Exact Solutions for Extreme Black Hole Magnetospheres

We present new exact solutions of Force-Free Electrodynamics (FFE) in the Near-Horizon region of an Extremal Kerr black hole (NHEK) and offer a complete classification of the subset that form highest-weight representations of the spacetime's SL(2,R) x U(1) isometry group. For a natural choice of spacetime embedding of this isometry group, the SL(2,R) highest-weight conditions lead to stationary solutions with non-trivial angular dependence, as well as axisymmetry when the U(1)-charge vanishes. In addition, we unveil a hidden SL(2,C) symmetry of the equations of FFE that stems from the action of a complex automorphism group, and enables us to generate an SL(2,C) family of (generically time-dependent) solutions. We then obtain still more general solutions with less symmetry by appealing to a principle of linear superposition that holds for solutions with collinear currents and allows us to resum the highest-weight primaries and their SL(2,R)-descendants.


Introduction and Summary
The electromagnetic field inside a magnetically-dominated relativistic plasma is generically described by the nonlinear equations of Force-Free Electrodynamics (FFE). These equations are believed to capture the behavior of a wide range of astrophysical systems, ranging from neutron stars to active black holes [1,2]. Force-free magnetospheres typically surround these objects, powering some of the most extravagant energy signals in the universe: pulsars (in the case of neutron stars) and active galactic nuclei, or quasars (in the case of black holes). FFE is expected to play a crucial role in efforts to elucidate the origin of such signals.
Though the significance of FFE to our understanding of magnetospheres was recognized several decades ago, it is only recently that the subject has attracted broader interest beyond the astrophysics community. The equations of FFE naturally arise in a variety of settings, from the modeling of the magnetic fields of our own Sun [3] to the realization last year of a force-free plasma in a laboratory on Earth [4]. Because the equations of FFE characterize ubiquitous physical phenomena, we hold that they should join the list of nonlinear equations considered fundamental in physics, a list that already includes the Einstein equations of gravity and the Navier-Stokes equations of fluid dynamics. In this spirit, the theory of FFE certainly merits investigation in its own right.
Few analytic solutions of FFE have been found [1,5,2,6,7,8,9,10,11]. Recently [12], a symmetrybased approach to FFE led to the discovery of several infinite families of force-free solutions in the Near-Horizon Region of Extreme Kerr black holes (NHEK). This geometry, which forms a spacetime in its own right (albeit a non-asymptotically flat one), presents an excellent opportunity to develop such an approach because it enjoys an enlarged isometry group: in NHEK, the U(1) time symmetry of the Kerr black hole is enhanced to an SL(2, R) conformal group [13], a symmetry group whose exploitation has already proved fruitful in the context of the Kerr/CFT correspondence [14,15].
In this paper, we pursue the line of investigation opened in [12] and apply the symmetry-based approach to FFE in the NHEK region more systematically. Along the way, we begin to uncover the beautiful structure of the force-free equations and develop three new solution-generating techniques: • First, one may apply a nonlinear transformation to known force-free solutions to obtain new solutions with current flowing in new directions. This mechanism, which we describe in section 5, relies on the nonlinear superposition of the purely magnetic and purely electric solutions derived in sections 3 and 4, and yields electromagnetic solutions with current flowing in the θ-direction, J θ = 0.
• Second, one may consider the transformation properties of the solutions under SL(2, C), the complexification of the SL(2, R) isometry group. While the action of SL(2, R) does not provide new solutions (but rather produces solutions related by finite SL(2, R) transformations), the complex SL(2, C) transformations yield new solutions that are related to the original ones by complex diffeomorphisms, and are therefore physically inequivalent. We work out many of the details in section 6.
• Third, one may exploit the previously-observed [12] phenomenon of linear superposition within certain infinite families of solutions to obtain more general solutions with less symmetry. We explain the origin of this phenomenon in appendix B and apply it to SL(2, R)-descendants in section 7. The SL(2, R) highest-weight solutions and their descendants may be construed as a mode expansion in conformal harmonics, akin to a spherical expansion, indicating the existence of solutions with full functional freedom, which we finally present in section 8.
These techniques enable us to derive the large class of solutions presented in Table 1, including some that describe stationary axisymmetric magnetospheres with F 2 > 0.
The outline of the paper is as follows. In section 2, we briefly review the geometry and symmetries of the NHEK region. Readers who are familiar with these details and are mostly interested in explicit solutions may skip this section. One key difference with [12] is a change of the spacetime embedding of the isometry group. We now adopt a more natural choice from the perspective of the Poincaré observer (which is the one descended from the Boyer-Lindquist observer in the scaling limit), and obtain precisely three families of solutions: a purely Magnetic Type, a purely Electric Type, and a mixed Electromagnetic Type. These are quickly derived from the ground up in Sections 3, 4 and 5, respectively. They all exhibit a complicated θ-dependence that we analyze in appendix A.
Subsequently, in section 6 we revisit the choice of embedding, and determine that it is paramaterized by a complexified SL(2, C) isometry group. One of its SL(2, R) subgroups is merely part of the isometry group and therefore acts trivially on the solutions, but the other "hidden" component of SL(2, C) generates highly nontrivial time-dependent solutions.
Vector potential A Properties 1 2JΓ

