Generalised Couch-Torrence Symmetry for Rotating Extremal Black Holes in Maximal Supergravity

The extremal Reissner-Nordstr\"om black hole admits a conformal inversion symmetry, in which the metric is mapped into itself under an inversion of the radial coordinate combined with a conformal rescaling. In the rotating generalisation, Couch and Torrence showed that the Kerr-Newman metric no longer exhibits a conformal inversion symmetry, but the radial equation arising in the separation of the massless Klein-Gordon equation admits a mode-dependent inversion symmetry, where the radius of inversion depends upon the energy and azimuthal angular momentum of the mode. It was more recently shown that the static 4-charge extremal black holes of STU supergravity admit a generalisation of the conformal inversion symmetry, in which the conformally-inverted metric is a member of the same 4-charge black hole family but with transformed charges. In this paper we study further generalisations of these inversion symmetries, within the general class of extremal STU supergravity black holes. For the rotating black holes, where again the massless Klein-Gordon equation is separable, we show that examples with four electric charges exhibit a generalisation of the Couch-Torrence symmetry of the radial equation. Now, as in the conformal inversion of the static specialisations, the inversion of the radial equation maps it to the radial equation for a rotating black hole with transformed electric charges. We also study the inversion transformations for the general case of extremal BPS STU black holes carrying eight charges (4 electric plus 4 magnetic), and argue that analogous generalisations of the inversion symmetries exist both for the static and the rotating cases.


Introduction
It was observed many years ago that the extremal limit of the Reissner-Nordström black hole exhibits a remarkable conformal inversion symmetry, in which an inversion of the radial coordinate, which maps the near-horizon region to the region near infinity, combined with a conformal rescaling, transforms the original metric back into itself [1]. Explicitly, consider the original extremal Reissner-Nordström metric written in the isotropic form where dΩ 2 2 is the metric on the unit 2-sphere. The horizon is located at r = 0. Performing the inversion to a new radial coordinater = Q 2 r , (1.2) one finds that the conformally-related metric ds 2 defined by is given by Thus after the conformal inversion, the resulting metric ds 2 is again the extremal Reissner-Nordström metric, and takes the identical form to the original metric (1.1) [1]. The radius of the inversion in (1.2) is equal to the electric charge Q (and hence also the mass).
The conformal inversion symmetry of the extremal Reissner-Nordström metric has been employed in a number of papers (see, for example, [2][3][4]) in order to relate some of the recent observations about the asymptotic behaviour of Klein-Gordon and other fields on the future horizon of an extremal black hole (see, for example, [5,6]) to the asymptotic behaviour of these fields at future null infinity.
It was shown also in [1] that the conformal inversion symmetry of the extremal Reissner-Nordström metric does not generalise to the rotating case, namely the extremal Kerr-Newman metric. It was, however, observed that if one considers a massless Klein-Gordon field Φ in the extremal Kerr-Newman background, then after performing a separation of variables the differential equation for the radial function exhibits a remarkable inversion symmetry. This involves a radius of inversion that depends not only on the black hole charge and rotation parameters but also on the separation constants ω and m that arise in the factorised solutions Φ(t, r, θ, ϕ) = R(r) S(θ) e −i ωt e i mϕ . (1.5) Thus the inversion radius is different for different modes.
In [4], conformal inversions of a class of more general static extremal black holes were investigated. Specifically, the static extremal 4-charge black hole solutions [7] of the fourdimensional STU supergravity theory were studied. STU supergravity comprises N = 2 supergravity coupled to three vector supermultiplets. 1 Thus there are four electromagnetic field strengths in total, each of which can carry, in general, independent electric and magnetic charges. The general 8-charge solution is quite complicated but the special case where each field strength carries just an electric charge is much simpler, with the metric being given by ds 2 = −H −1/2 dt 2 + H 1/2 (dr 2 + r 2 dΩ 2 2 ) , (1.6) 1 This is also a consistent truncation of ungauged supergravity theory with maximal (both N = 4 and N = 8) supersymmetry. Black hole solutions of STU supergravity are generating solutions of the full maximally supersymmetric ungauged supergravity theories.
