Integrability in conformally coupled gravity: Taub-NUT spacetimes and rotating black holes
aa r X i v : . [ h e p - t h ] N ov Integrability in conformally coupled gravity:Taub-NUT spacetimes and rotating black holes
Yannis Bardoux, Marco M. Caldarelli and Christos Charmousis , Laboratoire de Physique Th´eorique (LPT), Univ. Paris-Sud,CNRS UMR 8627, F-91405 Orsay, France Mathematical Sciences and STAG research centre, University of Southampton,Highfield, Southampton SO17 1BJ, United Kingdom Laboratoire de Math´ematiques et Physique Th´eorique (LMPT), Univ. Tours,UFR Sciences et Techniques,Parc de Grandmont, F-37200 Tours, France [email protected], [email protected], [email protected]
ABSTRACT
We consider four dimensional stationary and axially symmetric spacetimes for conformally coupledscalar-tensor theories. We show that, in analogy to the Lewis-Papapetrou problem in GeneralRelativity (GR), the theory at hand can be recast in an analogous integrable form. We give therelevant rod formalism, introduced by Weyl for vacuum GR, explicitly giving the rod structure of theblack hole of Bocharova et al. and Bekenstein (BBMB), in complete analogy to the Schwarzschildsolution. The additional scalar field is shown to play the role of an extra Weyl potential. Wethen employ the Ernst method as a concrete solution generating example to obtain the Taub-NUTversion of the BBMB hairy black hole, with or without a cosmological constant. We show that theanti-de Sitter hyperbolic version of this solution is free of closed timelike curves that plague usualTaub-NUT metrics, and thus consists of a rotating, asymptotically locally anti-de Sitter black hole.This stationary solution has no curvature singularities whatsoever in the conformal frame, and theNUT charge is shown here to regularize the central curvature singularity of the corresponding staticblack hole. Given our findings we discuss the anti-de Sitter hyperbolic version of Taub-NUT in fourdimensions, and show that the curvature singularity of the NUT-less solution is now replaced bya neighboring chronological singularity screened by horizons. We argue that the properties of thisrotating black hole are very similar to those of the rotating BTZ black hole in three dimensions. ontents
It is well known that solution generating techniques in General Relativity (GR) are very powerful,and seemingly difficult non-linear problems are often integrable (for a full discussion, see [1] andreferences therein). Typically one starts by considering spacetimes with special geometrical prop-erties, such as Petrov type D metrics [2], and as a result most common black hole spacetimes arefound to belong to this class. Or again one assumes the presence of symmetries in the shape ofKilling vectors, such as in the case of Weyl static and axisymmetric spacetimes [3], finding niceanalogies with ordinary Newtonian gravity (see [4] for a concise overview), leading to numerousmulti-black hole solutions. Assuming then stationary rather than static spacetimes, one obtainsLewis-Papapetrou metrics [5] where numerous formalisms, such as that of Ernst [6], enable thegeneration of rotating metrics from static ones (e.g. Kerr from Schwarzschild).Given a similar set of symmetries, an obvious question then is which of these solution-generatingtechniques can be generalized to higher dimensions and to more complex gravitating theories. Thisinterest is manifest from recent advances in string gravity/holography and the need for a genericand concise way of obtaining solutions in the presence of a cosmological constant, with additionalfields, and/or in higher dimensions. Furthermore, this interest is enhanced by four dimensionaltheories of gravity modification – like scalar-tensor theories – where exact solutions are in need inorder to understand issues, such as no hair theorems, and to test them at strong gravity scales.Many of the above mentioned integrability properties are however tied to four dimensional gravity.Typically, Petrov type D metrics allow the inclusion of a cosmological constant, as was shown bythe pioneering work of Carter [7], or an electromagnetic field, but one still does not know, forexample, the equivalent of the charged rotating solution of the Kerr-Newman black hole in higher2imensions . Furthermore, although some of the properties of the Weyl metrics survive in higherdimensions [10], it is found that the cosmological constant spoils the integrability properties of theWeyl and Papapetrou problems [11, 12] by introducing a non-trivial curvature scale in the action.In fact, it is found that the inclusion of a curvature scale does not permit a universal coordinatesystem in which integration techniques are possible. In other words, although generic methods,such as that of Ernst, go through [12], one has to adapt them each time to the solution sought,making therefore novel solutions difficult to find. A notable exception to this negative statementhas been given recently by Astorino [13]. One can also develop powerful analytical approximationtechniques such as the matched asymptotic expansion [14, 15] and the aforementioned blackfoldeffective theory [8] in order to study the geometric and physical properties of black holes whoseexact analytic form is unknown. In this article we will however restrict our attention to exactsolutions of the field equations and to methods to generate them.In practice, it turns out that the addition of massless matter fields enjoying some symmetry(possibly not shared by the full theory) sometimes provides better integrability properties than asimple cosmological constant, precisely because of the absence of an additional curvature scale. Thisis the case that we will study here, concerning four dimensional conformally coupled scalar-tensortheories. To be more precise, let us consider the following gravitational action, S [ g ab , φ ] = Z M √− g (cid:18) R − πG − Rφ − ∂ a φ∂ a φ − αφ (cid:19) d x (1)where G is the gravitational Newton constant, and we have included for later convenience thecosmological constant Λ and a self-interaction quartic potential with arbitrary coupling α . It isalso useful to introduce the effective gravitational constant˜ G ( φ ) = (cid:18) − πG φ (cid:19) − G. (2)The variation of the action (1) with respect to the metric g ab gives G ab + Λ g ab = 8 πGT φab , (3)with the scalar field energy-momentum tensor T φab = ∂ a φ∂ b φ − g ab ∂ c φ∂ c φ + 16 ( g ab (cid:3) − ∇ a ∇ b + G ab ) φ − αg ab φ . (4)Variation with respect to the scalar field gives, (cid:3) φ = 16 Rφ + 4 αφ . (5)This latter equation is invariant under the conformal transformation [16]( g ab , φ ) (cid:0) Ω g ab , Ω − φ (cid:1) (6) In such cases one has to resort to analytical perturbative methods, such as the blackfold approach [8], thatallowed the construction of approximate, higher-dimensional Kerr-Newman solutions, at least in some regimes [9]. M . Moreover, given this conformal symmetry,the trace of the energy-momentum tensor vanishes, T aa = 0. As a result, the equation of motionfor the scalar field (5) becomes (cid:3) φ = 23 Λ φ + 4 αφ . (7)Taking Λ = 0 and α = 0 