A Positive-Definite Energy Functional for the Axisymmetric Perturbations

arXiv:2105.12632v1 [gr-qc] 26 May 2021 OIIEDFNT NRYFNTOA FOR FUNCTIONAL ENERGY POSITIVE-DEFINITE A Date H XSMERCPRUBTOSOF PERTURBATIONS AXISYMMETRIC THE a 7 2021. 27, May : aie ee sepce ob nesnilfis tpi dealing, in step first essential an problem. be nonlinear to the with expected ultimately, lin- is (arbi- the level at of to stability earized such respect stability Establishing with dynamical holes perturbations. results black axisymmetric the Kerr-Newman our analyzing rotating) that rapidly in anticipate trarily We useful prove solutions. Sobole linearized may (weighted) the order higher of certain norms positive-definite) bound (conserved, turn, in order that, higher perturba ergies of to smooth well-known sequence sufficiently A correction a for order. tions, generate, the second to and one at allows therefrom perturbations technique induced order mass ADM first the for defined positive-definite (conserved, ergy the provide between program connection this natural particular a In such for systems. program covariant) stability’ (generally ‘linearization so-called the from derived conditions. setup, gauge our and in constraints negatively determined, elliptic entirely dimensional, by is four the metric whereas a Lorentzian map space base-space wave target in Riemannian the values hyperbolic) by (complex their captured curved entirely take are degrees which system ‘dynamical fields, this propagating rel- of the The freedom’ on of metric 3-manifold. Lorentzian quotient 2+1-dimensional of evant a system field to a the coupled to of fields geometries) Our reduction axisymmetric (Hamiltonian) (for equations. familiar equations field the linearized utilizes an the gravitational analysis (coupled, to an of solutions perturbations class a electromagnetic) these for for conservation constru its we functional establish results, energy uniqueness positive-definite the hole a for black (stationary) previously of developed the proofs exploiting identities, By Carter-Robinson charge. electric famous and momentum subex- (but angular large tremal) arbitrarily for allowing holes, black Kerr-Newman Abstract. h soitdlnaie qain r nlzdwt insight with analyzed are equations linearized associated The ICN OCIFADNSAT GUDAPATI NISHANTH AND MONCRIEF VINCENT ERNWA LC HOLES BLACK KERR-NEWMAN ecnie the consider We axisymmetric 1 ierprubtosof perturbations linear , aemap wave en- ) en- ct d d v s - 2 VINCENTMONCRIEFANDNISHANTHGUDAPATI

1. Introduction Impressive observational and experimental evidence has accumulated for the existence of black holes as dynamically stable entities in the Uni- verse. But are these the black holes predicted by general relativity? To conclude that they are would seem to hinge, in large measure, on the success of ongoing mathematical efforts to prove that the purely theo- retical, Einsteinian black holes are, themselves, dynamically stable. A natural first step in this direction would be to establish such stability at the level of linear perturbation theory—a long-standing research program that began with the pioneering work of Regge and Wheeler [65], Vishveshwara [72] and Zerilli [75] for the case of Schwarzschild perturbations and with the discovery, by Teukolsky [70, 71], of a separable wave equation for Kerr perturbations. Subsequently the coupled gravitational and electromagnetic perturbations of (electrically charged but non-rotating) Reissner-Nordström black holes were analyzed by Zerilli through working in a special gauge [76] and by one of us who developed a gauge-independent, Hamiltonian formalism for the perturbative study of such spherically symmetric ‘backgrounds’ [54, 55, 56]. A corresponding treatment of (charged and rotating) Kerr-Newman black holes has, up until now, been lacking. Indeed, as recently as 2006, Brandon Carter could write that the coupled system of electromagnetic and gravitational Kerr-Newman perturbations ‘has so far been found to be entirely intractable’ [16]. Much of the early work on black hole perturbation theory is summarized and extended in interesting ways in the classic monograph by Chandrasekhar [17] which, though it includes an independent derivation of the Reissner-Nordström results, devotes only a few pages to the unsolved, Kerr-Newman problem. The earlier, somewhat formal, ‘mode stability’ analysis for Schwarzschild perturbations has recently been upgraded to a genuine proof of linear stability by Dafermos, Holzegel and Rodnianski [23] and, independently, by Hung, Keller and Wang [48]. On the other hand, much of the recent work on Kerr stability has focused on analyzing the evolution of various, lower spin ‘probe’ fields propagating in given (Kerr) black hole ‘backgrounds’. Important results of this type have been obtained for scalar [1, 24, 25, 31, 32], electromagnetic [2] and wave map [50, 51] fields. The methods employed in the electromagnetic and wave map cases have required that the background black hole be ‘slowly rotating’ in a suitable sense whereas those ultimately developed for scalar field perturbations allow ‘arbitrarily rapid’ rotation (consistent with the preservation of an event horizon). AXISYMMETRIC PERTURBATIONS 3

For the actual gravitational perturbations of Kerr black holes Hol- lands and Wald have emphasized a crucial distinction between the analysis of axisymmetric versus fully non-symmetric metric perturbations that arises primarily because of the suppression of ‘superradiance’ in the axisymmetric case [45]. They have argued that the existence of a conserved, positive definite ‘canonical’ energy functional for axisymmetric, linear perturbations is in fact a necessary condition for Kerr stability. For non-rotating (spherically symmetric) backgrounds, on the other hand, the phenomenon of superradiance (whereby a black hole can absorb negative radiative energy) disappears (unless electro- magnetically charged fields are considered [10]) and the importance of distinguishing between axisymmetric and non-symmetric perturbations is largely dissolved. One of the main results of Refs. [54, 55, 56] was in fact the derivation of a conserved, gauge-invariant, positive definite energy functional for the coupled, dynamical, gravitational and electromagnetic perturbations of Reissner-Nordström black holes. Using totally different (‘Hertz potential’) methods Wald and Prabhu have recently announced that the conserved, ‘canonical’ energy formula for purely gravitational perturbations given by Hollands and Wald in [45] is indeed positive definite when specialized to a Schwarzschild background and they conjecture that a corresponding result should hold for axisymmetric Kerr perturbations [63]. Even for exclusively axisymmetric perturbations, though, a serious obstacle for the construction of a positive definite energy functional for Kerr (or Kerr-Newman) perturbations is the presence of an ‘ergo- region’ lying outside of any (rotating) black hole’s event horizon. This is the region in which the ‘time-translational’ Killing field of the unperturbed (Kerr-Newman) spacetime becomes spacelike and conventional local energy density expressions built from it can lose their definiteness. To a limited extent this shortcoming can be handled by introducing ‘weighted’ energy densities that, by exploiting timelike linear combinations of the ‘time-translational’ and rotational Killing fields of the background, interpolate between positive definite density expressions inside the ergo-region and exterior to it. But this technique does not seem to be capable of treating arbitrarily rapid rotation and, since such energies are not strictly conserved, needs additional, technically intricate, Morawetz type estimates for the extraction of uniform bounds on the fields and their derivatives. By imposing axial symmetry at the outset Dain and his collaborators applied well-known Kaluza-Klein reduction techniques to re-formulate the (fully nonlinear) vacuum field equations as a 2 + 1—dimensional 4 VINCENTMONCRIEFANDNISHANTHGUDAPATI