Stationary Axisymmetric
Same SL(2, C)-transformed analogues of the above solutions (all time-dependent) After this generalization, the three aforementioned families no longer have a definite type, and are more appropriately termed Type M, Type E and Type EM, respectively. As a helpful example, we show that for certain values of the hidden SL(2, C) parameters, the Type E solutions reduce to the family of non-null solutions presented in [12]. The full classification of highest-weight solutions, labeled by SL(2, C) parameters, is presented in Table 2.
In section 7, we investigate SL(2, R)-descendants and determine the conditions under which they may admit a principle of linear superposition. Invoking the theorems derived in Appendix B, we obtain a quick criterion for resummability, and then form the most general linear combinations of solutions. These are in general complex, but nonetheless their real and imaginary parts are often still solutions due to the collinearity of their currents. We can then list the most general physical solutions amenable to discovery through our symmetry-based approach, be they stationary or time-dependent, axisymmetric or not. Table 1 summarizes the outcome of this procedure.
The real force-free solutions we obtain contain at most two free functions, compared with the four independent functions required to specify initial data for the Cauchy problem in FFE. The initial data consists of six components of the electric and magnetic fields E, B, minus two degrees of freedom removed by the constraints ∇ · B = 0 and E · B = 0 [16,17].
Interestingly, we find that several of these solutions contain free functions on an AdS 2 subfactor of NHEK (which may be viewed as warped fibration of WAdS 3 over S 1 ). Only one of them is null (F 2 = 0), so it should therefore reproduce the null solutions previously found in [12] -we check this is indeed the case in section 8.3. Finally in section 8.4, we point out that though many of these solutions exhibit radial singularities at the horizon of NHEK, these can be pushed beyond the horizon by a conformal transformation from NHEK to near-NHEK, the geometry of the near-horizon near-extremal extremal Kerr black hole.

NHEK geometry
In this section we briefly review the Kerr black hole, as well as the geometry and symmetries of the NHEK region. We also lay out the symmetry approach to the force-free equations in NHEK.

Kerr black hole
The Kerr metric models astrophysically realistic rotating black holes. In Boyer-Lindquist coordinates (t,r,θ,φ) with natural units (we set c = G = 1), its line element is where J parametrizes the angular momentum of the black hole, M its mass, and There is an event horizon atr from which it follows that the Kerr solution has a naked singularity unless |a| ≤ M . This last bound is saturated by the so-called extreme Kerr solution, which carries the maximal angular momentum As long as J > 0, the Kerr black hole is spinning and develops an ergosphere, which is bounded by the hypersurfacer 2.2 NHEK region: the scaling limit In this paper we are interested in the region very close to the horizon of extreme Kerr. It is described by the so-called Near-Horizon Extreme Kerr (NHEK) geometry, which can be obtained by a near-horizon limiting procedure from the Kerr metric in usual Boyer-Lindquist coordinates (2.1). Following [13], define new dimensionless coordinates (t, r, θ, φ) by (2.6) In taking the limit λ → 0 while keeping these coordinates fixed, one is effectively "zooming" into the region near the horizon. This procedure yields the NHEK line element in Poincaré coordinates where t ∈ (−∞, ∞), r ∈ [0, ∞), θ ∈ [0, π], φ ∼ φ + 2π and Γ(θ) = 1 + cos 2 θ 2 , Λ(θ) = 2 sin θ 1 + cos 2 θ . (2.8) This metric may be viewed as a warped fibration (warped by Λ) of WAdS 3 over the S 1 parameterized by θ, with the (t, r) coordinates forming an AdS 2 subfactor. In contrast with the original Kerr metric (2.1), the NHEK geometry is not asymptotically flat. The event horizon of the original extreme Kerr black hole is now located at The boundary of the ergosphere reduces to two hypersurfaces of constant θ = θ ± e , with This boundary is an AdS 3 subfactor of NHEK.

Isometries of NHEK
Under the scaling limit described in the previous section, the original U(1) × U(1) Kerr isometry group is enlarged to an SL(2, R) × U(1) symmetry that governs the dynamics of the NHEK region. The U(1) rotational symmetry is generated by the Killing vector field The time translation symmetry becomes part of an enhanced SL(2, R) isometry group generated by the Killing vector fields (2.14) It is easily verified that these satisfy the SL(2, R) × U(1) commutation relations, namely: These symmetries do not leave the original Kerr horizon (2.9) invariant and mix up the inside and outside of the original black hole. Note that this parameterization of SL(2, R) is natural from the perspective of the Poincaré observer, and differs from the global basis {L 0 , L ± } employed in [12]we will return to this crucial point in section 6. Here and hereafter, it is understood that in this paper we use the same symbol (e.g. 'H+') to denote both a vector field and its associated 1-form, and rely on the context to distinguish between the two uses.