gives us a massless theory, for which we will examine the Lewis-Papapetrouproblem. The more general case will then be analysed in a later section.This massless scalar-tensor theory has interesting, non-trivial, and quite simple solutions, start-ing with the static BBMB solution [17] found by Bocharova et al. and later rediscovered by Beken-stein [16], and its interesting extension with a cosmological constant – the MTZ black hole –constructed by Martinez et al. [18]. Since these pioneering works, extensions of these black holeshave been found [19, 20, 21] (see also [22, 23] for C-metric solutions), most impressively includingthe general Petrov type D metric found by Anabalon and Maeda [24]. Scalar-tensor theories are ingeneral an interesting laboratory of gravity modification and this particular theory, although notparticularly phenomenologically oriented, gives non-trivial examples and has important integrabil-ity properties due to the conformal symmetry of the scalar. So, how far can we go within thistheory? In this paper we will study the Lewis-Papapetrou system associated to the above action(1) for Λ = 0 and α = 0. We will see that it is essentially an integrable problem, as it is in the caseof GR, as long as the scalar is massless and we do not have a cosmological constant. As a concreteexample, we will construct explicitly the Taub-NUT metric of this theory, and we will study itsproperties. It should be possible to recover it as a singular limit of the Petrov type D metrics foundin [24], but we will not explicitly discuss this limit here, for the solution will be found using theErnst method. Using the insight gained from massless case, the solution is easily extended to thecase of Λ = 0 and α = 0. The presence of the cosmological constant regularizes the solution, givinga rotating black hole geometry. This is the first explicit example of a rotating black hole withinthis theory.Taub-NUT spacetime has been described by Misner as the counterexample to almost everythingin gravity [25]. In its original form it is a vacuum solution to Einstein’s equations. A concisedescription of its most important properties is given in [26]. The solution was discovered by Taubin 1951 in its cosmological chart [27], and extended by Newman, Tamburino and Unti in 1963 inits static region [28]. The metric is usually written in the following form,d s = − r − mr − n r + n (d t + 2 n cos θ d ϕ ) + r + n r − mr − n d r + ( r + n ) (cid:0) d θ + sin θ d ϕ (cid:1) (8)where n is the NUT parameter. Due to the presence of the parameter n , the solution is notasymptotically flat but also has no curvature singularity at r = 0. The solution has two Killinghorizons given by the radial coordinates r ± = m ±√ m + n . In the region where the coordinate t isspacelike, it was noted in [27] that this solution describes a Big Bang at r = r − and a Big Crunch at r = r + , and that the spacelike slices have the topology of S . The stationary, spacelike NUT regionscontain unfortunately closed timelike curves (CTCs) as a consequence of the periodicity of t , whichis imposed in order to avoid Misner strings at θ = 0, π . The presence of CTCs generically discards4he Lorentzian signature version of these metrics, and in general only the Euclidean signaturesolutions are studied, which can give nut and bolt instantons. The NUT parameter n is associatedto the notion of gravitational magnetic mass. In fact, sections of constant r describe the geometryof a principal U (1) fibration over S where the coordinate t has the periodicity 8 πn (see e.g. [29]).The geometric cross-term A = 2 n cos θ d ϕ in (8) is analogous to the potential of the electromagneticfield generated by a magnetic monopole (the Dirac monopole) of charge proportional to n , up to agauge transformation. Thus, there is a dictionary between this U (1)-bundle over S and a magneticmonopole, introduced by Dirac in order to explain the quantization of electric charge [30]. In factthe periodicity imposed on the coordinate t is analogous to the Dirac quantization condition of theelectric charge. Moreover, we see that the potential A cannot be globally defined: two patches haveto be used to cover S . These singularities of the potential A at θ = 0 or θ = π are called Misnerstring singularities in the context of the Taub-NUT solution, and are analogous to the Dirac stringsingularities when we consider the field produced by a magnetic charge.In the next section we shall discuss stationary and axisymmetric spacetimes, solutions of (1).As we shall see, we have complete integrability of the Lewis-Papapetrou system in the same senseas in GR. We will then explicit the Ernst method for this theory and give as a concrete example theconstruction of the NUT charged BBMB solution. Having done this it will be straightforward toadd a cosmological constant, and Maxwell and axionic charges. We will show that the hyperbolicversion of Taub-NUT-AdS with conformal scalar is actually a rotating black hole with a well definedhorizon. Finally, in the concluding section we will discuss, in light of our findings, the simpler caseof hyperbolic Taub-NUT-AdS metric and argue that it is a rotating black hyperboloid membrane inAdS. The black hole curvature singularity, present for n = 0, is replaced by a chronology violatingregion covered by inner and outer horizons, in a way that is reminiscent of the three-dimensionalBTZ solution [31]. Consider stationary and axisymmetric metrics, solutions of the theory defined by the action (1) forΛ = 0 and α = 0. This means that we are assuming the existence of two commuting Killing vectors k and m such that the former is asymptotically timelike, and the latter is spacelike and has closedorbits. It is natural to impose the same symmetries to the scalar field φ , that is L k φ = L m φ = 0,where L X denotes the Lie derivative with respect to the fields X . In GR, all stationary andaxisymmetric metrics can be written in the Lewis-Papapetrou form, see for example [32]. Thiscan be extended to the case of a cosmological constant, but here, given that our metrics are notEinstein metrics, we are not ensured that the Frobenius conditions are still verified for the abovegravitational action (1). We provide a proof of these conditions in appendix A. Consequently,using the relations (88), assuming φ = πG , and skipping some details [32], we can introduce acoordinate system ( t, r, z, ϕ ) such that any stationary and axisymmetric solution of the theory (1)5an be recast in the Lewis-Papapetrou form,d s = − e λ (d t + A d ϕ ) + e ν − λ ) (d r + d z ) + α e − λ d ϕ , (9)where α , λ , ν , A and the scalar field φ are functions of variables ( r, z ). When A = 0, the geometriesare static and form the Weyl class of metrics. What is important for our solution-generatingpurposes is that the form of the field equations approaches that of the equivalent set-up in vacuumGR. Towards this aim, we define ε = ± G , carry out the following redefinitionsof the metric functions, β = ε (cid:18) − πG φ (cid:19) α, e ω = ε (cid:18) − πG φ (cid:19) e λ , e χ = ε (cid:18) − πG φ (cid:19) e ν , (10)and rewrite the