Einstein—wave map system for which the wave map target space is the hyperbolic plane [27, 28]. In this formulation the scalar wave map variables represent the truly dynamical gravitational wave degrees of freedom whereas the 2 + 1—dimensional Lorentzian metric to which they are coupled is fully determined by gauge conditions and elliptic constraints. After using this setup in elegant ways to study Penrose inequalities and black hole thermodynamics in the axisymmetric case, they linearized their system and applied it to the Kerr black hole stability problem. By utilizing an extension [27] of the classic Brill mass formula [13] for axisymmetric, vacuum spacetimes expressed in terms of the wave map variables they computed the first and second variations of this functional about a Kerr background and derived therefrom a conserved, positive definite energy functional for the linearized, purely gravitational perturbations of an extremal (i.e., maximally rotating) Kerr black hole. A key step in the logic of their derivation was the observation that, for fixed angular momentum (a strictly conserved quantity for axially symmetric evolutions), the extended Brill mass functional is minimized, for Cauchy data containing an apparent horizon, precisely at the initial data for an extremal, Kerr black hole. Through an application of Carter’s remarkable identity [15] (that played a fundamental role in the proof of the uniqueness of the Kerr family among stationary, asymptotically flat, vacuum black holes without naked singularities) they showed, by an explicit calculation, that the second variation of the extended Brill mass density functional was, up to a spatial divergence term, positive definite. Upon discarding the boundary integral that resulted from integrating this density over a Cauchy surface for the black hole’s domain of outer communications (DOC) they arrived at an energy expression for the (axisymmetric) linear perturbations of the extremal Kerr spacetime’s DOC that was both conserved and positive definite. On the other hand, even though the concept of extremality applies equally well to the Reissner-Nordström family (with electrical charge playing a role analogous to that of angular momentum for the Kerr case) no such limitation (to extremal black holes) was needed for the derivation of the earlier results which had been obtained by a somewhat analogous variational calculation. Partly for this reason the authors realized that it should be entirely feasible to remove this limitation in the rotating case and treat sub-extremal (as well as electrically charged) black holes. We present the results of our analysis herein by deriving an explicit, positive definite, conserved energy functional for the axisymmetric (coupled gravitational and electromagnetic) perturbations AXISYMMETRIC PERTURBATIONS 5 of arbitrary sub-extremal Kerr-Newman black holes. While the occurrence of a non-negligible electric charge for a black hole is of doubtful astrophysical significance, sub-extremal holes are certainly more astro- physically significant than extremal ones which, in fact, are thought to be unachievable via realistic natural evolutions, an expectation encoded in the ‘third law’ of black hole mechanics [11]. While it may not be strictly necessary for our program, we have found it very illuminating to appeal to a straightforward modification of the mathematical ‘machinery’ developed long ago for the study of the so-called linearization stability (LS) problem in general relativity [14, 33, 34, 7, 8, 9, 57, 58]. In particular, this technology (which was developed initially for the study of perturbations of spatially compact, ‘cosmological’ spacetimes) provides one with a rather clear understanding of a somewhat mysterious step in the Dain, et al analysis, wherein one multiplies the variations of the Brill energy density by an explicit weight factor that plays, for those authors, its desired role only in the extremal case. As we shall see the natural interpretation of this weight factor is that it serves as (a special case of) an element (C, Z) of the kernel of the adjoint operator of the linearized Einstein constraint map wherein C is the normal and Z the tangential projection (at an arbitrary Cauchy hypersurface for the unperturbed spacetime) of the (asymptotically timelike) Killing field of the background [34, 57, 58]. With this recognition of the significance of such Killing Initial Data Sets (or KIDS as they are now often called) one can remove the limitation to extremality and derive, by methods analogous to those given in [28] combined with Carter’s identity for the wave map variables, an energy functional that is both conserved and positive definite. In fact, by exploiting Robinson’s renowned generalization of Carter’s identity [69] together with the Kaluza-Klein reduced form of the axisymmetric Einstein-Maxwell equations to a still larger 2 + 1—dimensional Einstein—wave map system (now with complex hyperbolic space as the naturally occurring target), one can extend the aforementioned results to cover the coupled, axisymmetric gravitational and electromagnetic perturbations of fully general Kerr-Newman black holes. Although we shall not attempt to fully exploit it here, the recognition of the (spacetime covariant) geometrical significance of the kernel (C, Z) of the relevant adjoint operator allows one also to remove any apparent dependence upon the ‘slicing’ employed for the background spacetime and, in particular, to allow for hypersurfaces of the black hole’s DOC that could, for example, penetrate its (future) event horizon or intercept (future) null infinity (‘Scri’) or perhaps both. In the 6 VINCENTMONCRIEFANDNISHANTHGUDAPATI present paper though we shall, for simplicity, only deal with the Boyer- Lindquist type slicings that, in contrast to the above, are ‘locked down’ at the horizon’s bifurcation two-sphere and at spacelike infinity. These are actual Cauchy surfaces for the DOC’s of interest here and allow for a strictly conserved energy functional whereas energies defined with respect to the more general slicings mentioned above would normally decay through the occurrence of outgoing fluxes at the horizon and at Scri [45]. Another advantage of the use of the LS ‘technology’ is that it shows clearly how to relate the linearized energy expression obtained therefrom to a perturbation of the asymptotically defined ADM mass of the perturbed spacetime which, as we shall see, is necessarily induced at second order from the presence of non-vanishing energy at first order. The absence of such compensating boundary integral expressions in the spatially compact, ‘cosmological’ cases originally considered for the LS problem was what gave rise to the curious phenomenon of linearization instability wherein any linear perturbation with a non-vanishing Killing conserved quantity was shown to be ‘spurious’ in that it could not, even in principle, be extended to higher order [7, 8, 9, 14, 33, 34, 57, 58]. For the spatially non-compact problems of interest here such conserved energy integrals are not, of course, forced to vanish but, when non- vanishing and combined with suitable boundary conditions on the perturbations at the black hole’s event horizon, coerce a corresponding perturbation in the ADM mass at second order. Though we shall focus exclusively on the derivation of this fundamental energy expression herein, there is a well-known technique for generating, for sufficiently smooth perturbations, a sequence of higher order energy expressions by successively Lie differentiating the linearized field equations with respect to the (asymptotically timelike) Killing field of the background, essentially ‘time’ differentiating the unknowns sequentially and evaluating their ‘energies’, and then using the linearized equations to ‘trade’ time derivatives for spatial ones in defining the ultimate, higher order, energy expressions. Though we shall not pursue this strategy in detail herein we shall sketch, in the concluding section, it’s potential application for extracting (higher order) Sobolev type bounds upon the perturbations from the corresponding energy integrals. The derivation of such bounds would serve, through the application of standard Sobolev inequalities, to establish uniform boundedness of the perturbations and their derivatives and will be the subject of a subsequent article. A well-known difficulty in deriving such bounds arises through the natural occurrence of certain ‘weight factors’ in the higher order energies that degenerate at the horizon and AXISYMMETRIC PERTURBATIONS 7 thereby force the need for a more subtle analysis for the extraction of the desired Sobolev estimates. In section 2 we shall begin by focusing on the special case of purely electromagnetic perturbations of a Kerr background spacetime. These have the distinct advantage of allowing a straightforward representation in terms of (electromagnetic) gauge and infinitesimal diffeomorphism- invariant variables that satisfy an elegant system of partial differential equations derived directly from Maxwell’s equations in the axisymmetric case. Even for this problem, however, Robinson’s identity, specialized to the case at hand, is needed to handle the ergo-region difficulties and demonstrate positivity of the resulting, ‘regularized’ energy expression defined therein. By contrast, the linearized wave map variables for the more general Kerr-Newman problem analyzed in sections 3 and 4 are gauge-dependent (since they correspond to the perturbations of non-constant background scalar fields) and accordingly, for the elliptic gauges considered herein, satisfy ‘non-local’ evolution equations incor- porating the linearized lapse and shift variables. While one could have employed a non-elliptic gauge of ‘spacetime harmonic’ type (i.e., the analogue of Lorenz gauge for Maxwell’s equations) this would have significantly enlarged the system to be analyzed and thus the number of evolving variables to be controlled by energy arguments in an ultimate stability analysis. In our setup, however, only the independent dynamical, linearized wave map variables need to be controlled by the energy (and its higher order generalizations). Somewhat remarkably most of the elliptic problems involved in our formulation reduce to 2-dimensional flat space Poisson equations for which the relevant fundamental solution (Green’s function) is explicitly known. Indeed, this is true for all of the elliptic problems in the special case of what we shall call the 2+1—dimensional, maximal slicing gauge condition. For more general gauge conditions (such as 3+1—dimensional maximal slicing) the linear elliptic equation for the perturbed lapse function need not be of this elementary, explicitly solv- able type. The elliptic analysis needed for dealing with the linearized constraints and the imposed gauge conditions is developed in Appen- dices G, H and I while Appendix A presents the Kerr-Newman black hole solutions in the coordinate systems of interest and Appendix C reviews the Hamiltonian formalism for the (axial) symmetry-reduced Einstein—wave map system that is the main object of our study. Ap- pendix B reviews the global Cauchy problem for the linearized field equations specialized to a ‘hyperbolic’ gauge of Lorenz type whereas 8 VINCENTMONCRIEFANDNISHANTHGUDAPATI