As was shown in [4], under the inversioñ (1.8) one finds that the conformally related metric ds 2 given by ds 2 = Π 2 r 2 ds 2 (1.9) takes the form ds 2 = − H −1/2 dt 2 + H 1/2 (dr 2 +r 2 dΩ 2 2 ) , (1.10) Thus the metric ds 2 obtained by the conformal inversion is in the same class of 4-charge static extremal black holes as the original metric (1.6), but for new charges Q i related to the original charges Q i by (1.12) [4].
In the special case where all four charges are equal, the metric (1.6) reduces to the extremal Reissner-Nordström metric (1.1) and the conformal inversion gives back this metric again. A more general specialisation where the conformal inversion becomes an actual symmetry is if the charges are set equal in pairs; for example (1.13) and hence H = H [4].
One purpose of the present paper is to investigate whether the observation of Couch  We also study the general 8-charge rotating extremal black holes. As in the 4-charge specialisation described earlier, here too the massless Klein-Gordon equation can be separated and the behaviour of the radial wave equation under inversion can be investigated.
We find that, as in the 8-charge static BPS extremal black holes described above, although we can write down the conditions for the inversion of the radial coordinate to give rise again to a radial equation for a set of mapped charges, it does not appear to be possible to give an elegant formula for the mapped charges. Again, the reason is that the system of conditions for invertability does not fully constrain the mapped charges. As in the static case, we may nevertheless argue that solutions for the mapped charges will exist.
There also exists a class of extremal black holes in STU supergravity that does not obey the BPS conditions. A simple example that was investigated in [4] was the static extremal A convenient presentation of the rotating black holes in four-dimensional STU supergravity carrying four independent charges can be found in [8,9]. The metric can be written as [9]: (so that the horizon is located atr = 0), we find that R and S satisfy the equationŝ where λ is the separation constant and A straightforward calculation then shows that, writing H as H(r, δ i ), with s i = sinh δ i , etc., where and the redefined charge parametersδ i are related to the original parameters δ i by the Let us now define a (dimensionless) coordinate x = β/r. Viewing the radial function R as now being a function P (x), we see that it satisfies One can easily verify that β and C 0 are invariant under δ i →δ i , while It follows that and so if P (x, δ i ) is a solution of (2.9) then defining the function P (x) will solve the tilded equatioñ Thus a solution P (x) of the radial equation with charge parameters δ i maps into a solution of the inverted radial equation with charge parametersδ i given by (2.8).
Note that in the specialisation to pairwise-equal charges, such as δ 3 = δ 1 and δ 4 = δ 2 , one hasδ 16) and so in this case H(x, δ i ) = H( 1 x , δ i ) (since the function H is symmetrical in the charge parameters). The constant β in this pairwise-equal case is given by Thus in this case, and in its further specialisation to δ 1 = δ 2 (all charges equal, i.e. the Kerr-Newman solution studied by Couch and Torrence), the inversion is an actual symmetry of the radial equation.

Conformal inversion in the static limit
It is interesting to look at the limit where the extremal 4-charge metric (2.1) (with m = a) reduces to the metric of extremal 4-charge static black holes, first obtained as a BPS black hole solution in [7]. Since the physical charges in the extremal rotating metric are given by q i = a sinh 2δ i , one must send the charge parameters δ i to infinity at the same time as sending the rotation parameter a to zero, so as to hold the q i finite, and so one has fixed, and the metric becomes in the extremal static limit, where H i = 1 + q i r −1 . After the inversion, and using (2.8), one and henceq This is precisely the relation between untilded and tilded charges that was found in [4] (and which we summarised in the Introduction) for the 4-charge static metrics and the conformally-related inverted metrics. In that case, the transformation mapped the entire metric into another metric within the same 4-charge class. Thus the inversion symmetry (up to charge transformations) of the radial equation in the rotating case, which we exhibited above, becomes an inversion symmetry (up to charge transformations) of the entire metric in the static limit.