scalar field in term of a function γ such thattanh γ = r πG φ ! ε . (11)With these redefinitions, the above metric becomesd s = 14 (cid:0) e γ + εe − γ (cid:1) h − e ω (d t + A d ϕ ) + e χ − ω ) (cid:0) d r + d z (cid:1) + β e − ω d ϕ i . (12)We see that this metric is similar to the one of vacuum GR with the same symmetries – thewell-known Lewis-Papapetrou form (see [5, 33]) – but now we have an extra conformal factor in(12) which is simply the dimensionless effective gravitational constant ˜ G/G . Indeed, introducingcomplex coordinates u = r − iz and v = r + iz , the equations of motion (3)-(5) can be rewritten asthe following system of coupled differential equations: β ,uv = 0 (13) A ,uv − (cid:18) A ,u β ,v β + A ,v β ,u β (cid:19) + 2 A ,u ω ,v + 2 A ,v ω ,u = 0 (14) ω ,uv + 12 (cid:18) ω ,u β ,v β + ω ,v β ,u β (cid:19) + e ω β A ,u A ,v = 0 (15) χ ,uv + ω ,u ω ,v + 3 γ ,u γ ,v + e ω β A ,u A ,v = 0 (16) γ ,uv + 12 (cid:18) γ ,u β ,v β + γ ,v β ,u β (cid:19) = 0 (17)2 β ,u β χ ,u − β ,uu β = 2 ω ,u + 6 γ ,u − e ω β A ,u ( u ↔ v ) (18)Note that equation (16) can be deduced from the others.The important result here is the structure of the field equations in that they are quasi-identicalto those governing a pure Einstein geometry in standard Lewis-Papapetrou form [5, 33]. The onlydifference emanates from the presence of the field γ encoding the additional scalar in the action.6irst, note that β is again a harmonic function and can be set to β = r without any loss ofgenerality. Then we have a coupled system of equations for the pair ( ω, A ), which can be treated innumerous ways (see for example [1] and the multitude of references within). Once determined, thesefields can be substituted in the non-linear equations (18) governing χ . Here, the only differencewith the vacuum case is the extra linear equation (17) that determines γ . This equation is also ofthe Weyl form (15) (in absence of the A ,u A ,v term), and the resulting field γ sources the equations(18) for β . In other words, the field equations originating from the action (1) consist – for the givensymmetries – are closely related to corresponding Lewis-Papapetrou equations of ordinary GR, asdescribed in [1]. Following Astorino [22], one could have chosen to work in the minimal frame andconformally transform for each solution. Here instead, we give the Lewis-Papapetrou form directlyin the frame of interest, allowing to construct directly novel solutions using standard GR methodssuch as those of Ernst or Papapetrou. As we shall see, this has the advantage of preserving someof the GR intuition concerning the relevant sources to use. For a full list of solution generatingmethods that can be applied to the problem one can consult [1]. Our starting point will be to treatthe simpler seed solutions first in their Weyl form, and then to seek the stationary solutions.Towards this end we now look into the simpler sub-case of a static and axisymmetric spacetime,which is equivalent to putting A = 0 in (12) and in the field equations (13)-(18). We choose β = r by virtue of (13) and it follows that (15) and (17) are two Laplace equations written in cylindricalcoordinates, ω ,rr + 1 r ω ,r + ω ,zz = 0 and γ ,rr + 1 r γ ,r + γ ,zz = 0 . (19)In GR, only the Weyl potential ω is present. Here, we have two Weyl potentials, similarly to whathappens in higher dimensional spacetimes [10]. Furthermore, these equations are linear, and theWeyl potentials can therefore be superposed. Once γ and ω are determined, they can be substitutedinto (18) to determine the function χ which encodes the non-linearity of Einstein’s equations. TheWeyl problem is an integrable problem in GR and we thus have shown that this property is alsotrue for the theory (1). There are several ways to proceed in order to find solutions [1]; here,we simply outline Weyl’s original method, that makes a nice parallel to Newtonian gravity in thespatial dimensions.We can interpret the function ω as a solution of Poisson’s equation, ∆ ω = − πρ , with a sourceterm ρ which represents some Newtonian source on the axis of symmetry of the cylindrical space.It turns out that the presence of an event horizon corresponds to taking a localized linear massdistribution – a rod – located on the axis r = 0. The gravitational potential takes the standardform ω ( ~r ) = − Z ρ ( ~x ) | ~r − ~x | d x. (20)In particular, if we consider a uniform rod on the r = 0 axis, stretched between z = z and z = z in cylindrical coordinates, and with density σ ω per length unit, we find ω ( r, z ) = − σ ω Z z z d˜ z p r + ( z − ˜ z ) = σ ω ln p r + ( z − z ) − ( z − z ) p r + ( z − z ) − ( z − z ) ! (21)7his is the standard procedure by which a massive rod is shown to correspond to the Schwarzschildsolution in GR. Note that the field γ , representing the scalar field, solves a similar Poisson equation,with its source again localized on the axis of symmetry. As mentioned in the introduction, a notable solution to the equations of motion (3)-(5) is theBBMB solution [17, 16]. It is a static and spherically symmetric configuration, given by the metricd s = − (1 − m/ρ ) d t + d ρ (1 − m/ρ ) + ρ (cid:0) d θ + sin θ d ϕ (cid:1) (22)and the following scalar field, φ = r πG mρ − m . (23)This solution is in many ways special. It is a black hole dressed with a non trivial massless scalarfield and secondary hair (note the presence of only one integration constant m ). It has a welldefined geometry, identical to that of the extremal Reissner-Nordstr¨om black hole: the effect of theconformally coupled scalar field on the geometry is indeed to shape it in a similar way to an electriccharge at extremality. The scalar field however, unlike the electric field, explodes at the location ofthe event horizon (see [34] for a discussion on the properties of this solution). Scalar-tensor theoriesadmitting other than ordinary GR solutions are rather rare. This is what makes this black hole sospecial, and motivates the detailed study of the conformally coupled theory in question, defined bythe action (1). The rod structure of the BBMB solution bears an interesting analogy with ordinaryGR. This is manifest if we recast its metric (22) in the form (12). For definiteness, let us restrict tothe ρ > m region, where ˜ G >