Appendix D establishes the equivalence between our Hamiltonian formulation of the ‘twist potential’ wave map variables and the more conventional Lagrangian definition of these fields and Appendix E reviews the charge and angular momentum conservation laws in our formalism. Appendix F introduces the (Weyl-Papapetrou) gauge condition needed to determine our linearized shift field. Appendix J establishes the vanishing of a certain integral invariant the result of which is needed to justify our chosen (Weyl-Papapetrou) gauge condition. Appendix K analyzes maximal slicing gauge conditions in both the 2+1 and 3+1 dimensional sense, whereas Appendix L lays the foundation for relat- ing our formulation of the linearized field equations to that involving the perturbed Weyl tensor. It has seemed advisable to us to relegate some of these more technical discussions to appendices in order not to unduly interrupt the logical flow of the arguments given in the main body of the article. In section 5 we briefly discuss some possible further extensions of our work. In particular, we describe some of the modifications that would be needed for the inclusion of a (positive) cosmological constant and the corresponding derivation of a (conserved, positive definite) energy functional for Kerr-Newman-de Sitter spacetimes. A key point here is that the Robinson identity, which is normally applied to purely electrovacuum problems, only generates, thanks to a favorable sign in one of its terms that vanishes for electrovacuum backgrounds, a new term of positive sign in the presence of a positive cosmological constant. While the remarkable work of Hintz and Vasy has already demonstrated the stability of slowly rotating Kerr-de Sitter black holes with respect to fully nonlinear and non-symmetric perturbations [42] there may be some potential contribution of our approach to the study, at least at linearized level, of rapidly rotating Kerr-de Sitter solutions as well as to their Kerr-Newman-de Sitter generalizations. We propose to pursue this issue in a future work. Though our treatment of the U(1)—symmetric Einstein—wave map formalism is herein limited to linearized equations we remark that work by Choquet Bruhat and one of us applied this same setup (in the vacuum case) to establish the (fully nonlinear) stability of a family of (spatially compact) cosmological models in the temporal direction of cosmological expansion [19]. The future stability of a still different set of vacuum cosmological background solutions was proven, for fully non-symmetric perturbations, by Andersson and one of us by using energies of a (generalized) Bel-Robinson type [5, 6]. Separately, large data global existence for the (nonlinear) equivariant Einstein—wave map system was proven by Andersson, Gudapati and Szeftel [3] by AXISYMMETRIC PERTURBATIONS 9 building on the non-concentration of energy result established by one of us in [36]. An entirely different approach to Kerr mode stability, made possible through Whiting’s remarkable transformation of the Teukolsky equation [74], has recently been further developed by Andersson, Ma, Paganini and Whiting [4]. We also briefly discuss, in the concluding section, the potential application of our approach to the study of black holes in higher than 4 spacetime dimensions. It has long been realized, for example, that when an n—2 dimensional, commutative, spacelike isometry group is imposed upon the solutions of the Einstein [52] or Einstein-Maxwell [49] equations in n + 1 dimensions (with n> 3), these systems can be reduced, àla Kaluza-Klein, to another wave map system coupled to a Lorentzian 3—metric. In fact stationary black holes and more general black objects, at least in the vacuum, analytic case, can be proven to automatically admit such toroidal isometry groups when the associated angular momentum parameters are non-vanishing [60, 44]. Fur- thermore, generalizations of the Carter and Robinson identities have been systematically derived for the proofs of corresponding black hole uniqueness theorems [43, 46, 47, 21]. Thus all of the needed ‘machinery’ for the extension of our results to such higher dimensional problems seems already to be available. On the other hand, as pointed out by Hollands and Ishibashi, such a high dimensional toroidal isometry group is compatible with asymptotic flatness (in the standard sense n 1 for spacetimes with a well-defined ‘Scri’ diffeomorphic to S − R) only in 4 and 5 spacetime dimensions [43]. But the stability of the× famous 5-dimensional Myers-Perry rotating black hole solution [62] (and its electrovacuum generalization [41]) is an important unsolved mathematical problem whereas the instability of still higher dimensional, rotating black objects has, to a considerable extent, been established [30]. Thus we conjecture that our methods can be applied to shed light n 2 on these open questions at least for perturbations preserving the T − ‘axial’ isometry group of the chosen, axi-symmetric background. We propose to investigate this in detail in future work. The senior author is especially grateful to Abraham Taub for his penetrating insights on the relationship between variational methods and gravitational conservation laws, to Jerrold Marsden for his deep understanding of linearization stability problems and their geometrical significance and to Sergio Dain for his insightful work on black hole perturbations that significantly influenced the present article. This article is dedicated to their memories. 10 VINCENT MONCRIEF AND NISHANTH GUDAPATI

2. Pure Electromagnetic Perturbations of Kerr Spacetimes As is well-known and easily seen, linearization of the Einstein-Maxwell equations about an arbitrary, vacuum solution leads to a decoupled system of perturbation equations of which the electromagnetic component consists simply of Maxwell’s field equations formulated on the chosen (vacuum) background. The corresponding linearized Einstein component for the metric perturbation is homogeneous in this approximation and thus always compatible with taking the metric perturbation to vanish identically. Specializing the background to be a Kerr, black hole spacetime and demanding, for simplicity, that the metric perturbation be trivial we thus arrive at the important special case of analyzing Maxwell’s equations on a given Kerr background. With this aim in mind it is natural to look for a conserved, positive definite energy functional for Maxwell fields on the domain of outer communications (DOC) of an arbitrary Kerr black hole. As far as we know however, no such energy functional has heretofore been constructed, even for the case of purely axisymmetric perturbations, thanks to the well-known difficulties presented by the ergo-region that always surrounds a (rotating) black hole. Thus the solution to this problem that we present here (for the axisymmetric case) may be of interest in its own right as well as providing an example, in a somewhat simpler setting, of the full linearized Kerr-Newman energy functional construction that is the main aim of this paper. While one could simply specialize our comprehensive, Kerr-Newman construction to the case at hand it will perhaps prove more illuminating to start ‘from scratch’ and derive the pure Maxwell energy functional from first principles, leaving its reconciliation with our general, Kerr-Newman results until later (c.f., the discussion at the end of Appendix G). The action for electromagnetic fields on an arbitrary, 3+1-dimensional, globally hyperbolic spacetime M˜ R, (4)g , with M˜ a smooth, connected 3-manifold, is given, in Hamiltonian{ × } form (c.f. Eqs. (C.7–C.10) by:

