Gaussian Null Coordinates for Rotating Charged Black Holes and Conserved Charges
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Gaussian Null Coordinates for Rotating Charged Black Holes andConserved Charges
M. Cvetiˇc , , C.N. Pope , , A. Saha and A. Satz Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104, USA Center for Applied Mathematics and Theoretical Physics,University of Maribor, SI2000 Maribor, Slovenia George P. & Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA DAMTP, Centre for Mathematical Sciences, Cambridge University,Wilberforce Road, Cambridge CB3 OWA, UK Sarah Lawrence College, Bronxville, NY 10708, USA
Abstract
Motivated by the study of conserved Aretakis charges for a scalar field on the horizonof an extremal black hole, we construct the metrics for certain classes of four-dimensionaland five-dimensional extremal rotating black holes in Gaussian null coordinates. Weobtain these as expansions in powers of the radial coordinate, up to sufficient orderto be able to compute the Aretakis charges. The metrics we consider are for 4-chargeblack holes in four-dimensional STU supergravity (including the Kerr-Newman blackhole in the equal-charge case) and the general 3-charge black holes in five-dimensionalSTU supergravity. We also investigate the circumstances under which the Aretakischarges of an extremal black hole can be mapped by conformal inversion of the metricinto Newman-Penrose charges at null infinity. We show that while this works for four-dimensional static black holes, a simple radial inversion fails in rotating cases becausea necessary conformal symmetry of the massless scalar equation breaks down. We alsodiscuss that a massless scalar field in dimensions higher than four does not have anyconserved Newman-Penrose charge, even in a static asymptotically flat spacetime. ontents
In the last few years there have been many studies that have revealed that the horizon ofan extremal black hole is unstable to small perturbations. These may be perturbations ofthe black hole metric itself, or perturbations of matter fields propagating in the black holebackground. The simplest such examples arise by considering the perturbations of a scalarfield [1–4]. Perturbations of other fields, including linearised gravity, were considered in [5].The instabilities stem from the existence of conserved charges on the future horizon of theextremal black hole, which imply that physical perturbations do not decay at large values ofthe advanced time v . These conserved charges are known as Aretakis charges. They existquite generally for any black hole with an extremal horizon, but not for a non-extremalblack hole with a bifurcate horizon.General arguments for the existence of Aretakis charges in a black hole with an extremalhorizon can be given, making use of the general near-horizon form of the metric for suchblack holes [6]. (The near-horizon metric is given in eqn (2.1) below.) The metric inthis form is written using Gaussian null coordinates (GNC). The Aretakis charges can be2alculated explicitly for a given extremal black hole solution by casting the metric into theGaussian null form. This is straightforward for a simple static example such as the extremalReissner-Nordstr¨om solution, but it is rather less simple for a stationary metric such as theextremal Kerr solution. In such a case one cannot construct an exact expression for themetric in Gaussian null form, but fortunately it is sufficient to determine just the firstfew orders in a GNC expansion of the metric in powers of the radial coordinate measuringdistance away from the horizon. This procedure was carried out for the Kerr metric in [7,8].Essentially, the method used was to solve the equations for null geodesics, in an expansionin powers of the radial distance from the horizon.One of the main purposes of the present paper is to cast the metrics for certain classes ofrotating extremal supergravity black holes into the Gaussian null form, thus allowing one tocompute the conserved Aretakis charges in these spacetimes. Specifically, we carry out thisprocedure for the rotating extremal black holes in four-dimensional STU supergravity thatcarry four independent electric charges, and also for the general 3-charge rotating extremalblack holes in five-dimensional STU supergravity. Specialisations of these results encompassthe previously-derived