Dyonic STU Black Holes
The generalisation of the Couch-Torrence inversion symmetry of the separated radial equation to the general case of the 8-charge dyonic rotating extremal black holes of STU supergravity is rather complicated, and we shall not present it here. It becomes much more manageable in special cases, such as the case with 4 electric charges, which we discussed previously. Another case that is relatively straightforward is when the field strengths of STU supergravity are set equal in pairs, with each of the two remaining independent fields carrying independent electric and magnetic charges.
where R = r 2 − 2mr + a 2 − n 2 , the separation constant is λ, and the factorised solutions are taken to have the form Ψ = e −i ωt+i kϕ P (r) S(θ). (We follow [11] and use k rather than m for the azimuthal quantum number, since m is used here to denote the black-hole mass parameter.) The constant n is given by (this is the condition for the physical NUT charge N = m ν 1 + n ν 2 to be zero) and the functions W r and L r in the extremal case are given by where we have defined and the quantities ν 1 , ν 2 and D are given, in the pairwise-equal case, by The physical mass M , given in general by M = mµ 1 +nµ 2 [11], is given in the pairwise-equal case by M = mν 2 − nν 1 (since then µ 1 = ν 2 and µ 2 = −ν 1 ). The four physical electric and magnetic charges (Q 1 , Q 2 , P 1 , P 2 ) carried by the two independent field strengths F 1 and F 2 are given in terms of the boost parameters (δ 1 , δ 2 , γ 1 , γ 2 ) by [11] Viewing H, defined in (3.1), as a function of ρ, we find where Thus the function H has the inversion symmetry This implies an inversion symmetry of the radial equation, namely, that if we defineρ = β 2 ρ −1 , then the original radial equation (3.1) implies the inverted equation where Note that as in the case of the pairwise-equal specialisation of the 4 electric charge black holes discussed previously, the inversion is an actual symmetry in this pairwise-equal dyonic charge case. One can easily verify that if the magnetic charges are set to zero, the inversion here reduces to the previous pairwise-equal result, with the radius of inversion β reducing from (3.8) to (2.17).

Conformal inversion symmetry for static pairwise-equal dyonic black
holes he static limit of the extremal rotating dyonic black holes with pairwise-equal charges is achieved by sending m (and hence n and a) to zero while sending the boost parameters to infinity, so as to keep the physical charges in (3.6) finite and non-zero. This can be done by 12) and the taking the limit λ → ∞. The metric given in [11] becomes static, with The electric and magnetic charges in the static limit are given by 3 .
A straightforward calculation shows that if we perform the inversion 3 Note that P2 = −P1 in the static limit. Although this might appear not to be a generic pairwise-equal dyonic configuration it actually is, once one takes into account that there is an S-duality of the pairwise-equal STU supergravity under which (Q1, P1) and (Q2, P2) can both be rotated under 14) and therefore the metric obeys the conformal inversion symmetry where ds 2 is the same as the original metric ds 2 given in (3.13), only now written withr in place of r. This reduction to a 5-parameter canonical form was employed in [12] in the case of the static BPS extremal STU black holes, in order to construct a solution with 5 independent charge parameters. They carry charges (Q, P ) of the form (after changing to a duality frame that matches the choice in our previous discussions) For our present purposes it will be more convenient to re-introduce the redundancy of the additional three parameters, so that we can present the static extremal black holes in a symmetrical form with eight independent charge parameters; four electric and four magnetic. In order to do this we shall construct explicitly the action of the SL(2, R) 3 global symmetry on the STU supergravity fields and the charges, and then make use of the U (1) 3 compact subgroup in order to derive the general 8-charge solution from the 5-parameter solution presented in [12]. Because these steps are a little involved, we relegate them to appendices A and B.
The upshot from these calculations is that the metric of the general 8-charge static extremal black hole is where the coefficients α, β, γ and ∆ are obtained in eqns (B.8) in appendix B, and which for convenience we reproduce here: In the expression for γ, we have defined If the radial coordinate of the metric (4.2) is subjected to the inversion then the conformally rescaled metric ds 2 , defined by where V =r 4 +αr 3 +βr 2 +γr + ∆, will be in the same class of black hole metrics provided that there exists a mapping of the charges such that These four equations constitute the conditions that the eight mapped charges Q i and P i must satisfy, if the conformally inverted metric is to be interpretable as again being contained within the 8-charge family of static BPS extremal black holes.
Note that the inversion of the inversion will give back the original metric, up to a constant scale factor that we can always set to unity by normalisation. The inversion will then be an involution, and the third equation in (4.6) is automatically satisfied if the first and the fourth are satisfied. Thus we can choose to view the four conditions (4.6) as instead being described by the first, second and fourth equations in (4.6), together with the normalisation condition for the inversion to be an involution.