Maxwell 4 i′ Max i Max i′ (2.1) I := d x A′ N N A′ Ω { i E,t − H − Hi − 0 E,i } ZΩ where 1 g ′ ′ ′ ′ (2.2) Max := ij ( i j + i j ) H 2 µ(3)g E E B B and ′ ′ (2.3) Max := ǫ j k . Hi − ijkE B AXISYMMETRIC PERTURBATIONS 11

Here the Lorentzian metric (4)g has been expressed in ADM (Arnowitt, Deser, Misner) form (c.f., Eq. (C.3)) and, to ensure convergence, the integral has been restricted to an arbitrary compact domain, Ω M˜ R, having a piecewise smooth boundary. The ‘primes’ attached⊂ to the× Maxwell fields, superfluous for the moment, are intended to signify that we regard these as linear perturbations of an identically vanishing background. We now specialize M˜ R, (4)g to be the domain of outer communications of an arbitrary,{ × rotating} Kerr black hole and constrain the (perturbative) Maxwell fields under consideration to be axisymmetric and thus, relative to the coordinate systems discussed in Appendix A, to satisfy

∂ ∂ i′ ∂ i′ (2.4) A′ = = =0. ∂ϕ µ ∂ϕE ∂ϕB ∂ ∂ Here ψ = ∂ϕ together with ζ = ∂t are the axial and time translational Killing fields of the general Kerr solution and, as elaborated in Appen- dix C, it is natural to pass to the quotient space for the circle action generated by ψ and to formulate the Maxwell equations on the base manifold (with boundary) (2.5) V/U(1) = R M × b defined therein. Maxwell Variation of IΩ with respect to A0′ leads immediately to the i′ (Gauss law) constraint equation ,i = 0 which, under our axial sym- Ea′ a 1 2 metry assumption, simplifies to ,a = 0 with x = x , x while 3 E { } { } x = ϕ. On the simply connected space Mb one can always solve this constraint, without loss of generality, by introducing a potential function η′ and setting

a′ ab (2.6) = ǫ η′ . E ,b This follows from applying the Poincarélemma to the dual, closed one- a′ c c form ǫ dx and expressing it as the exact form η′ dx . Writing λ′ acE ,c for the azimuthal component, A3′ , of the ‘linearized’ vector potential one arrives at a′ ab (2.7) = ǫ λ′ B ,b for the corresponding magnetic field components. Linearizing the defining equations for the electromagnetic momentum variables u,˜ v˜ (defined through Eqs. (C.28),(C.31), (C.18) and { } a (C.19)) and recalling that the metric one-form βadx vanishes on the 12 VINCENT MONCRIEF AND NISHANTH GUDAPATI

Kerr background (compare Eqs. (A.11) and (C.13)), one ﬁnds that

3′ 3′ (2.8)u ˜′ = , v˜′ = . B −E Taking Ω to be invariant with respect to the circle action generated ∂ ∂ by ψ = ∂x3 = ∂ϕ and assuming that it projects to a domain in the quotient space of the form [t0, t1], with compact in Mb, one can reexpress the action integralD× as D (2.9) t1 ˜ ˜Maxwell 2 1 N 2γ 2 2 IΩ =2π dt d x u˜′η,t′ +˜v′λ,t′ e (˜u′) +(˜v′) (2) t0 − 2 µ g˜ Z ZD ( " 1 ˜ ab 2γ ab + Nµ(2)g˜ g˜ e− (η,a′ η,b′ + λ,a′ λ,b′ ) β0ǫ η,a′ λ,b′ 2 − #) t1 2 ab +2π dt d x (λ′v˜′),t +(Aa′ ǫ η,t′ ),b t0 − Z ZD where we have now exploited the parametrization introduced via Eq. (C.13) to denote the background metric components in ‘quotient space’ for- mat. The Kerr values for these metric components (in Weyl-Papapetrou coordinates) can be read off from Eq. (A.11) (upon taking the background charge Q to vanish). Since the second integral in (2.9) equates to a pure boundary term and thus makes no contribution to the field equations one may discard it and define, accordingly, the ‘reduced Maxwell action functional’ (2.10) t1 ˜Maxwell 2 JΩ := dt d x u˜′η,t′ +˜v′λ,t′ t0 Z ZD ˜ 1 N 2γ 2 2 1 ˜ ab 2γ e (˜u′) +(˜v′) + Nµ(2)g˜ g˜ e− (η,a′ η,b′ + λ,a′ λ,b′ ) − "2 µ(2)g˜ 2 ab Maxwell β ǫ η′ λ′ =(I˜ /2π) (boundary term). − 0 ,a ,b Ω − It may be helpful to note here that the metric functions employed a above, namely γ, gãb, βa, β0, N˜ , N˜ are related to the conventional ADM metric functions{ g , N i, N through} { ij } 2γ 2γ (2.11) e = gϕϕ, βa = e− gaϕ, (2.12) g˜ = e2γ g e4γ β β = e2γ g g g , ab ab − a b ab − aϕ bϕ a a ϕ a ϕ a 2γ (2.13) N˜ = N , β0 = N + N βa = N + N e− gaϕ, AXISYMMETRIC PERTURBATIONS 13

(2.14) N˜ = eγ N and that we write µ(2)g˜ for the 2-dimensional ‘volume’ element det gãb . Furthermore, in the standard coordinate systems discussed in Appen-| | a a p dix A, βa = 0 and N˜ = N = 0 for a metric of the Kerr type whereas the shift vector N i ∂ reduces to N ϕ ∂ β ∂ . ∂xi ∂ϕ −→ 0 ∂ϕ To this point no actual field equations have needed to be imposed on the background metric — the axisymmetric Maxwell equations for an ˜Maxwell arbitrary such background may thus be derived by variation of JΩ with respect to the (unconstrained) canonical variables (η′, u˜′), (λ′, v˜′) . For most of the arguments to follow, however, satisfaction{ of the vac-} uum field equations (specifically by the Kerr metric) will play an essential role. In terms of the twist potential ω, defined via Eqs. (C.24) and (C.18), the field equations satisfied by the Kerr metric are given, after setting η = λ = 0, by Eqs. (C.64)–(C.71). Of these, the most immediately relevant are ˜ ab ˜ ab 4γ (2.15) 4(Nµ(2)g˜ g˜ γ,a),b +2Nµ(2)g˜ g˜ e− ω,aω,b =0, ˜ ab 4γ (2.16) (Nµ(2)g˜ g˜ e− ω,a),b =0 and

c 1 ab ˜ | ˜ (2.17) N c = µ(2)g˜ g˜ N,a =0. | µ(2)g˜ ,b In addition Eq. (C.37), specialized to the (stationary, vacuum) case at hand, reduces to ˜ 4γ bc (2.18) β0,a + Ne− ǫab µ(2)g˜ g˜ ω,c =0. Explicit formulas for the relevant quantities appearing herein are given, in Boyer-Lindquist coordinates, by Eqs. (C.39)–(C.46), upon setting the charge Q to 0. Henceforth, in this section, we restrict the background metric to be speciﬁcally that of a Kerr black hole. In a suitable function space setting the Hamiltonian

