only case for which linearized stability has been investigated. The spherical symmetry of static solutions permits a scalar/vector/tensor decomposition of perturbations. For Reissner–Nordström solutions of d > 4 Einstein–Maxwell theory, decoupled equations governing each type have been obtained [164Jump To The Next Citation Point]. These have been used to prove analytically the stability of tensor and vector perturbations for d ≥ 5 and scalar perturbations for d = 5 [164Jump To The Next Citation Point]. Numerical studies have revealed that scalar perturbations are also stable for 6 ≤ d ≤ 11 [169].

7.2.2 Supersymmetric solutions: the Breckenridge-Myers-Peet-Vafa (BMPV) black hole

Black holes saturating a Bogomolnyi-Prasad-Sommerfield (BPS) inequality play an important role in string theory. The canonical example is the first black-hole entropy calculation [229], in which the BPS condition provided the justification for relating the Bekenstein-Hawking entropy of a classical black-hole solution to the statistical entropy of a microscopic brane configuration.

The only known asymptotically-flat BPS black-hole solutions occur for d = 4,5. For d = 5, one can obtain BPS black-hole solutions as a limit of the nonextremal charged rotating solutions just discussed. Starting from the six-parameter seed solution of [63] one can obtain a four-parameter BPS black hole: the BMPV black hole [17242]. The solution has equal angular momenta J1 = J2 = J and the four parameters are J and the three electric charges. The mass is fixed by the BPS relation. Note that one loses two parameters in the BPS limit; this is because the BPS limit and the extremal limit are distinct for rotating black holes - a BPS black hole is necessarily extremal, but the converse is untrue. As for the Myers–Perry solution, the equality of the angular momenta gives rise to a non-Abelian isometry group ℝ × U(1) × SU (2). Classical properties of the BMPV solution have been discussed in detail in [102Jump To The Next Citation Point106].

7.3 Black rings with gauge fields

7.3.1 Dipole rings

Black holes can only carry electric charge with respect to a two-form. However, higher-rank p-forms may also be excited by a black hole, even though there is no net charge associated with them. This occurs naturally for black rings. Consider a black string in d dimensions. A string carries electric charge associated with a three-form field strength H. The electric charge is proportional to the integral of ⋆H over a (d − 3)-sphere that links the string. Now consider a black ring with topology S1 × Sd−3 formed by bending the string into a loop and giving it angular momentum around the 1 S. This would be asymptotically flat and, hence, the charge associated with H would be zero. Nevertheless, H would be nonzero. Its strength can be measured by the flux of ⋆H through a Sd− 3 linking the ring, which is no longer a conserved quantity but rather a nonconserved electric dipole. Such a ring would have three parameters, but would only have two conserved charges (mass and angular momentum), hence it would exhibit continuous nonuniqueness.

Exact solutions describing such “dipole rings” have been constructed in d = 5 [79Jump To The Next Citation Point]. In d = 5, one can dualize a 3-form H to a two-form, so these solutions carry magnetic dipoles with respect to two-form field strengths. The dipole rings of [79Jump To The Next Citation Point] are solutions of 𝒩 = 1, d = 5 supergravity coupled to two vector multiplets, which is a theory with 3 U(1) gauge group that can be obtained by consistent truncation of the maximal d = 5 supergravity theory. These rings are characterized by their mass M, angular momentum J1 around the S1 of the horizon, and three dipoles qi. The angular momentum J2 on the S2 of the horizon vanishes. They have the same ℝ × U (1)2 symmetry as vacuum black rings. These solutions are seed solutions for the construction of solutions of maximal d = 5 supergravity with 27 independent dipoles obtained by acting with U Sp (8 ) as described above. One would expect the existence of more general dipole-ring solutions with two independent nonzero angular momenta, but these are yet to be discovered.

7.3.2 Charged black rings, supersymmetric black rings

Black rings can also carry conserved electric charges with respect to a 2-form field strength (the first regular example was found in [70], which can be regarded as having two charges and one dipole in a U (1)3 supergravity theory; see also [71Jump To The Next Citation Point]). Hence there is the possibility that they can saturate a BPS inequality. The first example of a supersymmetric black-ring solution was obtained for minimal d = 5 supergravity in [74Jump To The Next Citation Point] using a canonical form for supersymmetric solutions of this theory [101Jump To The Next Citation Point]. This was then generalized to a supersymmetric black-ring solution of the U (1)3 supergravity in [1275100Jump To The Next Citation Point]. The latter solution has 7 independent parameters, which can be taken to be the 3 charges, 3 dipoles and J1. The mass is fixed by the BPS relation and J2 is determined by the charges and dipoles. See [84] for more detailed discussion of these solutions.