0. To this end, we first trade the radial coordinate ρ for an auxiliarycoordinate u = ln ( ρ/m − t, r, z, ϕ ) throught thecoordinate transformation z = 2 m cosh u cos θ and r = 2 m sinh u sin θ. (24)As a result, the BBMB metric becomes of the form (12) with e ω = e − γ = p r + ( z + 2 m ) − ( z + 2 m ) p r + ( z − m ) − ( z − m ) (25)and e χ = p r + ( z − m ) p r + ( z + 2 m ) + ( z − m )( z + 2 m ) + r p r + ( z − m ) p r + ( z + 2 m ) . (26)Comparing with (21), we see that ω and γ are solutions of the Laplace equation with a sourcelocated at r = 0 between z = − m and z = 2 m with a density σ ω = 1 / σ γ = − / ρ = 2 m in the original coordinates( t, ρ, θ, ϕ ) used in (22) and (23). On the other hand, when m < ρ < m , where φ > πG , we8et u = − ln( ρ/m − ρ coordinate (22). Apart from this fact we find the same rod structure as before summed up infigure 1. Concerning the function χ , we verify in both cases that lim r → χ ( r, z ) = 0 for any z < − m and z > m according to (26), that there is no conical singularity on the axis.We can point out that all spherically symmetric and static solutions of (1), given in [35] andparametrized by three constants ( m, ǫ, δ ), can be generated using this rod structure with the fol-lowing densities σ ω = cos ǫ and σ γ = − sin ǫ √ and the same lengths 4 m (see [36] for the details). Thethird parameter δ , is a constant that we can always add to the Weyl potential γ in addition to thelogarithmic term associated to the rod while keeping the same asymptotic properties. In completeanalogy to GR, the C-metric version of the BBMB solution [20, 24] can be obtained by adding asemi-infinite ω -rod of density 1 / m mσ ω = 1 / ω -2 m mσ γ = − / γ Figure 1: Rod structure of the BBMB solution
So much for the Weyl formalism. When treating the full stationary and axisymmetric problem (13)-(18) one can make use of a very elegant method developed by Ernst, [6] which turns the coupledPDE’s (14)-(15) for the fields ( ω, A ) into a single PDE of a complex variable. Here, the form of(14)-(15) is identical to the one in pure GR so we can just iterate the method, by introducing anauxiliary field Ω( r, z ) such that( A ,r , A ,z ) = (cid:0) βe − ω Ω ,z , − βe − ω Ω ,r (cid:1) . (27)Then the complex function E = e ω + i Ω, called the
Ernst potential , verifies the
Ernst equation β ~ ∇ . (cid:16) β ~ ∇E (cid:17) = ~ ∇E .~ ∇E Re ( E ) , (28)where ~ ∇ = ~u r ∂ r + ~u z ∂ z is the gradient in the euclidean r - z plane. The real and imaginary parts ofthis equation correspond to (15) and (14) respectively. Redefining the Ernst potential with a newcomplex function ξ so that E = ξ − ξ + 1 , (29)9quation (28) takes the form [6]( ξξ ∗ −
1) 1 β ~ ∇ . (cid:16) β ~ ∇ ξ (cid:17) = 2 ξ ∗ ~ ∇ ξ.~ ∇ ξ. (30)This equation enjoys a U (1) invariance: if ξ is a solution, e iλ ξ is also a solution, for any real λ .According to (13), we adopt henceforth the choice β = r . It is then useful to introduce prolatespheroidal coordinates ( x, y ) defined by r = σ p x − p − y and z = σxy, (31)where x and y are dimensionless coordinates, and σ is a constant with the dimension of length. Inparticular, we get d r + d z = σ ( x − y ) (cid:18) d x x − y − y (cid:19) . (32)Knowing ξ , we can infer the real and imaginary parts of the Ernst potentials (cid:0) e ω , Ω (cid:1) , and determinethe function A using A ,x = σ (1 − y ) e − ω Ω ,y and A ,y = σ (1 − x ) e − ω Ω ,x (33)and finally deduce the function χ with χ ,x = 1 − y x − y ) e − ω (cid:2) x ( x − B xx + x ( y − B yy + 2 y (1 − x ) B xy (cid:3) , (34) χ ,y = x − x − y ) e − ω (cid:2) y ( x − B xx + y ( y − B yy + 2 x (1 − y ) B xy (cid:3) , (35)where B ij = (cid:0) e ω (cid:1) ,i (cid:0) e ω (cid:1) ,j + Ω ,i Ω ,j + 3 e ω − γ ) (cid:0) e γ (cid:1) ,i (cid:0) e γ (cid:1) ,j . (36)Summarizing, given the potentials ( γ, ξ ), we have shown that following step by step Ernst’s originalmethod, we can determine completely a stationary and axisymmetric solution of (1). A treatmentof this problem in vacuum, higher dimensional GR can be found for example in [37]. The Ernstequation in presence of a cosmological constant was studied in [12, 13], and used to construct aMelvin solution with a cosmological constant [13]. As a concrete example of the above method, we now construct the Taub-NUT extension of theBBMB solution. A similar construction is given for GR in [38] where one starts from Schwarzschildas the Ernst potential for the seed solution (see also [39] for a construction based on the Papapetroumethod). Here our seed solution is quite naturally BBMB, for which σ = 2 m . According to (24)and (31), we transform to prolate spheroidal coordinates ( x, y ) for the BBMB solution by setting x = cosh u and y = cos θ . In particular, the seed Ernst potential, which is real since the solution isstatic, is given by e ω = (cid:18) x − x + 1 (cid:19) / = x ± √ x − − x ± √ x − , (37)10y virtue of (25). As a result, ξ defined in (29) is ξ = x ± √ x −
1. Then, according to (30), ξ = e iλ ξ is also a solution of (30) and the resulting complex Ernst potential reads, E = √ x − x + cos λ + i sin λx + cos λ . (38)In turn, equations (33) give, A = 2 my sin λ. (39)As for the field γ representing the scalar field, we have a choice. One can keep the same Weylpotential as for BBMB (see figure 1) or, again, take something more general. It turns out thatkeeping the same scalar field as for the BBMB solution, that is e γ = (cid:18) x + 1 x − (cid:19) / (40)is the relevant choice. This evades extra singularities since (34) and (35) give e χ = x − x − y , (41)and this is the same field as the one for the BBMB solution since it is independent of the phase λ .In ( t, ρ, θ, ϕ ) coordinates, we obtain,d s = − ( ρ − m ) f ( ρ ) (d t + 2 m sin λ cos θ d ϕ ) + f ( ρ )( ρ − m ) d ρ + f ( ρ ) (cid:0) d θ + sin θ d ϕ (cid:1) (42)where f ( ρ ) = ρ + 2 m (cos λ − ρ + 2 m (1 − cos λ ) . (43)Finally, shifting ρ → ̺ = ρ + m (cos λ − n = m sin λ , andredefining the mass parameter through µ = √ m − n , we recast the solution into a more familiarform, d s = − ( ̺ − µ ) ̺ + n (d t + 2 n cos θ d ϕ ) + ̺ + n ( ̺ − µ ) d ̺ + (cid:0) ̺ + n (cid:1) (cid:0) d θ + sin θ d ϕ (cid:1) (44) φ = r πG p µ + n ̺ − µ . (45)Quite naturally, the BBMB solution is recovered when the NUT parameter vanishes. Fur-thermore, as for BBMB, the metric geometry of (44) is identical to that of the extremal chargedTaub-NUT solution first found in [40]. The Taub region of the metric, which is the non-stationary,timelike region situated in between the inner and outer horizons of the bald metric, is now absentgiven the extremal nature of the horizon. The solution is only locally asymptotically flat as asymp-totic infinity is deformed by the NUT charge. In fact, although the Riemann curvature decays atasymptotic infinity at the same rate as in the Schwarzschild solution or BBMB, the Taub-NUTversion is not a solution of a 1 /̺ Newtonian deformation about flat spacetime [41]. The scalar11eld explodes at the horizon location ̺ = µ as for the BMBB metric. Despite this singularity themetric (44) is regular for all ̺ . In fact, ̺ = 0 is not even a singular point for the metric! The mainpathology of Taub-NUT solutions is the presence of the Misner string either at θ = 0 or θ = π rendering the 1-form d ϕ ill-defined at either of these points. In order to get rid of the North polesingularity at θ = 0, we can define a new time coordinate t ′ = t + 2 nϕ . For the south pole at θ = π we take t ′′ = t − nϕ . Given the periodicity of ϕ by 2 π we now have to identify t ′ and t ′′ modulo8 πn . Hence, the only way out is to impose periodicity of the time coordinate by t → t + 8 πn . Theresulting one form in d ϕ is well defined but the price to pay are CTCs in the Lorentzian metric (45). We are therefore led to consider as usual the regularity of the euclidean metric. For that, set t → iτ and n → in on (44)-(45)d s = ( ̺ − µ ) ̺ − n (d τ + 2 n cos θ d ϕ ) + ̺ − n ( ̺ − µ ) d ̺ + (cid:0) ̺ − n (cid:1) (cid:0) d