expressions for the extremal Kerr metric in four dimensions [7] andthe extremal Myers-Perry metric in five dimensions [8]. The intermediate stages in thecalculations necessary for casting the supergravity black hole solutions into Gaussian nullform are quite involved, but the final results that we obtain, at the order that is sufficientfor calculating the Aretakis charges, are remarkably simple.An intriguing observation, in the case of the extremal Reissner-Nordstr¨om (ERN) metric[9, 10], is that by performing an inversion of the radial coordinate so that the horizon ismapped into future null infinity I + , and then extracting an overall conformal factor, theAretakis charges on the null horizon of the ERN metric can be mapped into conservedNewman-Penrose charges at null infinity in the conformally-inverted metric. In fact, thisconformal inversion of the ERN metric actually maps it into the ERN metric again, a resultthat had been obtained many years previously by Couch and Torrence [11].The mapping of Aretakis charges on an extremal horizon into Newman-Penrose chargeson I + was investigated in a more general setting in [12]. It was shown that the conformalinversion of a general extremal black hole, written in Gaussian null coordinates, gives rise toa metric that was called Weakly asymptotically flat (WAF) in [12]. This metric approachesMinkowski spacetime at infinity, but with rather weaker fall-off conditions than those ofan asymptotically flat spacetime written in Bondi-Sachs coordinates. It was shown in[12] that Newman-Penrose charges can be computed in the WAF spacetime obtained by3onformal inversion of the original extremal black hole, and in various static examples themapping between Aretakis and Newman-Penrose charges was exhibited. These generalisedthe mapping for extremal Reissner-Nordstr¨om that was found in [9, 10]. In particular,in the more general examples, such as multi-charge static extremal black holes in STUsupergravity, the conformal inversion of the original metric does not give back the samemetric again, unlike the ERN case.The discussion in [12] in principle applied also to stationary extremal black holes thatare not static. One can certainly again compute Newman-Penrose charges in the WAFmetric obtained by conformal inversion. As we shall discuss in the present paper, however,in the stationary case there is a lacuna in the argument that would be needed in order tolink the Newman-Penrose charges to the original Aretakis charges. Namely, in order to mapthe one into the other, it is necessary to be able to argue that the solutions of the masslessscalar wave equation in the original black hole metric and in the conformally inverted WAFmetric can be related by the necessary conformal transformation. This is fine as long asthe Ricci scalar, which enters in the conformally-invariant scalar equation ( (cid:3) − R ) ψ = 0,either is zero or else it goes to zero sufficiently rapidly in the asymptotic region. This iscertainly true in the case of the extremal Reissner-Nordstr¨om solution, where R vanishes,and also in the more complicated four-dimensional static supergravity black holes, where R vanishes sufficiently rapidly asymptotically. But, as we show later, in the weakened fall-offof the WAF metrics in the stationary case, the Ricci scalar does not fall off fast enough atinfinity, and this provides an obstruction to being able to relate the Aretakis charges to theNewman-Penrose charges of the conformally-inverted WAF metric, at least if we assumea simple inversion of the radial coordinate. In fact a manifestation of this problem wasforeshadowed in the results in [13], where the Aretakis and Newman-Penrose charges werecalculated in the case of the extremal Kerr black hole and its conformal inversion.The focus in [12], concerning the relation between Aretakis charges and Newman-Penrosecharges, was four-dimensional spacetimes. We also address in this paper the possible ex-tension of these considerations to more than four dimensions. We show that conservedAretakis charges for massless scalar fields exist in higher-dimensional extremal black holesalso. However, as we show, Newman-Penrose charges for a massless scalar field no longerexist when one goes beyond four dimensions. This happens simply because there is a termin the large- r expansion for the scalar equation written in Bondi-Sachs coordinates thatpresents an obstruction to the existence of conserved charges, and this term occurs with adimension-dependent coefficient ( n − n −