FFE in NHEK
We wish to solve the force-free equations for a 2-form F in the background of NHEK: To exploit the isometries of the NHEK region, we focus on solutions that lie in highest-weight representations of SL(2, R) and carry U(1)-charge. That is, we assume that (2.22) The most general 2-form satisfying these conditions is best expressed in tensor notation as (2.23) It is an angular momentum eigenstate with angular momentum m, and forms a highest-weight representation of SL(2, R) with highest weight h. Evidently, we must require that m ∈ Z to ensure that F is not multi-valued. This group-theoretic Ansatz for F completely fixes its dependence on the 3 coordinates (t, r, φ), thereby leaving it undetermined only up to six arbitrary functions of θ. Imposing the Bianchi identity (2.17), eliminates half of these functions: Thus, before even plugging this Ansatz into the force-free equations proper, F is already fixed up to three scalar functions of θ. Furthermore, recall that a 2-form F that solves the force-free equations (2.17)-(2.19) must necessarily be degenerate [10], in the sense that For this Ansatz with m = 0, this degeneracy condition takes the explicit form Schematically, this condition may be satisfied in one of three ways: • C(θ) = 0 -this choice entirely kills off the electric field and corresponds to the Magnetic Type solutions we will derive in section 3.
• Y (θ) = Z(θ) = 0 -this choice entirely kills off the magnetic field and corresponds to the Electric Type solutions we will derive in section 4.
• C(θ) = 0 and Y (θ) = 0 with which leads to a solution with both electric and magnetic fields turned on, corresponding to the Electromagnetic Type solutions we will derive in section 5.
Intuitively, the degeneracy condition F ∧ F = 0 states that the electric and magnetic fields are orthogonal, E · B = 0. As such, this precursor to the force-free equations may be satisfied by either setting E = 0, or B = 0, or E ⊥ B, thereby selecting one of the three cases listed above.

Derivation of Magnetic Type solutions
NHEK has an SL(2, R) × U(1) isometry group. It is natural to classify solutions to the force-free equations by their transformation properties under these symmetries.
The simplest possibility consists of a solution that is invariant under the action of SL(2, R) × U(1).
That is, suppose that in addition to the force-free equations (2.17)-(2.19), the 2-form F also satisfies There are several such solutions, though only one which is nonsingular and has J θ = 0. We defer its examination for the moment (as it will form the starting point of our later derivation of the Electric Type solutions in section 4) and consider instead the gauge potential which is fixed up to an arbitrary function S 0 (θ). As we will prove in section 3.4, this Ansatz solves the force-free equations (2.17)-(2.19) provided that S 0 (θ) satisfies the following second-order ODE: This equation has two solutions, one of which is badly singular in the range θ ∈ [0, π]. The other solution is a constant, which produces a vanishing current and therefore yields a trivial solution to the force-free equations, J = 0. We may express this gauge potential in terms of the isometries as where we introduced a scalar function Note that Φ h (r) is U(1)-invariant and forms a scalar highest-weight representation of SL(2, R) with highest weight h: We now relax the requirement of SL(2, R)-invariance. Instead, it is natural to search for solutions that form highest-weight representations of SL(2, R) with some highest weight h. That is, assume that in addition to the force-free equations (2.17)-(2.19), the 2-form F also satisfies the conditions for some h ∈ R. Consider as an Ansatz the gauge potential whose dependence on the coordinates has been split between two arbitrary functions X(t, r, φ) and S h (θ). Imposing the 3 conditions (3.9)-(3.11) entirely determines the scalar prefactor X to be Our Ansatz for the gauge potential then becomes As we will prove in section 3.4, this Ansatz solves the force-free equations (2.17)-(2.19) provided that S h (θ) satisfies the following second-order ODE: . For the special values h = 0 and h = 1, this equation for S h (θ) admits a constant solution and the gauge field becomes θ-independent. Observe also when h = 0, it reduces to the SL(2, R) × U(1)-invariant case considered in the previous section. The solution S h (θ) is given in appendix A.

SL(2, R) highest-weight U(1)-eigenstate solution
We next relax the requirement of U(1)-invariance by allowing the gauge field to carry a U(1) charge.
Suppose that in addition to the force-free equations (2.17)-(2.19), the 2-form F also satisfies where h, m ∈ R. As before, we could use as an Ansatz the SL(2, R) × U(1)-invariant gauge potential with a yet-to-be-determined scalar prefactorX(t, r, φ): However, it is not possible to satisfy all our conditions with a gauge field of this form, and an additional term dependent on Θ = 2JΓ dθ is needed. Instead, we consider the Ansatz whose dependence on the coordinates has been split between an arbitrary functionX(t, r, φ) and two arbitrary functions S h,m (θ) andS h,m (θ). Imposing the 3 conditions (2.20)-(2.22) entirely determines the scalar prefactorX to beX where we introduced a new scalar function (3.26) As we will prove in section 3.4, this Ansatz solves the force-free equations (2.17)-(2.19) provided that we defineS and that S h,m (θ) satisfies the following second-order ODE: This equation has the same symmetries as

Verification of Magnetic Type solutions
We now wish to prove that the Magnetic Type solutions examined above do indeed solve the forcefree equations (2.17)- (2.19). It suffices to check that One may then check that This current is only null when Λ(θ) = 1, corresponding to the boundary of the ergosphere (2.10): which proves that this is indeed a solution to (2.17)-(2.19). Note that for this (complex) solution, F 2 > 0 outside the ergosphere, where Λ 2 < 1:

Derivation of Electric Type solutions
We now turn to the derivation of the Electric Type solutions. Following the same approach as in the previous section, we attempt to classify solutions to the force-free equations by their transformation properties under the SL(2, R) × U(1) isometry group of NHEK.