θ + sin θ d ϕ (cid:1) (46) φ = r πG p µ − n ̺ − µ . (47)As before, to have no Misner string, the periodicity of τ is fixed at the value 8 πn . But now τ is justparametrising the circle which is fibered over the 2-sphere of coordinates θ and φ . Following theclassification of gravitational instantons given in [42] we now have a curvature singularity at ρ = n and hence no nut solutions can be defined unless µ = n . This is the case where the scalar fieldis set to zero and in fact we go back to the Ricci flat nut instanton. When µ > n the singularityat ρ = n is never reached since space closes off smoothly at ̺ = µ due to the extremal nature ofthe Lorentzian horizon. In this case the U (1) isometry generated by the Killing vector ∂ τ has atwo-dimensional fixed point set, a bolt, as does Euclidean Schwarzschild, given by g ττ = 0. Thefixed point, ̺ = µ , set are the 2-spheres. Finally, due to the extremal nature of the horizon there isno conical singularity (once the periodicity of τ is imposed). We therefore have a regular metric aslong as µ ≥ n . The solution is continuously related to the Ricci flat nut solution at µ = n . In thepresence of NUT charge therefore the transition in between the BBMB and Schwarzschild familyis smooth. The scalar explodes at ρ = µ but we can remedy this including a cosmological constant.This is what we will consider next. We now turn on the cosmological constant Λ and the self-interaction potential in (1). We will alsoadd in turn, the electromagnetic (EM) interaction, S em [ g ab , A a ] = − πG Z M √− gF ab F ab d x. (48) Note that hypersufaces orthogonal to t = const are space like only for sufficiently small θ . In other words thecausality inconsistencies of Lorentzian Taub NUT are not only a result of imposing time periodicity [41], they arethere in the metric element from the start. S ax = − X i =1 Z M (cid:18) − πG φ (cid:19) − H ( i ) ∧ ⋆ H ( i ) . (49)We add here two exact three-forms H ( i ) = H ( i ) abc d x a ∧ d x b ∧ d x c , ( i = 1 , B ( i ) such that H ( i ) = d B ( i ) . Although the EM interaction is minimallycoupled in four dimensions for the conformal frame the axions are non-minimally coupled to thescalar field φ . It is perhaps more illuminating to write this action with a minimal coupling for thescalar field. Performing the following conformal transformation,˜ g ab = (cid:18) − πG φ (cid:19) g ab (50)and a redefinition for the scalar field Ψ = q πG artanh (cid:18)q πG φ (cid:19) , the full action (49) becomes S = Z M p − ˜ g " ˜ R πG − X i =1 H ( i ) abc H ( i ) abc − F ab F ab πG − ∂ a Ψ ∂ a Ψ − U (Ψ) d x (51)where the scalar potential is given by U (Ψ) = Λ8 πG " cosh r πG ! + 9 α π Λ G sinh r πG ! . (52)Let us turn now to the equations of motion in the conformally coupled frame. Variation of thefull action (1), (48) and (49) with respect to the metric gives (cid:18) − πG φ (cid:19) G ab + Λ g ab = 8 πG (cid:18) − πG φ (cid:19) − X i =1 (cid:18) H ( i ) acd H ( i ) cdb − g ab H ( i ) cde H ( i ) cde (cid:19) + 2 (cid:18) F ac F cb − g ab F cd F cd (cid:19) + 8 πG (cid:18) ∂ a φ∂ b φ − g ab ∂ c φ∂ c φ (cid:19) + 4 πG g ab (cid:3) − ∇ a ∇ b ) φ − πGαg ab φ . (53)Then varying with respect to φ , A and B ( i ) , we obtain (cid:3) φ = R φ + 4 πG φ (cid:18) − πG φ (cid:19) − X i =1 H ( i ) abc H ( i ) abc + 4 αφ , (54) ∇ a F ab = 0 and ∇ a "(cid:18) − πG φ (cid:19) − H ( i ) abc = 0 (55)respectively. An important property of the field equations stemming from the conformal couplingis the following: taking the trace of the metric equations (53) and replacing it in the equation ofmotion for the scalar field (54) gives (cid:3) φ = 23 Λ φ + 4 αφ . (56)13his is precisely the equation of motion (7) which emanates from the theory (49) in the absenceof matter fields. Hence we can try to find a solution related to a hairy solution of the non-axionictheory. Finally, given the equation of motion for the axionic fields, it will be useful to introducepseudo-scalar fields χ ( i ) which are dual to the axionic fields in the following sense H ( i ) = (cid:18) − πG φ (cid:19) ⋆ d χ ( i ) . (57)We expect that the presence of the cosmological constant will permit to hide the divergenceof the scalar field [19]. Let us first switch off the axionic fields for the solutions are completelydifferent in nature for the latter case. In this section we consider (1) including a cosmological constant and an EM-field (48). Giventhe Taub-NUT-BBMB construction (44)-(45), it is not difficult to guess the solution including theadditional terms. The equations of motion (3)-(5) admit the following solution,d s = − V ( ̺ ) (d t + B ) + d ̺ V ( ̺ ) + ( ̺ + n )d σ k ) (58)with V ( ̺ ) = − Λ3 ( ̺ + n ) + (cid:18) k − n Λ (cid:19) ( ̺ − µ ) ̺ + n (59)and B = n cos θ d ϕ when d σ k =1) = d θ + sin θ d ϕ nθ d ϕ when d σ k =0) = d θ + θ d ϕ n cosh θ d ϕ when d σ k = − = d θ + sinh θ d ϕ (60)The scalar field and the electromagnetic potential read φ = r − Λ6 α p µ + n ̺ − µ (61)and A = q̺̺ + n (d t + B ) (62)respectively. Thus α Λ < n, µ, q )satisfy the following constraint q = (cid:0) µ + n (cid:1) (cid:18) k − n Λ (cid:19) (cid:18) π Λ G α (cid:19) . (63)For a vanishing NUT parameter, and for k = 1 , − α go to zero so that − Λ6 α tends to the value πG , we obtain theTaub-NUT-BBMB solution (44)-(45) for k = 1. 14n order for the metric to be regular we need to evade the Misner strings whenever theseare present. For spherical sections, k = 1, these are unavoidable given the presence of non zeroNUT charge. As a result CTCs will appear in the periodic orbits of time just like for usual NUTspacetime. One can consider a Euclidean signature metric as in the previous section. When k = 0however, the t -fibration on the plane is trivial and thus no Misner strings are present [43]. ForΛ < ̺ and θ the Killing vector ∂ϕ becomes timelike. Given that ϕ is periodic this means that we have closed timelike curves for ϕ = constant and large enough ̺ and θ . The situation is similar to the one encountered in G¨odel spacetime,[26]. For Λ > k = 0), the nature of the solution completely changes, for then, ̺ is actuallyeverywhere timelike and the solution is of cosmological nature, homogeneous and non-isotropic.The ̺ coordinate sweeps the whole real line and the solution is completely free of coordinate orcurvature singularities. The scalar field explodes at ̺ = µ . At large ̺ spacetime locally asymptotesde-Sitter. We are thus in the Taub region of Taub-NUT, although this region is now covering thewhole of spacetime. It is the hyperbolic case k = − T = t + 2 nϕ that g ϕϕ > ρ and θ as long as 4 n l <