As one can easily verify, in the special case of the solution with just four electric charges, the expressions (4.3) reduce to A relatively simple possibility is to restrict the three SL(2, R) matrices  for i = 1, 2 and 3. This then leaves just the three undetermined parameters a i to be solved, by requiring the remaining equations (the first three) in eqn (4.6) to be satisfied. Provided that there exist real such solutions for the a i , then the problem of finding the mapping of charges under the conformal inversion is solved, in the sense that it is reduced to solving three equations for three unknowns.
As a first example, we may consider the 4-charge case where the magnetic charges are all zero, which was studied in [4] and is described in the introduction of the present paper.
The mapped charges Q i are given in terms of the original charges Q i by (1.12), and as can easily be checked from the formulae in appendix A, they are produced by means of the SL(2, R) 3 transformations with the restricted form (4.9) where (4.10) As another example, we may consider the special case of pairwise-equal charges; without loss of generality we choose the case where the gauge fields labelled 1 and 3 are set equal, and likewise the gauge fields labelled 2 and 4. Thus we have The SL(2, R) 3 global symmetry of the full STU supergravity, described in appendix A, reduces to just the SL(2, R) 2 symmetry. In this case the coefficients α, β, γ and ∆ in (4.3) and so the metric function V becomes the perfect square (4.13) After the inversion (4.4) and conformal scaling (4.5), the charge transformation conditions (4.6) reduce simply toα = α , ∆ = ∆ . (4.14) As we mentioned previously, the conformal inversion is actually a symmetry in this pairwise-equal case, and correspondingly, as can be seen from (4.14), one solution for the transformed (tilded) charges is simply to take them to be equal to the original charges.
However, it is interesting to note that we can also find other solutions to the conditions (4.14) in which the charges are non-trivially transformed. One way to do this is by taking From the transformation (A.24) we therefore find that eqns (4.14) are satisfied if a 2 is chosen so that The transformed charges are given explicitly by Another, inequivalent, way of solving (4.14) in this pairwise-equal example is to use From the transformation (A.24) we now find that eqns (4.14) are satisfied if The transformations of the charges in this case correspond to a different way of solving the constraint equations (4.14). This reflects the fact that the constraints provide an undetermined system of equations. In this pairwise-equal specialisation, we have the two constraint equations (4.14) and the four unknowns ( Q 1 , Q 2 , P 1 , P 2 ). The general solution would give a family of transformed charges characterised by two continuous parameters. We have exhibited above two discrete members within this family, in addition to the "trivial" member where the charges are untransformed.
In the 4-charge and the pairwise-equal examples above, it was possible to present explicit expressions for solutions to the constraint equations. As mentioned previously, this does not appear to be possible in the general case with 8 independent charges. For example, we can always look for solutions for the transformed charges by considering the subset of SL(2, R) 3 transformations described by (4.9). The fourth constraint in (4.6) is automatically satisfied because ∆ is invariant under SL(2, R) 3 , and so the remaining 3 constraints in (4.6) will imply a discrete set of solutions for the 3 unknowns (a 1 , a 2 , a 3 ). The three equations are polynomials in the a i parameters, but seemingly they are of too high a degree to be explicitly solvable. Of course for any specified set of 8 original charges one can compute numerically the corresponding a i parameter values that satisfy the constraints, and so in this sense the problem is fully solvable. There is, however, one obstacle that can arise, namely, that it might happen that all of the solutions for the a i parameters turn out to be complex. We have looked at numerous examples of "randomly chosen" sets of original charges (Q 1 , Q 2 , Q 3 , Q 4 , P 1 , P 2 , P 3 , P 4 ), and we find that sometimes real solutions for the a i

Inversion Symmetry of Radial Equation for 8-Charge Rotating Black Holes
We use the expressions and notation given in the paper [11] by Chow and Compère. It can be seen that the metric will be extremal if a 2 = m 2 + n 2 , and then the radial function R 4 The mapped charges should also, like the original ones, be non-negative, since otherwise the conformally inverted metric ds 2 defined in (4.5) would have naked singularities outside the horizon at r = 0.