SL(2, R) × U(1)-invariant solution
As before, we start with the simplest possibility, which consists of a solution that is invariant under the action of SL(2, R) × U(1). That is, suppose that in addition to the force-free equations (2.17)-(2.19), the 2-form F also satisfies The only such solution that is nonsingular and has J θ = 0 is obtained from the Ansatz which is fixed up to an arbitrary function P 0 (θ). As we will prove in section 4.3, this Ansatz solves the force-free equations (2.17)-(2.19) provided that P 0 (θ) satisfies the following second-order ODE: This equation has two solutions, one of which diverges at the poles θ ∈ {0, π}. The other solution is a constant, and it leads to an interesting solution to the force-free equations with J = 0. We may express this gauge potential in terms of the isometries and the scalar Φ as We now relax the requirement of SL(2, R)-invariance and search for solutions that form highestweight representations of SL(2, R) with some highest weight h. That is, assume that in addition to the force-free equations (2.17)-(2.19), the 2-form F also satisfies the conditions (2.20)-(2.22) for some h ∈ R. Consider as an Ansatz the gauge potential whose dependence on the coordinates has again been split between two arbitrary function X(t, r, φ) and P h (θ). As before, imposing the 3 conditions (2.20)-(2.22) entirely determines the scalar prefactor X to be Our Ansatz for the gauge potential then becomes As we will prove in section 4.3, this Ansatz solves the force-free equations (2.17)-(2.19) provided that P h (θ) satisfies the following second-order ODE: For the special values h = 0 and h = 1, this equation for P h (θ) admits a constant solution and the gauge field becomes θ-independent. Observe also when h = 0, it reduces to the SL(2, R) × U(1)-invariant case considered in the previous section. The solution P h (θ) is given in appendix A.

Verification of Electric
(4.10) One may then check that The associated current This current is always spacelike: which proves that this is indeed a solution to (2.17)-(2.19). Note that for this (complex) solution, F 2 < 0 everywhere, as expected of a purely electric solution: We next relax the requirement of U(1)-invariance by allowing the gauge field to carry a U(1) charge.
That is, suppose that in addition to the force-free equations (2.17)-(2.19), the 2-form F also satisfies (2.20)-(2.22) for some h, m ∈ R. Consider as an Ansatz the gauge potential whose dependence on the coordinates has again been split between two arbitrary functionsX(t, r, φ) and P h,m (θ). As before, imposing the 3 conditions (2.20)-(2.22) entirely determines the scalar prefactorX to beX Our Ansatz for the gauge potential then becomes In this particular case, the gauge potential reduces to One may then check that Using the explicit formula (4.21), this further simplifies to Likewise, using (4.21), the associated current which proves that this is indeed a solution to (2.17)-(2.19). Note that for this (complex) solution, F 2 = 0 everywhere -this is related to the condition that the current is null.

h = 0 case
In this particular case, the gauge potential reduces to One may then check that Using the explicit formula (4.21), this further simplifies to Likewise, using (4.21), the associated current which is always spacelike: Since ω ∧ ω = 0 for any form ω, it is evident that which proves that this is indeed a solution to (2.17)-(2.19). Note that for this (complex) solution, F 2 < 0 everywhere, as expected of a purely electric solution:

Derivation of Electromagnetic Type solutions
We now turn to the derivation of the Electromagnetic Type solutions, which also share the same transformation properties under the SL(2, R) × U(1) isometry group of NHEK.
It is quite difficult to obtain these solutions directly. However, one may construct them as nonlinear superpositions of the purely electric and purely magnetic solutions. This mechanism yields solutions with current flowing in the θ-direction, J θ = 0.

SL(2, R) highest-weight U(1)-invariant solution
In the foregoing discussion, we derived two SL(2, R) highest-weight U(1)-invariant solutions to the force-free equations, one of which was purely magnetic, and another which was purely electric, In free Maxwell electrodynamics, where the equations are linear, one could then obtain electromagnetic solutions by forming linear combinations of these potentials. However, this cannot work in the context of force-free electrodynamics, where the equations are nonlinear. Nonetheless, we can explore the possibility of nonlinear superpositions. To that end, consider as an Ansatz the potential where the functions U h (θ), V h (θ) need not be identical to the functions P h (θ), S h (θ) introduced in sections 3 and 4. Since they are (as of now) unconstrained functions of θ, this is not a linear combination, but instead a truly nonlinear superposition (with nonlinearity in θ). More generally, one could consider including an additional factor in the θ-direction; that is, one could Ansatz where the dependence of the last term on the coordinates (t, r, φ) is fixed to be Φ h (r) in order to One may then check that The associated current J EM where κ(θ) and λ(θ) are given by This current is not null: Note that for this (real) solution, F 2 = 0: For this to be a solution to the force-free equations, we must now impose the force-free condition This relation implies the weaker degeneracy condition F ∧ F = 0, from which we immediately see that we must require for some constant D. (It is especially nice that we can eliminate V h (θ) from a purely algebraic condition.) For the t component of the force-free equations to be satisfied (in order to ensure F µt J µ = 0), we must then demand that for some constant C. Using (5.14) and (5.15), the force-free equations (2.17)-(2.19) may be satisfied provided that U h (θ) satisfies the following second-order nonlinear ODE: The solution can be succinctly written as with U h defined by (5.16). This is likely related to the solution recently published in [11]. Recall that we derived two SL(2, R) highest-weight U(1)-eigenstate solutions to the force-free equations, one of which was purely magnetic, and another which was purely electric, where the dependence of the additional Θ factor on the coordinates (t, r, φ) is fixed to be Φ h (r)Ψ m (φ) in order to ensure that the potential A EM h,m (U h,m , V h,m , T h,m ) is still an SL(2, R) highest-weight U(1)-eigenstate. Again, note that this is a nonlinear superposition in θ because the functions U h,m (θ), V h,m (θ) need not be identical to the functions P h,m (θ), S h,m (θ) introduced in sections 3 and 4. Following the same procedure as before, we obtain the Ansatz with χ h,m (θ) as in (3.28). As we will prove momentarily, this Ansatz solves the force-free equations (2.17)-(2.19) provided that U h,m (θ) satisfies the following second-order ODE: When m = 0, A EM h,m (U h,m ) reduces to the purely electric U(1)-invariant solution A E h (S h ) discussed previously, but when the U(1) charge is non-vanishing (m = 0), the solution is no longer strictly electric. The corresponding 2-form gauge connection F EM h,m = dA EM h,m is One may then check that which is not null: Though it is not quite obvious, one can explicitly check that which proves that this is indeed a solution to (2.17)-(2.19). In general, for this (complex) solution F 2 < 0. However, outside the ergoregion and when m is large, it is possible to achieve F 2 > 0: 6 Hidden SL(2, C) symmetry Let us now reexamine one of our solutions, such as the general one of Magnetic Type, It is a striking fact that every part of this gauge potential is entirely determined by the Killing vectors {H 0 , H ± , W 0 }. In particular, the scalar functions multiplying these vector fields are themselves defined by their transformation properties under the isometries. For instance, Φ h (r) is obtained by demanding that it lie in a scalar highest-weight representation of SL(2, R) of highest weight h.
However, the Killing vectors {H 0 , H ± , W 0 } only form a particular embedding of the SL(2, R) × U(1) Lie algebra into the NHEK spacetime. This suggests that a different spacetime embedding, which may lead to a different gauge potential, would still produce a force-free solution.
In this section, we will examine all such spacetime embeddings. The ones related to the Poincaré basis {H 0 , H ± , W 0 } by an SL(2, R) transformation will generate physically equivalent solutions obtainable by simple coordinate transformations (finite SL(2, R) isometries), but those obtained from the action of SL(2, C), the complexification of SL(2, R), will produce new solutions. In other words, we will prove the existence of a hidden symmetry of FFE and exploit it to generate new solutions.