4) that is absent in n = 4 dimensions but not4hen n ≥ The metric near the horizon of an extremal black hole in any dimension n can always bewritten in Gaussian null coordinates, where it takes the form [6] ds = L ( x ) h − ρ F dv + 2 dvdρ i + γ IJ ( dx I − ρ h I dv )( dx J − ρ h J dv ) , (2.1)where F , γ IJ and h I depend on the radial coordinate ρ and on the coordinates x I on the(spherical) horizon, which is located at ρ = 0. Near the horizon we may assume F ( ρ, x ) = 1 + ρ F ( x ) + ρ F ( x ) + · · · ,γ IJ ( ρ, x ) = ¯ γ IJ ( x ) + ρ γ (1) IJ ( x ) + ρ γ (2) IJ ( x ) + · · · ,h I ( ρ, x ) = h I ( x ) + ρ h I ( x ) + ρ h I ( x ) + · · · . (2.2)We shall consider the Aretakis charges for a scalar field ψ obeying the massless Klein-Gordon equation (cid:3) ψ = 0. The solutions can be taken to have the small- ρ expansion ψ ( ρ, v, x ) = ψ ( v, x ) + ρ ψ ( v, x ) + ρ ψ ( v, x ) + · · · . (2.3)From this, and the form of the metric expansion, it follows that on the horizon one has [12] ∂∂v h ∂ψ∂ρ + 12 ψ ∂ log γ∂ρ i + h I ∂ I ψ + 1 √ ¯ γ ∂ I (cid:16) √ ¯ γ L ¯ γ IJ ∂ J ψ (cid:17) = 0 . (2.4)If this is integrated over the horizon, with measure √ ¯ γ d n − x , the final term gives zerosince it is a total derivative. The coordinates x I on the ( n − ϕ on the 2-sphere), and latitude type coordinates (like θ on the 2-sphere). The azimuthal coordinates are associated with Killing vectors. Crucially,for the extremal black hole metrics we shall be considering and as we shall see in detaillater, Gaussian null coordinates can be chosen so that h I is zero for the index values I corresponding to the latitude type coordinates. It follows that h I ∂ I ψ can be written as( √ ¯ γ ) − ∂ I ( √ ¯ γ h I ψ ), and thus this term is also a total derivative that integrates to zero.The upshot is that the quantity Q A = Z √ ¯ γ d n − x h ∂ψ∂ρ + 12 ψ ∂ log γ∂ρ i , (2.5) In principle it would suffice to assume weaker asymptotic conditions on the metric functions as ρ ap-proaches zero (as discussed, for example, in [12]), but in practice these are the ones that arise in the blackholes we shall be considering. ∂ v Q A = 0.In the subsequent sections we shall calculate the Aretakis charge for the case of certainrotating charged extremal black holes in four-dimensional STU supergravity, and for thegeneral 3-charge extremal rotating black holes in five-dimensional STU supergravity. Thekey part of the calculations involves constructing the expressions for the black hole solutionsin Gaussian null coordinates, up to the necessary order in the expansion in powers of ρ .A technique for constructing Gaussian null coordinates for an extremal black hole metrichas been described in [7, 8]. Essentially, one writes down the equations for null geodesicsin the extremal metric, for which first integrals exist for the time and the azimuthal coor-dinate(s). The equations for the remaining coordinates cannot be integrated explicitly, soone then expands these in power series in the affine parameter λ along the geodesics. Thegeodesic equations are then integrated order by order in λ , imposing certain transversalityconditions in the process. Finally, a change of variable from λ to ρ brings the metric intothe desired form (2.1). The 4-charge four-dimensional STU supergravity black holes that we shall be consideringhere were constructed in [14]. A convenient presentation for our purposes can be foundin [15]. The metric can be written as ds = − ¯ ρ − mrW ( dt + B (1) ) + W (cid:16) dr ∆ + dθ + ∆ sin θ d ˜ φ ¯ ρ − mr (cid:17) , (3.1) B (1) = 2 ma sin θ ( r Π c − ( r − m ) Π s )¯ ρ − mr d ˜ φ , ¯ ρ = r + a cos θ , ∆ = r − mr + a ,W = R R R R + a cos θ + h r + 2 mr X i s i + 8 m (Π c − Π s ) Π s − m ( s s s + s s s + s s s + s s s ) i a cos θ ,R i = r + 2 ms i , Π c = Y i c i , Π s = Y i s i , s i = sinh δ i and c i = cosh δ i . The physical mass M and the four physical charges Q i are given by M = m + m X i s i , Q i = 2 ms i c i = m sinh 2 δ i . (3.2)The metric is extremal when m = a , which we assume from now on. This implies ∆ =( r − a ) . Defining a new azimuthal coordinate φ = ˜ φ − a (Π c + Π s ) t (3.3)so that ∂/∂t is the Killing vector that becomes null on the horizon, we may write theextremal metric in the form ds = W − ( Adt + 2 Bdtdφ + Cdφ ) + W h dr ( r − a ) + du − u i , (3.4)where u = cos θ .In algebraic computations the parameterisation of the charges in terms of the fourparameters s i is not ideal, since the relation to the c i involves square roots, namely c i = q s i . We have found it more convenient here to work instead with the five parameters( α, β, γ, Π c , Π s ), where α = s + s + s + s , β = s s + s s + s s + s s + s s + s s ,γ = s s s + s s s + s s s + s s s , Π c = c c c c , Π s = s s s s . (3.5)These are related by the identity γ = Π c − Π s − − α − β , (3.6)and in practice we find it most convenient to use ( α, β, Π c , Π s ) as the four independentquantities that parameterise the four charges. In terms of ( α, β, γ, Π c , Π s ) the charge-dependent quantities W and B (1) in the metric (3.1), subject to the extremality condition m = a , are given by W = r + 2 aα r + 2 a (2 β + x ) r + 2 a (4 γ + x ) r + a [16Π s + 4(2Π c Π s − s − γ ) u + u ] , B (1) = 2 a sin θ ( r Π c − ( r − a ) Π s )¯ ρ − ar d ˜ φ . (3.7) In the case where the charges are set equal, with δ i = δ , the solution reduces to the Kerr-Newman blackhole, with ¯ r = r + 2 m sinh δ being the standard Kerr-Newman radial coordinate. We shall discuss theextremal Kerr-Newman metric in Gaussian null coordinates in the next subsection. t and φ are taken to be˙ t = − CW ( r − a ) (1 − u ) , ˙ φ = BW ( r − a ) (1 − u ) , (3.8)where a dot denotes a derivative with respect to the affine parameter λ . We now expandthe r and u coordinates as power series in λ , with r = a + X n ≥ R n ( y ) λ n , u = y + X n ≥ X n ( y ) λ n , (3.9)where the affine parameter λ vanishes on the horizon. Substituting these expansions in tonull geodesic constraint g µν ˙ x µ ˙ x ν = 0 and the Euler-Lagrange equation for u allows us tosolve iteratively for the R n ( y ) and X n ( y ) coefficients in (3.9). We find R ( y ) = 2(Π c + Π s ) s ( y ) , X ( y ) = y (1 − y a s ( y ) , (3.10) R ( y ) = 2(Π c + Π s )(1 − y ) [4(Π c − Π s ) + (9 + 7 α + 4 β )Π s − (5 + 5 α + 4 β )Π c − (Π c − Π s ) y ] a s ( y ) , where s ( y ) = 1 + 2 α + 4 β + 8 γ + 16Π s + 2(1 + α − γ + 4Π c Π s − s ) y + y , = Y i (1 + 2 s i ) + 2 (cid:16) X i s i − X i Ψ = 0 in the weakly asymptotically flatmetric (5.2) in n dimensions has a large- r expansion of the formΨ( r, u, x ) = Ψ ( u, x ) r − γ + Ψ ( u, x ) r − γ − + · · · , γ = n − . (5.10)If the ( n − h IJ in the WAF metric (5.2) is expanded as h IJ ( r, u, x ) = ω IJ + h (1) IJ r − + · · · , (5.11)we may choose a coordinate gauge where √ h = ζ ( r ) √ ω , with ζ ( r ) = 1 + ζ r − + ζ r − + · · · . (5.12)18ubstituting (5.10) into (cid:3) Ψ = 0, evaluated in the WAF metric background, we find that atthe leading order in the large- r expansion, ∂∂u [2Ψ + ζ Ψ ] − ( n − n − − n − C I ∂ I Ψ + n − D I ( C I Ψ ) + D I D I Ψ = 0 , (5.13)where C I has an expansion of the form C I = C I r − + C I r − + · · · , and where D I denotesthe covariant derivative in the ω IJ metric.In n = 4 dimensions, we can obtain a conserved charge by integrating (5.13) over the2-sphere with metric ω IJ : Q NP = Z √ ω d x [2Ψ + ζ Ψ ] . (5.14)In a general dimension n = 4 there are two obstructions to obtaining a conserved charge.Firstly, the term n − C I ∂ I Ψ is not a total derivative in general. Note, however, that if C I vanishes when I lies in the direction(s) associated with latitude type coordinates onthe sphere (i.e. directions that are not associated with Killing vectors), then this term canbe rewritten as the total derivative n − D I ( C I Ψ ), since the remaining, azimuthal, spherecoordinates are associated with Killing directions. As we saw earlier, C I will indeed vanishin the non-azimuthal directions in the case of the WAF metrics obtained by conformalinversion of extremal black hole metrics, since h I = 0 in those directions in the black holemetrics. This still leaves the problem of the term − ( n − n − Ψ in (5.13). This termimplies that there can be no Newman-Penrose charge for a massless scalar obeying (cid:3) Ψ = 0in any dimension higher than n = 4.It is perhaps worth remarking that in the case of a WAF metric obtained by conformalinversion of a static spherically symmetric extremal black hole in n ≥ /r at large distance and thus it would make a contribution in the NP chargecalculation at the leading order in a large- r expansion if one were to add an R Ψ term to themassless