1. Therefore thereare no CTCs whatsoever. This, to our knowledge, is a novel property for Taub NUT metrics andis obtained due to the presence of the conformally coupled scalar field. This spacetime geometryis very peculiar and interesting for the radial coordinate ̺ can be “extended” to range from −∞ to + ∞ starting and ending at a hyperbolic slicing of AdS. In this way the metric has up to fourKilling horizons, but no spacetime singularity whatsoever! I As a result we can take µ > V ( ̺ ) and for ̺ ∈ R , ̺ + = ¯ l r − n ¯ l − µ ¯ l ! (64) ̺ − = ¯ l − r − n ¯ l − µ ¯ l ! (65) ̺ ++ = ¯ l − r − n ¯ l + 4 µ ¯ l ! (66) ̺ −− = ¯ l − − r − n ¯ l + 4 µ ¯ l ! (67)according to the values of µ , n et ¯ l = √ l − n . The metric has no other singular points whatsoever.The case of n = 0 is a black hole and has been studied in [19]. Note that the solution presentedhere is, for n = 0 regularizing the Λ − BBMB solution which has a spacetime curvature singularityat ̺ = 0. We have thus a rotating solution which at the price of changing the asymptotic structureregularizes its static counterpart metric. As often the Taub NUT family of metrics presents acounterexample to our usual intuitive understanding of black hole solutions. In fact, we use the termblack hole since there is a singularity in the scalar field at ̺ = µ . We have that V ( ̺ ) = µ + n l > ̺ −− ≤ ≤ ̺ ++ ≤ µ ≤ ̺ − ≤ ̺ ext ≤ ̺ + which is verified for 0 ≤ µ ≤ ¯ l − n ¯ l . We havenoted as ̺ ext the case ̺ − = ̺ + saturating the above bound. It is maybe more relevant from thepoint of view of singularity theorems to study this metric in the Einstein frame . If we make aconformal transformation to the Einstein frame,(50), there is a genuine curvature singularity wherethe conformal transformation is singular , ̺ = µ + η p µ + n , where we note for convenience η = − π Λ G α ≥
1. In this case in order for ̺ − < ̺ < ̺ + we must have, − η − (1 + η ) q η − n ¯ l (1 − η ) (1 − η ) < µ ¯ l < − η + (1 + η ) q η − n ¯ l (1 − η ) (1 − η ) (68)which in the Einstein frame guarantees a black hole. For hyperbolic sections, but Λ > >
0. The maximal NUT charge isattained when n Λ = k . Given the constraint (63) the electromagnetic field vanishes but not thescalar field which is again singular at ̺ = µ . The spacetime geometry is that of de-Sitter and antide-Sitter for the relevant sign of Λ. We now switch on the axionic fields (49). Let us start by presenting the bald solution, that is witha vanishing scalar field, φ = 0. For a negative cosmological constant, Λ = − /l , the theory (49)admits the following solution,d s = − V ( ̺ ) [d t + n ( x d y − y d x )] + d ̺ V ( ̺ ) + ( ̺ + n ) (cid:0) d x + d y (cid:1) (69)with lapse function V ( ̺ ) = 1 ̺ + n (cid:20) − m̺ + 1 l (cid:0) ̺ + 6 n ̺ − n (cid:1) + p (cid:0) n − ̺ (cid:1) + q (cid:21) (70)and the following electromagnetic potential A = q̺̺ + n [d t + n ( x d y − y d x )] . (71)For p = 1, we notice that the lapse function is identical to the Taub-NUT AdS-Reissner-Nordstr¨omsolution with a hyperbolic base manifold [43]. Here, however, we have a flat horizon induced bythe presence of two homogeneously distributed 3-forms H ( i ) . These axionic fields are generated bythe following scalar fields, defined in (57), χ (1) = p √ πG x and χ (2) = p √ πG y. (72) Given that our solution does not couple to matter the choice of physical frame can be in either of frames. For a discussion see for example [21] and note also the change of sign in the the effective Newton’s constant(2)
16o construct explicitly the corresponding axionic fields, we use (57) to obtain, H (1) = p √ πG ( − d t + ny d x ) ∧ d ̺ ∧ d y, (73)and H (2) = p √ πG (d t + nx d y ) ∧ d ̺ ∧ d x. (74)First, for p = 0, we get the Taub-NUT Reissner-Nordstr¨om solution with a flat base manifoldbriefly discussed in the euclidean time in [43] without the electromagnetic interaction. Secondly,for n = 0, this solution is the axionic planar charged black hole of [44]. Turning on the parameter n makes the metric non-static, and the resulting black hole rotates on the plane of its horizon.Indeed the fibration is trivial and consequently there is no Misner string singularities. There is noperiod associated to the coordinate time t . However, it is easy to note that using polar coordinates( θ, ϕ ) defined by ( x, y ) = ( θ cos ϕ, θ sin ϕ ) that the norm g ϕϕ changes sign for large enough ̺ and θ . This means that we will always have closed timelike curves for in the 2 π periodic φ orbits. Infact these CTCs can appear arbitrarily close to the outer horizon ̺ = ̺ h for large enough radialhorizon coordinate θ . In fact for any constant ̺ > ̺ h once θ > θ = ρ + n V ( ρ ) n we have CTCs inconstant ( θ , ̺ ) orbits of ϕ . Hence these metrics are pathological.The hairy version is not far better. For a negative cosmological constant, Λ = − /l , we have,d s = − V ( ̺ ) [d t + n ( x d y − y d x )] + d ̺ V ( ̺ ) + ( ̺ + n ) (cid:0) d x + d y (cid:1) (75)with V ( ̺ ) = ̺ + n l + (cid:18) n l − p (cid:19) ( ̺ − µ ) ̺ + n (76)and the following electromagnetic potential and scalar field A = q̺̺ + n [d t + n ( x d y − y d x )] and φ = 1 √ αl p µ + n ̺ − µ (77)respectively. Using polar coordinates ( θ, ϕ ) defined by ( x, y ) = ( θ cos ϕ, θ sin ϕ ), we notice that for p = 1 the metric is that of the charged Taub-NUT-BBMB solution (58)-(59) with a hyperbolic basemanifold corresponding to k = − H ( i ) as usual. These axionic fields are generated by thefollowing scalar fields, defined in (57), χ (1) = p √ πG x χ (2) = p √ πG y, (78)and as derived in the previous subsection, it is straightforward to show that the axionic fields takethe following form H (1) = p √ πG (cid:18) − πG φ (cid:19) ( − d t + ny d x ) ∧ d ̺ ∧ d y, (79) H (2) = p √ πG (cid:18) − πG φ (cid:19) (d t + nx d y ) ∧ d ̺ ∧ d x. (80)17inally, the integration constants ( n, µ, q, p ) must satisfy the following constraint q = (cid:0) µ + n (cid:1) (cid:18) n l − p (cid:19) (cid:18) − πG αl (cid:19) . (81)First, for p = 0, we get the charged Taub-NUT-BBMB solution (58)-(59) with a flat basemanifold corresponding to k = 0 in (60) analysed in the previous section. Secondly, for n = 0, thissolution is the axionic planar charged black hole with a conformal hair of [21]. Consequently, thesolution presented here is the generalization of that solution with the introduction of a rotatingparameter n . Again this solution has the same CTC pathologies as its bald counterpart. In this article, we have considered scalar-tensor theories of a particular nature: the scalar field isconformally coupled to the four dimensional Einstein-Hilbert action and although the full action isnot conformally invariant the theory has nice integrability properties due to the partial conformalinvariance. We have shown that the Lewis-Papapetrou metrics and the Weyl problem are integrablein the same way as for vacuum General Relativity [1]. The field equations are augmented by anan additional Weyl potential associated to the conformally coupled scalar field. As such, solutiongenerating methods can be employed in more or less the same manner as for GR, and solutions ofthe action (1) can thus be obtained with relative ease. Using the Ernst method [6], we then derivedthe generic family of Taub-NUT solutions for this theory. This procedure was first developed byReina and Treves to obtain the relevant Taub-NUT metrics in General Relativity [38]. We thenderived topological versions by introducing a cosmological constant. The hyperbolic version turnsout to be free of closed timelike curves and represents a rotating black hole in the Einstein frame.The solution in the conformal frame is totally free of curvature