Automorphism group of SL(2, R) × U(1)
To explore this idea, we need to find all the possible embeddings of the SL(2, R) × U(1) Lie algebra into the NHEK spacetime. In other words, we wish to find all the Killing vector fields {H 0 ,H ± ,W 0 } that satisfy the SL(2, R) × U(1) commutation relations, We already know one such embedding, {H 0 , H ± , W 0 }. All other embeddings are by definition related to it by an automorphism of the SL(2, R) × U(1) Lie algebra. Since SL(2, R) × U(1) is its own automorphism group (because it admits no outer automorphisms), these other embeddings can be parameterized by four SL(2, R) × U(1) parameters {α, β, γ, δ} as follows: 3)
For real parameters α, β ∈ R this transformation is equivalent to the coordinate transformation 2r , (6.14) The transformation {t, r, θ, φ} → {t , r , θ, φ } leaves the form of the metric unchanged, and is therefore an isometry † . To be more exact, it is a "finite SL(2, R) transformation" and as such, it must take solutions of the force-free equations into other (equivalent) solutions.
However, in principle nothing prevents us from allowing instead complex parameters α, β ∈ C. This amounts to considering complex automorphisms of SL(2, R) or, more precisely, automorphisms of its complexification SL(2, C). The force-free equations (2.17)-(2.19) will still be satisfied (they are in some sense holomorphic), albeit by complex gauge potentials. These new solutions will be truly different from the original ones, as they will be related by a complex diffeomorphism.
We have thus found a hidden symmetry, stemming from the complex part of SL(2, C). To determine what it is, recall that SL(2, C) ∼ = SL(2, R) × SL(2, R). (6.16) * For reasons that will become clear in section 6.3, the dependence on δ dropped out. The dependence on γ also dropped out naturally because e γH + is γ-independent. Thus the U(1) subgroup of SL(2, R) generated by γ acts trivially, a point we revisit in greater detail in section 6.3.
Modding out by the real SL(2, R) automorphisms (because they generate physically equivalent solutions) then leaves us with SL(2, C)/SL(2, R) ∼ = SL(2, R). (6.17) However, we should be a bit more careful because, as we will show momentarily, this SL(2, R) has a U(1) subgroup generated by γ that induces trivial dilations. When the three parameters {α, β, γ} of SL(2, R) are complexified to six parameters {α R , α I , β R , β I , γ R , γ I } of SL(2, C), the complex parameter γ C = γ R + iγ I will still generate a trivial dilation, though this time it will also multiply by a phase in addition to the rescaling factor. As such, we must remove an additional trivial generator, corresponding to the trivial complex U(1) phase. We are then left with an additional SL(2, R)/U(1) symmetry group.
In conclusion, our fundamental solutions depend on two labels (h, m) that indicate which representation of SL(2, R) × U(1) they lie in. In addition, they also depend on two parameters α, β ∈ C that generate an extra SL(2, R)/U(1) symmetry. Therefore, the solutions exhibit an enlarged SL(2, C) symmetry group, . " (6.18)