(2.19) H˜ Maxwell := d2x N Max + N ϕ Max , { H Hϕ } ZMb where now (2.20) ˜ 2γ Max ϕ Max 1 Ne 2 2 1 ˜ ab 2γ ab N +N ϕ = (˜u′) +(˜v′) + Nµ(2)g˜ g˜ e− (η,a′ η,b′ + λ,a′ λ,b′ ) β0ǫ η,a′ λ,b′ , H H (2 µ(2)g˜ 2 − ) will be well-deﬁned and yield the symmetry-reduced Maxwell equations in the form of Hamilton’s equations for the canonical pairs (η′, u˜′), (λ′, v˜′) . { } 14 VINCENT MONCRIEF AND NISHANTH GUDAPATI

These latter are readily found to be: (2.21) δH˜ Maxwell Ne˜ 2γ η,t′ = = u˜′, δu˜′ µ(2)g (2.22) δH˜ Maxwell Ne˜ 2γ λ,t′ = = v˜′, δv˜′ µ(2)g˜ (2.23) ˜ Maxwell δH ˜ ab 2γ ˜ ab 4γ u˜,t′ = =(Nµ(2)g˜ g˜ e− η,a′ ),b + Nµ(2)g˜ g˜ e− ω,aλ,b′ , − δη′ (2.24) ˜ Maxwell δH ˜ ab 2γ ˜ ab 4γ v˜,t′ = =(Nµ(2)g˜ g˜ e− λ,a′ ),b Nµ(2)g˜ g˜ e− ω,aη,b′ − δλ′ − wherein we have exploited the background equation (2.18) to reexpress the resulting forms, eliminating β0 in favor of the twist potential, ω. Though H˜ Maxwell would seem to be a natural candidate for the energy functional we are seeking to construct, its density (2.20) can be shown to attain negative values inside the Kerr ergo-region leaving positivity of the total energy in doubt. To see this explicitly assume for deﬁnite- ness that the Kerr rotation parameter a is positive and evaluate the Hamiltonian density (2.20) in Weyl-Papapetrou coordinates ρ, z , for which { }

ab ∂ ∂ ∂ ∂ ∂ ∂ (2.25) µ(2) g˜ + , g˜ ∂xa ⊗ ∂xb −→ ∂ρ ⊗ ∂ρ ∂z ⊗ ∂z taking (locally deﬁned) Cauchy data of the form

(2.26)u ˜′ =v ˜′ =0 with η′ and λ′ satisfying the Cauchy Riemann equations,

(2.27) λ′ = η′ and λ′ = η′ , ,ρ ,z ,z − ,ρ within some open subset of the ergo-region. With these substitutions the Hamiltonian density reduces to Max ϕ Max 2γ 2 2 (2.28) N + N (Ne˜ − + β ) (λ′ ) +(λ′ ) H Hϕ −→ 0 ,ρ ,z 2γ and one has β0 + Ne˜ − < 0 inside the ergo-region. To treat the case a < 0 one need only reverse the roles of η′ and λ′ to generate a similar, negative result. Thus whereas for a single, axisymmetric ∂ scalar field the troublesome term in the shift vector, β0 ∂ϕ , drops out of the corresponding Hamiltonian density this is not true for the pair AXISYMMETRIC PERTURBATIONS 15 of electromagnetic scalars η′ and λ′ for which the shift induces the natural, Maxwellian coupling between them. Note that, by virtue of the background field equation (2.18) one can write ab ab β0ǫ η,a′ λ,b′ = β0ǫ η,b′ λ,a′ (2.29) − ab ˜ 4γ ab =(β0ǫ η,b′ λ′),a + Ne− µ(2)g˜ g˜ ω,aη,b′ λ′ and use this identity to replace the ‘shift term’ in the Hamiltonian density (2.20) by a term involving the background gravitational ‘twist potential’ ω together with a spatial divergence. Since the latter inte- grates to a pure boundary expression that will not contribute to the equations of motion we may discard it and define an alternative Hamil- tonian, HAlt, given by

(2.30) HAlt := d2x Alt {H } ZMb where (2.31) ˜ 2γ Alt 1 Ne 2 2 1 ˜ ab 2γ ˜ 4γ ab = (˜u′) +(˜v′) + Nµ(2)g˜ g˜ e− (η,a′ η,b′ + λ,a′ λ,b′ )+ Ne− µ(2)g˜ g˜ ω,aη,b′ λ′ H (2 µ(2)g˜ 2 ) As we shall see later this arises as a special case of the general Kerr- Newman perturbational Hamiltonian that we shall derive below in Ap- pendix G. At ﬁrst sight though it appears to amount to a step back- wards since, if we exploit the freedom to shift the (undiﬀerentiated) Alt λ′ by an additive constant, we could make locally negative even outside the ergo-region! H Now, however, we are in the fortuitous position of being able to exploit Robinson’s identity which, specialized to the case of a vacuum background and purely electromagnetic perturbations and reexpressed in our notation, reads: (2.32)

ab 2γ ab 4γ 1 2 1 2 Nµ˜ (2) g˜ e− (η′ η′ + λ′ λ′ )+2Nµ˜ (2) g˜ e− ω η′ λ′ + L (η′) + (λ′) g˜ ,a ,b ,a ,b g˜ ,a ,b 1 2 2 1 ∂ ab 4γ ab 2γ 2 2 λ′η′L + 2Nµ˜ (2) g˜ e− ω η′λ′ + N˜(µ(2) g˜ )(e− ) (η′) +(λ′) − 2 2 ∂xb − g˜ ,a g˜ ,a 1 2γ ab n 2γ 2γ 2γ 2γ o = Ne˜ µ(2) g˜ ∂ (e− λ′)∂ (e− λ′)+ ∂ (e− η′)∂ (e− η′) 2 g˜ a b a b 1 2γ ab 2γ 2γ 2γ 2γ + Ne˜ − µ(2) g˜ (η′ + λ′e− ω )(η′ + λ′e− ω )+(λ′ η′e− ω )(λ′ η′e− ω ) 2 g˜ ,a ,a ,b ,b ,a − ,a ,b − ,b 16 VINCENT MONCRIEF AND NISHANTH GUDAPATI where

2γ e− ab 4γ ab (2.33) L := 4(Nµ˜ (2) g˜ γ ) +2Ne˜ − µ(2) g˜ ω ω 1 2 g˜ ,a ,b g˜ ,a ,b n o and

ab 4γ (2.34) L := (Nµ˜ (2) g˜ e− ω ) 2 − g˜ ,a ,b

Note that L1 and L2 both vanish when the background field equations (2.15)–(2.16) are enforced. Thus for a vacuum background Robinson’s identity lets us replace the indefinite potential energy density in Alt (i.e., the first two terms appearing in (2.32)) with a spatial divergencH e and a sum of positive terms. Again discarding the boundary term resulting from the integrated divergence, we define our ultimate, regulated, Maxwell Hamiltonian as the integral over M of the density, Reg, thus constructed, setting b H