singularities and the scalar fieldis singular only in a region hidden behind an event horizon. We then found the bald and hairyaxionic solutions, but those are plagued by CTCs which can be even arbitrary close to the horizon.An interesting and yet unresolved problem would be to determine the Kerr family of the action (1).Unfortunately here, a direct application of the Ernst method to conformally coupled scalars fails,primarily because the Weyl potential for the BBMB solution is different from that of Schwarzschildblack hole, and as a result spheroidal coordinates do not seem to be quite adapted to the problem.This remains an open problem.Given the hindsight from our research in Taub-NUT metrics, the case of hyperbolic slicingstands out as the most interesting. Indeed it is in this case that the negatively curved horizonpermits to do away with the CTC pathology of Taub-NUT. It is interesting therefore to dwell a biton the simplest of cases, namely that of Einstein-Hilbert with a negative cosmological constant. In18his theory, the relevant hyperbolic Taub-NUT-AdS solution can be conveniently written as [43],d s = − f ( ρ ) (cid:18) d t + 4 n sinh (cid:18) θ (cid:19) d ϕ (cid:19) + d ρ f ( ρ ) + (cid:0) ρ + n (cid:1) (cid:0) d θ + sinh θ d ϕ (cid:1) , (82) f ( ρ ) = ρ + n l + ( l − n )( n − ρ ) − ml ρ ( ρ + n ) l . (83)It is worth recalling that setting n = 0, one recovers the hyperbolic AdS black hole [45, 46]. Thecoordinates ( θ, φ ) parametrize a hyperboloid, and upon appropriate quotients, Riemann surfaces ofany genus higher than 1. For m ≥ − l √ , the spacetime contains a black hole, with an event horizonin ρ + , the largest root of the function f ( ρ ), and a curvature singularity for ρ = 0. This singularity istimelike when m <
0, because of the presence of an inner Cauchy horizon, and spacelike otherwise.Saturating the lower bound for the mass parameter one obtains the extremal ground state, withvanishing temperature. On the other hand, for m = 0 the spacetime is just AdS in a hyperbolicslicing; the horizon is still present and has the interpretation of an acceleration horizon [46]. Uponcompactification of the hyperboloid, it becomes a genuine event horizon though.What happens when we switch the parameter n on? First, it regularizes the metric, in thesense that all its curvature invariants now stay bounded, and the radial coordinate ρ ranges overthe whole real axis. But then again, it makes the spacetime prone to CTCs. The Euclidean sectionhas been studied in [43], where it was shown that it has a bolt where f ( ρ ) has its largest root, andno NUT. The U (1) fibration being trivial, there are no Misner strings [43]. Likewise, the Lorentzianspacetime – on which we focus our interest here – has no Misner string singularity, but, like for itsspherical cousin, there are ‘large’ naked CTCs when | n | > l/ | n | ≤ l/ m [47]. Therefore, for this range of the parameter n , causality is preserved in the exterior region ofthe spacetime, and the metric describes a black hole, with an event horizon and an inner Cauchyhorizon. The latter encloses a static region of spacetime containing a causality-violating core withCTCs that, contrary to the ‘large’ ones found when n > l/
2, are contractible and harmless, sincethey live beyond the Cauchy horizon.More precisely, let us consider an asymptotic observer in the ρ > , and let us define m ± = ± l √ (cid:12)(cid:12)(cid:12)(cid:12) − n l (cid:12)(cid:12)(cid:12)(cid:12) r − n l . (84)Then, we see that when 0 < | n | ≤ l/ (2 √
3) the solution is a black hole with two horizons and aninnermost CTC core for m ≥ m − . When the equality sign holds, the two largest roots of f ( ρ )merge and the horizon is extremal. On the other hand, if the inequality is violated, there are nakedCTCs outside the event horizon, for large enough θ . They are confined close to the event horizon,and their radial position ρ cannot be arbitrarily large. For l/ (2 √ ≤ | n | ≤ l/ m ≥ m + , with two horizons hiding the CTC core, an extremal blackhole if m = m + , and naked CTCs if m < m − , with the only difference being that in the range This covers all cases because the Taub-NUT-AdS metric is left invariant under the ρ
7→ − ρ , m
7→ − m operation. − < m < m + the function f ( ρ ) as no roots at all . All these horizons are located at the zeroesof f ( ρ ) and are Killing horizons.So, what is the interpretation of the parameter n ? It is not a NUT parameter, since thespacetime carries no NUT charge. Its effect is to make the metric non-stationary, with g tφ ∝ n . Assuch, observers experience frame dragging, and there is rotation in the spacetime. This is confirmedby the holographic stress tensor [48], that can easily be extracted from the metric by changing toFefferman-Graham coordinates (see e.g. [49]). The result, not surprisingly, assumes the perfectfluid form, T ab = m πGl (3 u a u b + h ab ) , (85)with energy density ε = m/ (4 πGl ) and pressure P = m/ (4 πGl ), with the expected equation ofstate for AdS black holes [50]. Here u = ∂ t is a unit timelike vector, and the boundary metric h ab itself is non static [51],d s bdy = − d t + 4 n sinh (cid:18) θ (cid:19) d φ ! + l (cid:0) d θ + sinh θ d φ (cid:1) , (86)but contains no CTC if | n | ≤ l/
2. As a result, in addition to the energy density ε , there is a non-vanishing angular momentum density T tφ ∝ mn [47]. In the coordinates we are using, the angularvelocity of the event horizon vanishes, Ω h = 0. However, this just reflects the coordinate systemwe are using; in asymptotically AdS spacetimes the conjugate variable to the angular momentumthat enters the thermodynamics is the angular velocity of the horizon relative to the boundaryΩ = Ω h − Ω ∞ [52]. The boundary metric (86) is indeed rotating, and one findsΩ = − nl + 4 n + ( l − n ) cosh θ . (87)We are therefore in presence of a rotating black hyperboloid membrane, as first suggested in [51].Since the event horizon has infinite extension, it is not possible to integrate the mass and angularmomentum densities to obtain finite charges. Moreover, due to the rotation, it is not possibleto compactify the hyperboloid to a smooth Riemann surface to cure the divergence. Also, theblack membrane is not rotating uniformly: the rotation is concentrated in the central region of thebrane, θ ≈
0, with an exponentially vanishing tail. This property explains why these spacetimescan avoid CTCs outside the horizon, and is as well reflected in the mass and angular momentumthat are also non-uniformly distributed. Hence, even locally, it is difficult to formulate the first lawof thermodynamics, and to check that the entropy follows the Bekenstein-Hawking formula. It ishowever easy to verify that in the limit of large black holes, ρ + /l ≫
1, one recovers the expectedhydrodynamic behavior [53], and with it a local thermodynamic interpretation. Indeed, the Euler Note that from the perspective of an observer in the ρ < m > m + or m < m − : in that region there will be a black hole when there are naked CTCs in the ρ > ρ > m − < m < m + , there are no horizon at all for l/ (2 √ < | n | ≤ l/