Most general highest-weight solutions
Let us revisit the earlier example of We are now ready to perform an SL(2, C) automorphism to obtain a new gauge potential and that we couple the SL(2, R) and U(1) actions by letting But now observe that the transformation generated by γ has a trivial action: This is just a rescaling of the gauge potential by a constant, and we are free to counter it by multiplying through by δ −1 = e γ . This results iñ Table 2: Complete classification of vector potentialsÃ that define solutionsF = dÃ to the forcefree equations (6.4)-(6.6) that form representations of the SL(2, R) × U(1) isometry group of NHEK. In every entry, it is understood that γ = 0 inH + .
We now understand the benefit of working with the {α, β, γ} parameterization of the SL(2, R): it renders manifest the fact that SL(2, R) has a U(1) subgroup (generated by γ) whose action is a trivial dilation. We are free to ignore it, leaving us with a nontrivial SL(2, R)/U(1) action parameterized by α and β. It may seem surprising that the U(1) isometry parameterized by δ does not produce new solutions and that δ must instead be related to an SL(2, R) parameter. The reason for this is that "finite U(1) transformations" are really the angular shifts φ → φ = φ + C. Observe that ∂ φ = ∂ φ , so W 0 is invariant under this transformation. This symmetry does appear in (6.12), where it allows us to matchΨ to Ψ through the normalization factor C.
As a final note, we find by the procedure (6.7) described in the previous section solutionsÃ M , A E andÃ EM . These will no longer be purely magnetic, electric or electromagnetic, so instead we shall henceforth refer to them as Type E, Type M and Type EM, respectively. These three families of solutions are labeled by SL(2, C) parameters and exhaust the space of highest-weight force-free solutions. They are displayed in Table 2.

A helpful example
Consider as another example the SL(2, C) transformation obtained by setting which sends the real basis {H 0 , H ± } of SL(2, R) into a complex basis where {L 0 , L ± } is precisely the global basis of SL(2, R) used in our previous work [12], It also sends Φ h Ψ m , a scalar highest-weight representation of SL(2, R) with respect to the {H 0 , H ± } basis, intoΦ hΨm , a new scalar highest-weight representation of SL(2, R) with respect to {L 0 , L ± }: (6.28) Knowing this, we can form the analogue of the axisymmetric Electric Type solution, in this new basis of SL(2, R) by following the procedure (6.7) -the result is Recalling the transformation of the NHEK geometry to global coordinates, we see that in global coordinates (τ, ψ, θ, ϕ),Φ becomes Φ(τ, ψ) = 2e −iτ sin ψ, (6.32) which agrees with the definition ofΦ(τ, ψ) in [12] up to a factor of 2 that cancels the extra factor in (6.30). Thus (6.30) explicitly matches the solution described in [12]. Its physical properties, which we analyzed therein, are different from the ones of A E h (P h ). In particular, recall from that paper thatÃ E h (P h ) had F 2 = 0, but with oscillating sign. Meanwhile A E h (P h ) has F 2 < 0. Since F 2 is a geometric invariant of U(1) gauge fields, this proves the physical inequivalence of these solutions.

General proof
We wish to demonstrate thatÃ M h,m ,Ã E h,m andÃ EM h,m really solve the force-free equations (2.17)-(2.19) for all values of α, β ∈ C. This is in fact very simple, and we outline the steps of the proof.
First we recall that when α = β = 0,Ã M h,m ,Ã E h,m andÃ EM h,m reduce to the Magnetic, Electric, and Electromagnetic Type solutions A M h,m , A E h,m and A EM h,m , which were proved to solve the force-free equations in sections 3.4, 4.3 and 5, respectively.
Next, we consider α, β ∈ R. Turning on nonzero α and β is equivalent to performing a finite SL(2, R) transformation on these solutions. The resulting gauge potentialsÃ M h,m ,Ã E h,m andÃ EM h,m are therefore related by isometries to solutions of the force-free equations, from which it results that they are also (physically equivalent) solutions.
Finally, we observe that the force-free equations, when considered separately, are all linear. They are therefore insensitive to complexification, and it is in the sense we can call them "holomorphic". As such, a complex finite SL(2, R) transformation will still produce solutions, soÃ M h,m ,Ã E h,m and A EM h,m must still solve the force-free equations even when α, β ∈ C. To be more explicit, let's once again consider the case of the Magnetic Type solutionÃ M h,m . Recall from (3.33) and (3.34) that when α = β = 0, we find that the potentialÃ M h,m = A M h,m leads to The force-free equations are then obviously satisfied because F h,m ∧J h,m ∝ H + ∧H + = 0. Likewise, when α and β are nonzero, we find that the more general potentialÃ M h,m leads tõ The force-free equations are still obviously satisfied because F h,m ∧J h,m ∝H + ∧H + = 0.

SL(2, R)-descendants and resummation of solutions
In this section we examine the SL(2, R)-descendants of the solutions with highest weight (primaries).
In the Poincaré basis, the descendants will no longer be annihilated by H + , so they will acquire a time-dependence. Also, the electric or magnetic character of a primary is generally different than that of its solutions. For instance, the purely electric solutions have electromagnetic descendants.