(2.35) HReg = d2x Reg , {H } ZMb with

(2.36) ˜ Reg 1 N 2γ 2 2 := e (˜u′) +(˜v′) H 2 µ(2)g˜ 1 ab γ 1 γ 2γ γ 1 γ 2γ + Nµ˜ (2) g˜ ∂ (e− λ′) (e− η′)e− ω ∂ (e− λ′) (e− η′)e− ω 2 g˜ a − 2 ,a b − 2 ,b γ 1 γ 2γ γ 1 γ 2γ + ∂ (e− η′)+ (e− λ′)e− ω ∂ (e− η′)+ (e− λ′)e− ω a 2 ,a b 2 ,b 1 1 4γ γ 2 γ 2 + 2γ γ + e− ω ω (e− λ′) +(e− η′) 2 ,a ,b 2 ,a ,b Note that HReg can be more compactly expressed in terms of the ‘rescaled’ canonical pairs (η′, u˜′), (λ′, v˜′) defined by η′ γ λ′ γ (2.37) η′ := , u˜′ := e u˜′, λ′ := , v˜′ = e v˜′ eγ eγ AXISYMMETRIC PERTURBATIONS 17 for which the regulated density becomes simply (2.38) ˜ Reg 1 N 2 2 = (˜u′) +(˜v′) H 2 µ(2)g˜ 1 ab 1 2γ 1 2γ + Nµ˜ (2) g˜ ∂ λ′ η′e− ω ∂ λ′ η′e− ω 2 g˜ a − 2 ,a b − 2 ,b 1 2γ 1 2γ + ∂ η′ + λ′e− ω ∂ η′ + λ′e− ω a 2 ,a b 2 ,b 1 1 4γ 2 2 + 2γ γ + e− ω ω (λ′) +(η′) 2 ,a ,b 2 ,a ,b Hamilton’s equations, which now take the form δHReg δHReg (2.39) η′ = , λ,t′ = , ,t δu˜′ δv˜′ δHReg δHReg (2.40) u˜,t′ = , v˜,t′ = , − δη′ − δλ′ regenerate the Maxwell equations (2.21)–(2.24) given previously but now with a positive definite Hamiltonian. Using these equations to compute the time derivative of Reg one arrives at H (2.41) Reg ∂ 2 ab 1 2γ 2 ab 1 2γ = N˜ g˜ u˜′ η′ + λ′e− ω + N˜ g˜ v˜′ λ′ η′e− ω H,t ∂xb ,a 2 ,a ,a − 2 ,a which leads one to define the divergence-free vector density current, JReg, via (2.42) 0 := Reg JReg H (2.43)

b 2 ab 1 2γ 1 2γ := N˜ g˜ u˜′ η′ + λ′e− ω +˜v′ λ′ η′e− ω JReg − ,a 2 ,a ,a − 2 ,a with ∂ ∂ (2.44) J = 0 + b Reg JReg ∂t JReg ∂xb satisfying, by construction, ∂ ∂ (2.45) 0 + a =0. ∂tJReg ∂xa JReg 18 VINCENT MONCRIEF AND NISHANTH GUDAPATI

The regularity conditions at the axis satisfied by the (rescaled) canonical variables (η′, u˜′), (λ′, v˜′) together with their asymptotic behaviors at the Kerr event horizon and at infinity are discussed in detail in Ap- pendix B. Appealing to these results it is straightforward to apply the continuity equation (2.45) to show that the total electromagnetic ‘energy’, defined by HReg, is strictly conserved. This energy could only differ in value from those defined by HAlt and HMaxwell by possible boundary contributions at (spacelike) infinity or at the bifurcation 2- sphere lying in the black hole’s horizon. It has, however, the significant analytical advantage over these latter quantities of being manifestly positive definite. Combining Eqs. (2.21), (2.22), and (2.6)–(2.8) one finds that ˜ 2γ 3′ Ne a′ (2.46) η,t′ = B , η,b′ = ǫab , µ(2)g˜ E ˜ 2γ 3′ Ne a′ (2.47) λ,t′ = E , λ,b′ = ǫab , − µ(2)g˜ B and thus that, in any open connected domain of R M in which ′ ′ b the (projected) electromagnetic field components ( i ,× i ) all vanish, E B the potentials η′ and λ′ must both be spacetime constants. Since the ′ ′ Maxwell fields i and i propagate causally on the domain of outer communicationsE of a KerrB black hole it is easily verified that these fields, projected to the quotient space R Mb, propagate causally with respect to the induced, 2+1-dimensional× Lorentz metric (3)k defined by (2.48) (3)k := N˜ 2dt dt +˜g (dxa + N˜ adt) (dxb + N˜ bdt) − ⊗ ab ⊗ (c.f. Eq. (C.13)). It follows that any non-constant disturbance in the (3) potentials η′ and λ′ must propagate causally on (R Mb, k). Another way of verifying the causal propagation× of energy in this quotient space is to calculate the flux density of the current JReg across (3) an arbitrary null hypersurface in (R Mb, k) with (future directed) µ ∂ × µ (3) µ ν null normal ℓ ∂xµ , i.e., to evaluate ℓµ Reg = kµνℓ Reg for an µ ∂ − J − J arbitrary tangent field ℓ ∂xµ satisfying 0 (3) µ ν (2.49) ℓ > 0, kµνℓ ℓ =0. Appealing to the defining equations (2.38), (2.42) and (2.43) and recalling that N˜ a = 0 for the metrics of interest herein it is straightforward µ to verify directly that this flux density, ℓµ Reg, is always non-negative and thus that the corresponding energy− canJ only flow causally through such null hypersurfaces. AXISYMMETRIC PERTURBATIONS 19

For the coordinate systems discussed in Appendix A it is well-known that null geodesics originating in the Kerr black hole’s DOC cannot reach infinity or the event horizon in finite (coordinate) time but only in the limit as t . Projected to the quotient space this result −→ ±∞ implies, in particular, that Cauchy data (η′, u˜′), (λ′, v˜′) specified at { } t = t0 and having compact support in Mb at this ‘initial’ instant will evolve in such a way as to preserve this property t. In other words the support of these fields, evaluated at any time t,∀ will remain bounded and disjoint from the horizon. For these solutions in particular it is evident that the various ‘energies’, H˜ Maxwell, HAlt and HReg that we have defined will all coincide. Note however that whereas the densities N Max +N ϕ Max and Alt H Hϕ H both vanish at any point at which i = i = 0, this need not be true Reg E B i ∂ i ∂ of . Indeed, as we have seen, the fields ∂xi and ∂xi could both vanishH throughout an entire open subset of ER M implyingB only that × b η′ and λ′ be spacetime constants in this domain. Unless both these constants also vanish the density Reg will be strictly positive (though time independent) throughout thisH region. In other words Reg is ac- i ∂ i ∂ H tually non-local in the fundamental fields ∂xi and ∂xi . One could nevertheless express it explicitly in termsE of these fieldsB by applying the methods of Appendix H to solve the two Poisson equations,

ab cd a′ (2.50) (µ(2) g˜ η′ ) =(µ(2) g˜ ǫ ) g˜ ,a ,b g˜ ac E ,d and

ab cd a′ (2.51) (µ(2) g˜ λ′ ) =(µ(2) g˜ ǫ ) g˜ ,a ,b g˜ ac B ,d that follow from the deﬁning formulas (2.6) and (2.7). Alternatively ′ one could appeal to the exactness of the one-forms ǫ a dxc and ′ ac ǫ a dxc and integrate the deﬁning equations E ac B a′ (2.52) η′ = ǫ , ,c ac E and

a′ (2.53) λ′ = ǫ ,c ac B along conveniently chosen paths from (say) points on the axis where these potentials both vanish (c.f., the discussion in Appendix E). To summarize the results of this section, we have proven the following: Theorem 1. Maxwell’s equations for the axisymmetric, purely electromagnetic perturbations of an arbitrary Kerr black hole are derivable 20 VINCENT MONCRIEF AND NISHANTH GUDAPATI from a Hamiltonian (HReg defined by Eqs. (2.35)–(2.37)) that is positive definite and strictly conserved but non-local when expressed in terms of the electric and magnetic fields. Since this work was completed one of us (N.G.) has shown how to derive a corresponding (positive definite energy) result, in the presence of a positive cosmological constant, for the axisymmetric, Maxwellian perturbations of Kerr-de Sitter black holes [38]. Independently, Wald and Prabhu have derived positive definite energy functionals for the axisymmetric, electromagnetic perturbations of Kerr black holes that are not only compatible with ours in 4-dimensions but which apply as well to sufficiently symmetric electromagnetic perturbations of higher dimensional black holes [64].