2, whereas the CTC core is sandwiched between a ρ < ρ > < | n | < l/ (2 √ ε + P = T s is recovered at leading order in an l/ρ + expansions, with deviations appearingat the same order as for a Kerr-Newman-AdS black hole (see e.g. [54]), if the entropy density istaken to be a h / G , with a h the area density of the horizon.Further evidence that this is a rotating black membrane comes from a closer look at the Kerr-AdS black hole. Surprisingly, this black hole has an ultraspinning limit, typically associated tohigher dimensional black holes. Indeed, when taking the rotation parameter a → l , and simultane-ously zooming into the pole in an appropriate way, one reaches a finite limiting metric that is simply(82) with parameter n = l/ n = l/
2) the hyperboloid rotates uniformly Ω = 1 / l , and the boundaryalways rotates at subluminal speed, Ω < /l . Finally, notice that the n = l/ m = 0 metric issimply AdS in rotating coordinates. Switching on the mass parameter m , the metric develops anhorizon and becomes a rotating black hole, similarly to what happens with Kerr(-AdS) black holes.On the other hand, keeping m = 0 and increasing n the geometry deviates from the AdS spacetimecontrary to what happens with spherical Kerr(-AdS) metric. This is nevertheless explained by theobservation that the m = 0, n = 0 metric contains an accelerated horizon.Summarizing, the parameter n is a rotation parameter , analogous to the parameter a appearingin the Kerr metric. At n = 0, we have a static black hole spacetime with a curvature singularityat ρ = 0. When 0 < n ≤ l/
2, the black membrane is put in rotation, and the central spacetimesingularity disappears and is replaced by a central CTC core. Finally, when n > l/ under-rotating
BMPV black holes, while the metrics with l/ (2 √ ≤| n | ≤ l/ m − < m < m + resemble the over-rotating BMPV black holes [55]. It wouldbe interesting to check if these over-rotating menbrane solution also exhibits the ‘repulson’-likebehaviour characteristic of over-rotating black holes [55, 56]. A similar situation is in fact presentfor the spinning BTZ black hole, where CTCs appear for negative radial coordinate [31]. Theproposed strategy is to cut off the spacetime when one meets the velocity of light surface, at g ϕϕ = 0 (see however [57]). Anyway, the inner Cauchy horizon is likely unstable to perturbations,that would replace it with a genuine singularity, cutting off the CTC core. Finally, all theseproperties survive when a Maxwell field is turned on.In conclusion, we believe that it would be interesting to investigate further the properties ofthese black holes, clarify their role in the AdS/CFT correspondence [51, 58], and understand theirrelation to the other family of known rotating hyperbolic black membranes found in [59], and theirTaub-NUT generalization [60]. Note added:
During the final writing stage of this project, ref. [61] appeared in the arXiv ,where the authors also report the Taub-NUT-BBMB solution (44)-(45).21 cknowledgements
It is a pleasure to thank Joan Camps, Roberto Emparan, Dietmar Klemm, Harvey Reall, SimonRoss and Robin Zegers for useful discussions. YB would like to thank the SISSA for hospital-ity during the later stages of this work. This work was partially supported by the ANR grantSTR-COSMO, ANR-09-BLAN-0157. MMC acknowledges support from a grant of the John Tem-pleton Foundation. The opinions expressed in this publication are those of the authors and do notnecessarily reflect the views of the John Templeton Foundation.
A Note on proof of Frobenius condition
Consider stationary and axisymmetric metrics of (1) for Λ = 0 and α = 0. This means that weassume the existence of two Killing vectors: a Killing vector field k which is asymptotically timelikeand a spacelike Killing vector field m whose orbits are closed curves. In addition, we require thatthey commute, [ k, m ] = 0. It is natural to impose the same symmetries to the scalar field φ , thatis L k φ = 0 and L m φ = 0 where L X denotes the Lie derivative with respect to a vector field X .In order to write the metric in the Lewis-Papapetrou form, see for example [32], we have to verifythat the Frobenius conditions are still true for the above gravitational action (1). Indeed, in orderto show the Lewis-Papapetrou form in vacuum, or in the presence of a cosmological constant, onehas to use the fact that spacetime is an Einstein metric. This is not true here. We have to thereforedemonstrate that the corresponding one-forms satisfy the Frobenius conditions k ∧ m ∧ d k = 0 and k ∧ m ∧ d m = 0 (88)even in the presence of the conformally coupled scalar field φ in order to simplify the form of themetric . Let us consider the twist 1-forms ω ( k ) = 12 ∗ ( k ∧ d k ) and ω ( m ) = 12 ∗ ( m ∧ d m ) (89)associated to k and m respectively, with the sign ∗ denoting the Hodge star operator. Thereforethe condition (88) is equivalent to i m ω ( k ) = 0 and i k ω ( m ) = 0 where i is the interior product. Letus focus on demonstrating the former relation, i m ω ( k ) = 0.Since the metric and the scalar field are invariant under the flow of k , the equations of motion(3)-(5) infer the following relation after some algebra, (cid:18) − πG φ (cid:19) k [ a R ( k ) b ] = 4 πG (cid:0) k [ a ∇ b k c ] (cid:1) ∇ c φ, (90)where we have introduced the Ricci 1-form R ( k ) with component R ab k b . Then, using componentlanguage, the definition (89) of the twist form associated to k gives ǫ abcd ω d ( k ) = − k [ a ∇ b k c ] . More-over, k verifies the identity [32] d ω ( k ) = ∗ (cid:16) k ∧ R ( k ) (cid:17) , (91) Thus we correct the argument about the Frobenius conditions given in [36] where the stationary and axisymmetricproblem is studied. k [ a R ( k ) b ] = − ǫ abcd ∇ c ω d . Then, eq. (90) gives the differentialof the twist form associated to k ,d ω ( k ) = ω ( k ) ∧ d (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) − πG φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (92)when φ = πG . After that, using the Cartan identity L m = d ◦ i m + i m ◦ d and the fact that L m ω ( k ) = 0 (see [62] for a justification), we haved i m ω ( k ) = − i m d ω ( k ) = − (cid:0) i m ω ( k ) (cid:1) d (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) − πG φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + i m (cid:20) d (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) − πG φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:21) ω ( k ) . (93)The second term vanishes in virtue of L m φ = 0, and we thus obtaind (cid:20)(cid:18) − πG φ (cid:19) i m ω ( k ) (cid:21) = 0 . (94)Consequently, (cid:0) − πG φ (cid:1) i m ω ( k ) is a constant function, and the same holds for (cid:0) − πG φ (cid:1) i k ω ( m ) .Without loss of generality, we take these constants to be zero since there is a rotation axis on which m has to vanish. Therefore, we have shown that the Frobenius conditions (88) are true as long as φ = πG . The situation where φ = πG corresponds to an infinite effective gravitational constant˜ G according to (2). References [1] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt,
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