SL(2, R)-descendants as solutions
First, we need to determine whether SL(2, R)-descendants are also solutions to the force-free equations (2.17)-(2.19), as this is not automatically the case. Given a highest-weight solutionÃ h,m , its k th SL(2, R)-descendent is defined asÃ In other words, descendants are obtained by acting with SL(2, R) infinitesimally. Note that in contrast to the compact group SO (3), which has finitely many descendants that form the spherical harmonics Y ,m (θ, φ), the noncompact SL(2, R) group generates an infinite tower of descendants, so k ∈ Z.
Finite SL(2, R) transformations (isometries) always transform solutions of an equation into other solutions. However, at the infinitesimal level, this is in general true only for linear equations. The force-free equations, being nonlinear, offer no guarantee of having this property and in fact they generally do not. They will nonetheless have it in certain circumstances, which we characterize in appendix B. In particular, when descendants have currentJ h,k,m = L kH

General resummation of solutions
More generally, linear superpositions of force-free solutions with collinear currents also form solutions. This is demonstrated in Theorem 3 of appendix B for sums over SL(2, R)-descendants, and in Theorem 2 for arbitrary solutions. Hence we can infer that the following are solutions: • Linear combinations of Type M primariesÃ M h,m , since these solutions all have collinear currents J M h,m ∝H + . The same is not true of Type M descendants.
• Linear combinations of Type E primaries and their descendantsÃ E h,k,m , since these solutions all have collinear currentsJ E h,k,m ∝ W 0 . If m = 0 then h and k are arbitrary, but if m = 0 then either h = 0 or h = 1 (and k is still arbitrary).
The Type EM solutions (both primaries and descendants) all have non-collinear currents and therefore do not admit a principle of linear superposition.

Real solutions
The solutions we have classified so far have nice representation-theoretic properties, but are not necessarily physical -for instance, many of them stem from complex vector potentials and are therefore physically inadmissible. In this section, we extract real (physical) solutions from the classification given in Table 2 and attempt to construct more general solutions. In particular, several solution subspaces admit a linear superposition principle. It is therefore possible to combine solutions with different highest weights, resulting in general solutions with no definite transformation properties. The resulting form of these solutions is akin to an expansion in "conformal harmonics".

Stationary solutions
The highest-weight condition (2.20) in the Poincaré basis amounts to demanding stationarity. Therefore, in order to obtain stationary solutions, we must only consider highest-weight primaries in this basis, for their SL(2, R)-descendants will yield time-dependent solutions, as will the highestweight towers obtained from other SL(2, R) embeddings.

Stationary axisymmetric solutions
For axisymmetric solutions, we further restrict our attention to U(1)-invariant solutions with m = 0. By taking the real and imaginary parts of A M h,0 and superposing them with different weights h, we obtain a very general stationary and axisymmetric solution of Magnetic Type: where c(h), d(h) are arbitrary real functions, and S h (θ) = S 1−h (θ) is defined by (3.29): The solution S h (θ) is given in appendix A. Because of its symmetry under h → 1 − h, keeping both terms in the integrand is redundant. Without loss of generality, one may set d(h) = 0, leaving as the resummed form of the magnetic primaries. Likewise, we can perform a resummation of the electric primaries, resulting in a very general stationary and axisymmetric solution of Electric Type: where c(h), d(h) are arbitrary real functions, and P h (θ) = P 1−h (θ) is defined by (4.9): The solution P h (θ) is given in appendix A. As for the electromagnetic primaries, they are not linearly compatible and do not admit a superposition principle. The most general real, stationary and axisymmetric solution of Electromagnetic Type is therefore given by (5.17): with U h defined by (5.16).

Stationary non-axisymmetric solutions
By taking the real and imaginary parts of A M h,m and superposing them with different weights (h, m), we obtain a very general stationary non-axisymmetric solution of Magnetic Type: where χ h,m (θ) is defined as in (3.28), Here, c(h), x(m) and y(m) are arbitrary real functions, and S h,m (θ) = S 1−h,m (θ) is defined by (3.29): The solution S h,m (θ) is given in appendix A. As before, the symmetry under h → 1 − h allowed us to eliminate the d(h)r h−1 term. This is the resummed form of the magnetic primaries. Likewise, we can perform a resummation of the electric primaries with h = 1 or h = 0, resulting in and respectively. In both cases, P m is given by (4.21) as In the electromagnetic case with m = 0, we were unable to obtain any real solutions at all because taking the real and imaginary part of J EM h,m changes its direction.

Time-dependent solutions
Time-dependent solutions may be obtained in two ways: either by using SL(2, C) automorphisms, or by acting with H − on the highest-weight primaries to obtain SL(2, R)-descendants which are no longer annihilated by H + and are therefore time-dependent. This section explores the latter method -the automorphisms are then obtained following the procedure (6.7) elaborated in section 6.
First, by taking the real and imaginary parts of the SL(2, R)-descendants of Magnetic Type L k H − A M h,m and superposing them with different weights (h, k), we obtain a very general non-stationary solution, is a completely arbitrary function. Note that this solution is no longer of purely Magnetic Type, since time-varying magnetic fields produce electric fields -instead, A M t describes an electromagnetic field configuration. Also, observe that this solution is still axisymmetric, since descendants with m = 0 are not resummable. This is thus the most general resummed form of the magnetic primaries ‡ . Likewise, we can perform a resummation of the electric descendants, resulting in where P h (θ) = P 1−h (θ) is defined by (4.9): The solution P h (θ) is given in appendix A. Since this equation is only invariant under h → 1 − h, we may only superpose solutions with highest weights h and 1 − h: for some arbitrary real coefficients C and D. As for the electromagnetic descendants, we obtain where C and D are arbitrary real coefficients, while is a completely arbitrary function on the AdS 2 subfactor of NHEK. As a final note, recall from section 7 that the SL(2, R)-descendants of the Type M and Type EM highest-weight solutions do not form solutions. We evaded this limitation and were nonetheless able to obtain solutions A M t and A EM t by modifying the θ-dependence of the complex highest-weight solutions. ‡ Incidentally, this is the only null solution (F 2 = 0) we can obtain in this approach. Unsurprisingly, it is related to the family of null solutions from [12] by the same SL(2, C) automorphism that we already discussed in section 6.4.