In the examples we found, when the charges come out to be real they are also non-negative. It should be noted also that for the BPS extremal black holes to be regular, without naked singularities, the quartic invariant ∆ should be positive (see, for example, [13]). will be given by R = (r − m) 2 . We define a new radial coordinate ρ = r − m that vanishes on the horizon. The separation of variables for a solution ψ of ψ = 0 is carried out in [11] by writing ψ = e −i ωt+i kϕ Φ r (r) Φ u (u). Their separated equations for Φ r (r) and Φ u (u) are presented in eqn (9.17) of [11]: (We have renamed their separation constant as C cc rather than C, since they already use C for the quantity defined in eqn (5.5) of [11].) The functions W r , L r , W u and L u are given in [11] (in fact L u = 0 when, as in our case, we choose the physical NUT parameter N to be zero). The equations ( with U = a 2 −ū 2 , and this is completely independent of the charge parameters.
The radial equation now takes the form where H(ρ) is given by 4) and the various quantities ν 1 , ν 2 , µ 1 , C, D and M are defined in [11].
We may now seek an inversion symmetry of the radial equation. Thus we transform to a new radial coordinateρ such that where β is a constant to be determined, and test to see whether where the function H is the same in form as the function H given in (5.4), but using redefined charge parameters. 5 Thus we have Assuming that β is universal, that is to say, that it is invariant under the transformation of the charge parameters, we then have, by comparing the various powers of ρ orρ in the proposed relation (5.6) that From the first two equations in (5.8) we have Since we are assuming ω does not transform, and since this relation should hold for all frequencies ω, it follows thatã = a ,ν 2 + 2 D = ν 2 + 2D . From [11], M = m µ 1 + n µ 2 , and since we must set n = −m ν 1 /ν 2 so that the physical NUT charge N = m ν 1 + n ν 2 is zero, we have together with the transformed version where all quantities are tilded. Equations (5.11) therefore imply and hence Collecting the results so far, we see from the last equation in (5.8), from (5.10) and from (5.14) that the three quantities should all be invariant under the transformation of the charge parameters that accompanies the inversion (5.5). A natural guess for the inversion transformation, which would reduce to the known 4-charge case γ i = 0 discussed in section 2 (and its duality partner where instead δ i = 0), and would also reduce to the known pairwise-equal dyonic case discussed in section 3, is to tryδ One can in fact verify that the three quantities X 1 , X 2 and X 3 defined in (5.15) are indeed invariant under (5.16). However, there is one further condition contained in the set of equations (5.8), since until now we just extracted the one condition (5.14) from the third and fourth equations in (5.8). The remaining condition can be found by noting that the extremality condition a 2 = m 2 + n 2 implies a 2 = m 2 (1 + ν 2 1 /ν 2 2 ) (and its tilded version), and using this in the third and fourth equations of (5.8) leads to Straightforward calculation reveals that while this is indeed consistent with (5.16) in the 4-charge specialisation or in the pairwise-equal specialisation (as it must be, since those cases were already fully verified), it is not consistent in the general 8-charge case. Thus the transformation of the δ i and γ i charge parameters in the general case must be more complicated than the guess in (5.16). However, since the number of conditions that must be satisfied in order to achieve H(ρ) = H(ρ) is smaller than the number of unknown transformed charge parameters (δ i ,γ i ), we can conclude that it must be possible to solve for such (δ i ,γ i ), even if we cannot present the solution in a universal and elegant form.

Concluding Remarks
Studies of general rotating black holes in maximally supersymmetric ungauged supergravity theories, (often referred to as STU black holes), revealed their numerous intriguing properties which often stem from, and provide an intriguing generalization of, properties of Kerr-Newman black holes in Einstein-Maxwell gravity. Furthermore, the extremal black holes of that type are endowed with further enhanced symmetry properties, again generalising those of extremal Kerr-Newman black holes. For example, there has been substantial progress in recent studies of the Aretakis charge both for extremal Reissner-Nordström black holes [5,6] and extremal Kerr black holes [14] as well as recent generalizations to extremal static 4-charge black holes [4] and rotating ones [15] in STU supergravity. Furthermore, in [15] Aretakis charges for five-dimensional extremal STU black holes were derived, while generalisations to related conserved charges of extremal static p-branes were given in [16].
In this paper we focused principally on another type of symmetry, namely the Couch- In the present paper we further generalised this symmetry from the extremal static 4charge black holes [4] to the extremal rotating 4-charge black holes in STU supergravity.