3. An Energy Functional for Axisymmetric Kerr-Newman Perturbations In this section we construct a conserved, positive definite energy functional for linear, axisymmetric perturbations of arbitrary Kerr- Newman black holes. While we focus, for technical reasons, on the most astro-physically relevant, sub-extremal cases our main calculational results are equally applicable to extremal black holes for which they generalize, to the electro-vacuum framework, those given in Ref. [27]. To set the stage for our derivation we first recall how similar ideas were developed, long ago, for the special case of (charged but non- rotating) Reissner-Nordström black holes.

3.1. Background on Reissner-Nordström Perturbations. The derivation of a conserved, positive definite energy functional for the coupled (electromagnetic and gravitational) dynamical perturbations of Reissner-Nordström (RN) black hole spacetimes was given by one of us in Refs. [54, 55, 56]. It followed from computing the second variation of the Einstein-Maxwell action functional about a Reissner-Nordström black hole background and restricting the resulting expression to the latter’s domain of outer communications (DOC). It has been realized from the time of Jacobi that such a 2nd variation functional serves, in turn, as an action for the corresponding linearized equations — in the present context for those of the linear perturbations of an arbitrary RN background. By exploiting the spherical symmetry of the RN geometry and expanding the perturbations in Regge-Wheeler tensor harmonics one was able to carry out an explicit canonical transformation to a new set of variables wherein a certain (unconstrained, gauge invariant) subset of AXISYMMETRIC PERTURBATIONS 21 canonical pairs was found to represent the radiative, dynamical degrees of freedom. This was complemented by a further subset comprised of the (equally gauge-invariant) linearized constraints and their (gauge- variant but unconstrained) canonically conjugate partners. The linearized lapse, shift and electromagnetic ‘scalar potential’ served as La- grange multipliers in the perturbative action functional where, paired with the (gauge group generating) linearized constraints, they could be adjusted to fix the evolution of gauge-dependent canonical variables. The Hamiltonian emerging naturally from this formulation of the linearized field equations (for the radiative, dynamical degrees of freedom unconstrained by the Birkhoffian rigidity of the complementary, purely spherically symmetric perturbations) was found, by explicit calculation, to be positive definite, conserved and to bound a naturally associated Sobolev norm of the gauge-invariant dynamical variables. That a positive definite energy functional emerged rather straightforwardly from this analysis was however due, in no small measure, to the absence of an ergo-region in a Reissner-Nordström black hole’s DOC. An interesting feature of the resulting field equations, found independently by Zerilli who derived them by working in a special gauge [76], was that certain specific linear combinations of the perturbative, gravitational and electomagnetic variables decoupled from one another and satisfied single component wave equations of Regge-Wheeler-Zerilli type. On the other hand the heavy reliance on the use of tensor spherical harmonics seemed to limit the application of the aforementioned methods to spherically symmetric backgrounds. While there is no particular difficulty involved in computing the 2nd variation of the Einstein- Maxwell action about the more general, Kerr-Newman backgrounds of principal interest herein, a demonstration that the resulting ‘canonical energy’ expression is, at least for purely axisymmetric perturbations, actually positive definite has not heretofore been realized. By exploiting the reducibility of the axisymmetric field equations to a 2+1-dimensional Einstein—wave map system (c.f., Appendix C), computing the 1st and 2nd variations of the corresponding field equations (c.f., Appendix G) about a Kerr-Newman black hole background (c.f., Appendix A) and applying Robinson’s identity to the resulting energy expression, we shall derive below an energy functional with the desired properties. This result will subsume that for purely electromagnetic perturbations of Kerr backgrounds, given in section 2, as a special case and now incorporate the coupled gravitational and electromagnetic perturbations of general Kerr-Newman black hole spacetimes in linear approximation. Since our strategy for deriving the desired energy 22 VINCENT MONCRIEF AND NISHANTH GUDAPATI expression will entail an extension of certain mathematical methods developed for the study of the so-called linearization stability problem for Einstein’s equations we briefly review some of the central ideas of that analysis in the following subsection. These will not only show the way for deriving the desired energy formula but also that for interpreting its geometrical significance.

3.2. Brief Review of the Linearization Stability Problem. At around the same time that this early work on Reissner-Nordström perturbations was being carried out some seemingly unrelated technology was being developed for the rigorous analysis of what came to be called the ‘linearization stability problem’ for Einstein [9, 33, 34, 57, 58], Einstein-Maxwell [7] and, still more generally, Einstein-Yang-Mills [9] spacetimes having compact Cauchy hypersurfaces. These studies dealt with the fact that, for spatially compact solutions of the relevant field equations that admitted one or more globally defined Killing vector fields, one could show that the associated linearized equations admitted so-called ‘spurious solutions’ that were not tangent to any differ- entiable curve of exact solutions to the corresponding, nonlinear field equations. Such spurious solutions could be identified and excluded precisely by demanding that the Noether-like conserved quantities for the perturbation problem — one for each Killing field of the background — vanish. This additional condition supplemented the linearized field equations themselves with a (non-vacuous) set of conserved, gauge invariant quadratic integral restrictions upon the first order perturbations that was eventually shown to have a natural geometric interpretation [8, 9, 34]. The geometrical meaning of this result was that it showed that the manifold structure of the solution space of the (nonlinear) constraint equations broke down at any point corresponding to Cauchy data for an exact solution admitting nontrivial, global Killing symmetries (or, in the case of the Einstein-Yang-Mills equations, generalized gauge symmetries [9]). Roughly speaking the space of solutions to the nonlinear constraints was shown to exhibit a conical singularity at any such point and the role of the supplementary quadratic conditions on the linear perturbations was to restrict them to actually be tangent to this conical structure [9, 33, 34, 57, 58]. This geometrically nontrivial conclusion did not carry over to the case of non-compact Cauchy surfaces since, roughly stated, certain boundary integrals linear in the second order perturbations now arose to ‘take up the slack’ and allow the otherwise spurious first order perturbations to tangentially approximate curves of exact solutions while AXISYMMETRIC PERTURBATIONS 23 forcing the boundary contributions at second order (which were absent in the compact case) to take on certain specific values. More precisely the conserved quadratic integral expressions in the first order perturbations — one for each global Killing field of the background exact solution — were no longer coerced to vanish, since their actual values could always now be compensated by those of the (2nd order) boundary integral expressions. One could forcibly recover the conical singularity structure in the solution space for the nonlinear constraints only by artificially restricting certain asymptotically defined, conserved quantities for the nonlinear problem (e.g., the ADM mass for asymptotically flat solutions) to have fixed values. A key step in the development of this linearization stability technology was the proof (originally for the vacuum case) that the kernel of the adjoint of the linearized constraint operator was precisely the space of Cauchy data for the Killing fields of the corresponding, vacuum spacetime [9, 33, 34, 57, 58]. Each such Killing initial data set (or KID as they are nowadays known) consists of a function C and a spatial vector field Z such that the pair (C,Z) provides the normal and tangential components (at the chosen Cauchy hypersurface) of a Killing vector field on the resulting Einstein spacetime. For the special case of a timelike Killing field ζ, formulated in coordinates for which ∂ ζ = ∂t , the pair (C,Z) is nothing other than the lapse and shift of the spacetime metric expressed in those coordinates. While the published literature does not seem to include precisely the analogous (non-vacuum) case of interest to us here we shall simply verify the needed result, for the problem at hand, by a direct calculation rather than appeal to general theory. In this regard a key role will be played by the fact that the reduced lapse function, N˜, for Kerr- Newman backgrounds (c.f., Eqs. (C.13) and (C.42)) is harmonic (c.f., Eq. (C.71)). Since the corresponding reduced shift field vanishes, the pair (N,O˜ ) will prove to provide precisely the needed kernel of the adjoint to the linearized constraint operator. It is straightforward to show that, in the t, ϕ, ρ,¯ z¯ coordinate system defined in Appendix A, this kernel takes the{ simple} form (¯ρ, O).