Survey of Solutionland
In the previous two sections, we extracted real solutions of the force-free equations from our classification in Table 2 of the highest-weight solutions in the Poincaré basis {H 0 , H ± }. We summarize these results in Table 1.
A similar analysis may be performed for the highest-weight solutions with respect to other bases of SL(2, R), such as the global basis {L 0 , L ± } used in [12] and reviewed in section 6.4. All such solutions will be non-stationary, including the primaries, but they will be of the same form as the Poincaré basis solutions displayed in Table 1.
They can be obtained by following the procedure (6.7) detailed in section 6, and replacing the coordinates (t, r) by their analogues (t , r ) given in (6.13)-(6.14) wherever they appear. As an example, the Poincaré basis solution belongs to an SL(2, R)/U(1) family of solutions parameterized by (α, β), where g t ± 1 r = g − 2r √ 2(1 + αβ) + βt βr ± β 2 (8.22) is in general complex for α, β ∈ C. However, by current collinearity, we may take the real or imaginary part of g and still obtain a real physical solution. The same can be done with the null (F 2 = 0) solution In particular, transforming it to the global basis {L 0 , L ± } of SL(2, R) using the SL(2, C) transformation discussed in section 6.4, it reproduces the family of null solutions first found in [12] provided f t ± 1 r , θ = t ± 1 r P (θ)Λ(θ) √ 2 , withP (θ) an arbitrary function.

Near-Horizon Near-Extremal Kerr black hole (near-NHEK)
Finally, we would like to address the r −h singularities that pervade our force-free solutions in NHEK, as they may seem unphysical. By a conformal transformation, we can map the NHEK geometry to a near-NHEK spacetime describing the near-horizon region of a near-extremal Kerr black hole. This transformation maps our force-free solutions in NHEK to force-free solutions in near-NHEK, and in the process pushes the radial singularity in Φ h (r) = r −h behind the near-NHEK horizon, as illustrated by the Penrose diagram in Figure 1. We defer the investigation of our force-free solutions in near-NHEK to future work. This equation is manifestly invariant under h → 1−h, so P h (θ) = P 1−h (θ). By performing a suitable coordinate transformation to a new variable z = sin 2 θ, we may put it in the form of a generalized Heun equation, namely where α + β + 1 = γ + δ + and q is an accessory parameter. In our case, these parameters are given by γ = 1, δ = 1/2, = −1, α = −h/2, β = (h − 1)/2 and q = 2. The 4 regular singular points of this equation are located at z = z 0 , with z 0 ∈ {0, 1, a = 2, ∞}. The corresponding roots (t 1 , t 2 ) of the indicial equation are (0, 1 − γ), (0, 1 − δ), (0, 1 − ) and (α, β), respectively.
Because there is no other singularity within this interval, the power series solutions obtained from the Frobenius method will converge everywhere on z ∈ [0, 1].
Since γ = 1, the roots are repeated: t 1 = t 2 = 0. Hence only one of the two independent solutions to the equation is nonsingular. Though it has no closed form expression, it may be expanded as P h (θ) = ∞ n=0 a n sin 2n θ, where a n+1 = B n a n + C n a n−1 , (A.4) and B n = 6n 2 − h(h − 1) 4(n + 1) 2 , C n = − (2n − h − 2)(2n + h − 3) 8(n + 1) 2 . (A.5) Per the above discussion, this power series converges everywhere on the domain of interest θ ∈ [0, π]. Moreover, it renders manifest the reflection symmetry of P h about the θ = π/2 plane. Finally, B n and C n are also invariant under h → 1 − h, as expected.

A.2 Analysis of S h,m (θ)
In order to determine the θ-dependence of the highest-weight solutions of Type M, we need to determine the behavior of the function S h,m (θ) and solve (3.29), which may be rewritten as The equations of FFE are nonlinear. As such, while their solutions are of course mapped to other solutions under finite symmetry transformations, this is not in general the case for infinitesimal symmetries. However, it is still sometimes possible, as explained by the following where in the last line we used the assumption that J 1 ∝ J 2 , so that ι J 1 F 2 ∝ ι J 2 F 2 = 0 and ι J 2 F 1 ∝ ι J 1 F 1 = 0.
Theorem 2 explains why the primaries of different towers, which are not related by an infinitesimal symmetry but rather differ by their highest weight, can also be superposed as long as their currents are collinear. But we also need to explain why towers of descendants can be superposed. This is explained by the following Theorem 3: (This is a generalization of Theorem 1.) Suppose that F is a solution to the system of equations (2.17)-(2.19), and that K is a Killing vector. Moreover suppose that L K J ∝ J , or in other words that these currents are collinear: L K J J . Then for any polynomial of arbitrary degree d ∈ N, a n x n , (B.1) the 2-form P (L K ) F also solves (2.17)-(2.19). The proof follows from Theorems 1 and 2.
The upshot of this analysis is that force-free solutions with collinear currents can be linearly superposed.