We showed that in this case the radial equation for the separable massless Klein-Gordon We expect that a generalisation of the Couch-Torrence symmetry may persist also for general extremal black holes of STU supergravity in five dimensions. Again the separability of the massless Klein-Gordon equation will play an important role. We defer further consideration of this case to future work.
We would like to conclude by emphasising that our analysis focused on conformal inversion transformations for extremal BPS black holes in STU supergravity for which the asymptotic values of the scalar fields were set to zero. It would, of course, be interesting to study conformal inversion transformations for such black holes with non-zero asymptotic values of the scalar fields. This explicit dependence on the asymptotic scalar fields is currently being investigated [19]. Such generalisations of the black hole solutions would in turn allow for generalisations of the Couch-Torrence transformations that could also involve scalar field transformations. Furthermore, one would be able to address other types of inversion transformations, including those studied in [20].

A Bosonic Lagrangian of STU Supergravity in Symmetrical Form
Here, we present the bosonic sector of the STU supergravity Lagrangian, in a form where the four field strengths enter symmetrically. We take the Lagrangian as given in appendix B of [17], except that the field strengths called F + in that paper will be called F − here, to match conveniently with the conventions of Freedman and Van Proeyen [18].
In the notation of [18], but written in the language of differential forms, their eqn (4.66) becomes where f R AB and f I AB denote the real and imaginary parts of f AB (i.e f AB = f R AB + i f I AB ). Now F ≡ −i * F , and * 2 = −1 in four-dimensional spacetime when acting on 2-forms, so * F = i F and hence we have Note that the field strengths F A are simply the exterior derivatives of potentials, F A = dA A .
Note also that f AB is symmetric in A and B.
As in [18] we define F ± = 1 2 (F ± F ), and hence Noting that for any 2-forms X and Y we have we see that the wedge product of any self-dual 2-form with any anti-self-dual 2-form is zero: From the above, it follows that the Lagrangian (A.2) can be written as and hence In this form, we can compare with eqn (B.7) in [17] (with the understanding that the roles of + and -superscripts on F are exchanged as mentioned previously), and hence read off from (B.9) of [17] that the matrix f , with components f AB , is given by β 1 = e ϕ 2 +ϕ 3 χ 2 χ 3 + i e ϕ 1 χ 1 , β 2 = e ϕ 1 +ϕ 3 χ 1 χ 3 + i e ϕ 2 χ 2 , β 3 = e ϕ 1 +ϕ 2 χ 1 χ 2 + i e ϕ 3 χ 3 , The field equations following from (A.2) are dG A = 0, where the 2-forms G A are read off from varying L(F ) with respect to F A : and so It then follows that G ± A ≡ 1 2 (G A ∓ i * G A ) are given by Note that the Bianchi identities dF A = 0 and the field equations dG A = 0 can be written (A.14) (ℜ and ℑ denote the real and imaginary parts.) The Bianchi equations and equations of motion are invariant under the transformations  where A, B, C and D are real constant 4 × 4 matrices, provided that the scalar fields transform appropriately: We have Since the transformed fields must also obey (A.13), this implies that the scalar matrix f must transform according to [18] i It is important that f ′ , like f , must be symmetric and so this implies that the matrices A, B, C and D must obey the relations These are precisely the conditions for the matrix S defined in (A.16) to be an element of Sp(8, R), obeying [18] S T ΩS = Ω , where Ω = The scalar field Lagrangian is and this is invariant under the SL(2, R) 1 × SL(2, R) 2 × SL(2, R) 3 , where the three SL(2, R) act in the standard way: . After some algebra, we find Note that SL(2, R) 3
It is also helpful to note that tan(θ 1 + θ 2 + θ 3 ) = P 1 + P 2 + P 3 + P 4 Q 1 + Q 2 + Q 3 + Q 4 . (B.7) After some algebra, we can now read off the general expressions for the coefficients in (B.3) for the general 8-charge solutions. We find In the expression for γ, we have defined P ij = P i P j + ( k P k )/(P i P j ) and Q ij = Q i Q j + ( k Q k )/(Q i Q j ) (so P 12 = P 1 P 2 + P 3 P 4 , etc.).
Note that ∆ in ( where V =r 4 +αr 3 +βr 2 +γr + ∆, will be in the same class of black hole metrics provided that there exists a mapping of the charges such that