The most general, stationary, black-ring solution of the 3 U (1) supergravity theory is expected to have nine parameters, since one would expect the three charges, three dipoles, two angular momenta and the mass to be independent. This solution has not yet been constructed. Note that the general non-BPS solution should have two more parameters than the general BPS solution, just as for topologically-spherical rotating black holes. The most general, known, nonextremal solution [72Jump To The Next Citation Point] has seven parameters, and was obtained by applying solution-generating transformations to the dipole ring solutions of [79Jump To The Next Citation Point]. This solution does not have a regular BPS limit, and there is no limit in which it reduces to a vacuum black ring with two angular momenta.

It has been argued that a nine parameter black-ring solution could not be a seed solution for the most general black-ring solution of maximal d = 5 supergravity [181Jump To The Next Citation Point]. By acting with U Sp (8), one can construct a solution with 27 independent charges from a seed with 3 independent charges or one can construct a solution with 27 independent dipoles from a seed with three independent dipoles. However, one cannot do both at once. If one wants to construct a solution with 27 independent charges and 27 independent dipoles from a seed solution with three independent charges, then this seed must have 15 independent dipoles, and hence (including the mass and angular momenta) 21 parameters in total. The seed solution for the most general supersymmetric black ring in maximal d = 5 supergravity is expected to have 19 parameters [181].

7.4 Solution-generating techniques

[16] develops solution-generating techniques in minimal d = 5 supergravity, based on U-duality properties of the latter. By applying one such transformation to the neutral doubly-spinning black ring of Section 5.1.2, they obtain a new charged ring solution of five-dimensional supergravity. However, this solution suffers from the same problem of Dirac–Misner singularities that [71Jump To The Next Citation Point] described when a neutral single-spin ring seed is used. It appears that the problem could be solved, like in [72], by including an additional parameter that is then tuned to cancel the pathologies. It also seems possible that, like in [71], the neutral doubly-spinning black ring is a good seed for black rings with two charges, one dipole, and two independent angular momenta in the U(1)3 supergravity theory.

Besides the solution-generating techniques based on string theory and supergravity (sequences of boosts and dualities) there have been a number of analyses of the Einstein–Maxwell(-dilaton) equations leading to other techniques for generating stationary solutions. [146] studies general properties of the Einstein–Maxwell equations in d-dimensions with d − 3 commuting Killing-vector fields. [233] shows how four-dimensional, vacuum, stationary, axisymmetric solutions can be used to obtain static, axisymmetric solutions of five-dimensional dilaton gravity coupled to a two-form gauge field. Also, within the class of five-dimensional stationary solutions of the Einstein–Maxwell(-dilaton) equations with two rotational symmetries, solution-generating techniques have been developed in [175] and [251253]. Note that many of these papers do not take into account the Chern–Simons term present in d = 5 supergravity. This term is relevant when both electric and magnetic components of the gauge field are present. Thus, it can be ignored for electrically-charged static solutions, which do not give rise to magnetic dipoles. It does not play a role, either, for the dipole rings of [79Jump To The Next Citation Point], which has allowed a systematic derivation of these solutions [252254].

7.5 Multiple-black-hole solutions

The inclusion of electric charge makes it considerably easier to construct multiple-black-hole solutions than in the vacuum case discussed above. In d = 4, there exist well-known static solutions describing multiple extremal Reissner–Nordström black holes held in equilibrium by a cancellation of electric repulsion and gravitational attraction [131Jump To The Next Citation Point]. Similar static solutions can be constructed in d > 4 [198]. However, although the d = 4 solutions have smooth horizons [131Jump To The Next Citation Point], the d > 4 solutions have horizons of low differentiability [24622].

In d = 4,5, multiple-black-hole solutions can be supersymmetric. Supersymmetry makes it easy to write down stationary solutions corresponding to multiple BMPV black holes [102]. However, the regularity of these solutions has not been investigated. Presumably they are no smoother than the static solutions just mentioned. Although electromagnetic and gravitational forces cancel, one might expect spin-spin interactions to play a role in these solutions, perhaps leading to even lower smoothness, unless the spins are aligned. Note that, in general, superposition of these black holes breaks all symmetries of a single central BMPV solution except for time-translation invariance.

Supersymmetric solutions describing stationary superpositions of multiple concentric black rings have also been constructed [99100]. The rings have a common center, and can either lie in the same plane, or in orthogonal planes. The superposition does not break any symmetries. This may be the reason that these solutions have smooth horizons.

Turning to nonextremal solutions, one would certainly expect generalizations of the solutions of section 5.3 with nontrivial gauge fields. A solution describing a Myers–Perry black hole with a concentric dipole ring is presented in [255].

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