wave equation (cid:3) Ψ = 0. If the coefficient of this term were chosen appropriately, itcould be arranged to cancel the term − ( n − n − Ψ in (5.13), thus allowing the existenceof a conserved NP charge. (See eqn (5.9) for the calculation of the Ricci scalar termfor the conformal inversion of the five-dimensional extremal Reissner-Nordstr¨om metric.)However, as may be readily checked, the coefficient of R Ψ that would be needed to achievethis cancellation appears to have no other related significance. In particular, it is not equalto the coefficient that would be needed for the conformally invariant scalar operator.19 Conclusions In this paper, we have constructed the metrics in Gaussian null coordinates for certainclasses of extremal rotating black holes in supergravity theories, as expansions in the radialcoordinate at a sufficient order to be able to calculate the conserved Aretakis charges on thehorizon. Specifically, we did this for the extremal rotating black holes in four-dimensionalSTU supergravity that carry four independent electric charges (with the special case of theKerr-Newman black hole when the four charges are equal), and also for the general extremalrotating 3-charge black holes in five-dimensional STU supergravity. We then obtained theexplicit expressions for the simplest of the Aretakis charges for a massless scalar field ineach case.We also investigated the possibility of relating the Aretakis charge on the horizon ofthe extremal black hole to the Newman-Penrose charge at I + in the metric obtained byperforming an inversion of the radial coordinate, after the extraction of an appropriateconformal factor. This relation was studied for four-dimensional spacetimes in [12], wherevarious examples of the mapping were obtained for classes of static extremal black holes.In the present paper we showed that such a mapping becomes problematical for extremalrotating black holes, because after conformal inversion the resulting weakly asymptoticallyflat metric has a Ricci scalar whose fall-off at large r is sufficiently slow that one cannottreat the massless scalar equation (cid:3) ψ = 0 as being equivalent to the conformally-invariantequation ( (cid:3) − R ) ψ = 0 for the purpose of calculating the Newman-Penrose charge. Thismeans that the ability to relate the solutions for the scalar field in the original extremalmetric and in the conformally-inverted metric is lost in the case of rotating black holes, atleast if we consider just a simple ρ → /r inversion. In turn, one cannot by this meansrelate the Aretakis and Newman-Penrose charges for extremal rotating four-dimensionalblack holes.As we then discussed, the situation becomes worse in dimensions n > 4. The extremalblack holes (static or rotating) still admit conserved Aretakis charges for a massless scalarfield, but there are no Newman-Penrose charges for a massless scalar in any asymptoticallyflat spacetime of dimension n > 4. Thus it appears that the mapping between Aretakis andNewman-Penrose charges is exclusively a four-dimensional phenomenon.There remain a number of directions for further study. Firstly, it would be of interestto generalise the construction of the extremal rotating four-dimensional STU black holesin Gaussian null coordinates to the general case of eight charge parameters (independentelectric and magnetic charges carried by each of the four gauge fields. The solution for20he 8-charge rotating black holes is given in [17].) It would also be of interest to studythe analogous conserved Aretakis and Newman-Penrose charges for higher-spin fields in thecharged supergravity black hole backgrounds. Examples would include Maxwell fields, andalso perturbations of the background metrics themselves. We are grateful to Hadi Godazgar, Mahdi Godazgar, Carmen Li and James Lucietti forhelpful discussions. The work of M.C. is supported in part by the DOE (HEP) Awardde-sc0013528, the Fay R. and Eugene L. Langberg Endowed Chair (M.C.) and the SlovenianResearch Agency (ARRS No. P1-0306). The work of C.N.P. is supported in part by DOEgrant DE-FG02-13ER42020. References [1] S. Aretakis, Stability and instability of extreme Reissner-Nordstr¨om black hole space-times for linear scalar perturbations I , Commun. Math. Phys. , 17 (2011),doi:10.1007/s00220-011-1254-5, arXiv:1110.2007 [gr-qc].[2] S. 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