3.3. An ‘Alternative’ Energy Functional and its Regulariza- tion. While a fully general, ‘canonical’ energy density for non-symmetric perturbations of Kerr-Newman black holes could readily be constructed by reinstating the terms that were omitted in the static background (RN) limit analyzed in Ref. [54], this would have little hope of directly yielding a positive definite total energy. By constraining the study to 24 VINCENT MONCRIEF AND NISHANTH GUDAPATI axisymmetric perturbations, however, and transforming the field equations to the 2+1-dimensional Einstein—wave map form reviewed in Appendix C we are led to introduce the ‘alternative’ energy density, ‘ Alt’, defined in Eqs. (G.27) and (G.30) of Appendix G. E This expression is itself indefinite though since in particular it reduces to Alt for the purely electromagnetic perturbations of Kerr backgrounds.H The single negative term in the kinetic energy density, N˜ 2ν (2) 2 e √ h(τ ′) , can, however, be disposed of by imposing a suitable − 2 time-coordinate gauge condition. The simplest choice, τ ′ = 0, corresponds to enforcing 2+1-dimensional, maximal slicing1 whereas setting 2ν p˜′e− (3.1) τ ′ = − 4√(2)h implies the imposition of maximal slicing in the ‘lifted’, 3+1-dimensional sense. Though the latter choice leaves a negative term in the kinetic density it is easily seen to be dominated by the complementary, positive terms leaving a net positive definite kinetic energy density. N˜ 2 (2) Let us abbreviate by 2 D (q, h) (q′, q′) the potential energy density given explicitly by V · ˜ ˜ N 2 (2) N (2) ab 2γ 2 D (q, h) (q′, q′) := √ h h 4γ′ γ′ +2e− (γ′) (η η + λ λ ) 2 V · 2 ,a ,b ,a ,b ,a ,b ( 2γ 2γ 4e− γ′(η η′ + λ λ′ )+ e− (η′ η′ + λ′ λ′ ) − ,a ,b ,a ,b ,a ,b ,a ,b 4γ 2 +8e− (γ′) (ω,a + λη,a)(ω,b + λη,b) 4γ 8e− γ′(ω + λη )(ω′ + λη′ + λ′η ) − ,a ,a ,b ,b ,b 4γ + e− (ω,a′ + λη,a′ + λ′η,a)(ω,b′ + λη,b′ + λ′η,b)

4γ (3.2) + e− (ω,a + λη,a)(2λ′η,b′ ) and by

4 (2) ab 2γ ab K1 := (N˜√ h h γ,a),b + Ne˜ − h (η,aη,b + λ,aλ,b) √(2)h 4γ ab (3.3) +2Ne˜ − h (ω,a + λη,a)(ω,b + λη,b),

1 (2) ab 4γ (3.4) K2 := N˜√ h h e− (ω,a + λη,a) , √(2)h ,b 1 (2) ab 2γ ab 4γ (3.5) K3 := (N˜√ h h e− η,a),b + Nh˜ e− λ,b(ω,a + λη,a), √(2)h

1Here τ ′ designates the ﬁrst variation of the 2+1-dimensional mean curvature, τ, deﬁned by Eq. (K.1). Its 3+1-dimensional analogue is given by Eq. (K.2) AXISYMMETRIC PERTURBATIONS 25

1 (2) ab 2γ ab 4γ (3.6) K4 := (N˜√ h h e− λ,a),b Nh˜ e− η,b(ω,a + λη,a) √(2)h − the wavemap expressions which vanish for any Kerr-Newman background (c.f., Eqs. (C.65–C.68)). Finally, denote by (1)V,..., (8)V the 1-forms deﬁned explicitly by { }

(3.7)

(1) 4γ 1 2γ V := 2γ′ + e− (ω′ + λη′)(ω + λη )+ e− (η′η + λ′λ ), a ,a ,a ,a 2 ,a ,a

(3.8) (2) 2γ 1 2γ 2γ V := e− (ω′ + λη′) + e− (η′λ λ′η )+2e− γ′(ω + λη ), a − ,a 2 ,a − ,a ,a ,a (3.9)

(3) 1 2γ 2γ λ′ V := e (η′ ) e γ′(η )+ (ω + λ(η )) , a 2 ,a − ,a 2 ,a ,a

(3.10)

(4) 1 2γ 2γ η′ V := e (λ′ ) e γ′(λ ) (ω + λ(η )) , a 2 ,a − ,a − 2 ,a ,a

(3.11) (5) 1 V := (η′λ λ′η ), a 4 ,a − ,a

(3.12) (6) 1 1 2γ V := (η′ ) η′γ + e− λ (ω′ + λη′), a 2 ,a − ,a 2 ,a

(3.13) (7) 1 1 2γ V := (λ′ ) λ′γ e− η (ω′ + λη′), a 2 ,a − ,a − 2 ,a

(3.14) (8) 1 V := 2γ (ω′ + λη′) (η′λ λ′η ) 2γ′(ω + λη ). a ,a − 2 ,a − ,a − ,a ,a 26 VINCENT MONCRIEF AND NISHANTH GUDAPATI

In this notation Robinson’s identity [69] takes the form:

(3.15)

2 (2) ∂ (2) ab 4γ 2 ND˜ (q, h) (q′, q′)+ N˜√ h h 2e− γ (ω′ + λη′) V · ∂xb − ,a 2γ 2 2 4γ e− γ (η′) +(λ′) + e− (ω′ + λη′)(λ η′ λ′η ) − ,a ,a − ,a 4γ 4γ e− (ω + λη )η′λ′ +4e− γ′(ω + λη )(ω′ + λη′) − ,a ,a ,a ,a 2γ 2γ + 2e− γ′η′η,a +2e− γ′λ′λ,a √(2) h 2γ 2 1 2 2 + K e− (ω′ + λη′) + (η′) +(λ′) 2e2γ 1 2 (2) + √ hK [η′λ′ 4γ′(ω′ + λη′)] 2 − (2) 2γ + √ hK λ′e− (ω′ + λη′) 2γ′η′ 3 − (2) 2γ √ hK4 η′e− ( ω′ λη′) 2γ′λ′ − − − ˜ (2) ab (1) (1) (2) (2) N√ h h Va Vb + Va Vb ≡ 6γ (3) (3) 6γ (4) (4) +2e− Va Vb +2e− Va Vb 4γ (5) (5) 2γ (6) (6) + 12e− Va Vb +2e− Va Vb 2γ (7) (7) 4γ (8) (8) +2e− Va Vb + e− Va Vb

It follows immediately that, if we drop the terms that vanish by virtue of the background ﬁeld equations (i.e., set K1 = K2 = K3 = K4 = 0), then we can express the potential energy density occurring in Alt as the sum of a spatial divergence and a positive deﬁnite quadratic expressionE in the one-forms (1)V,..., (8)V . AXISYMMETRIC PERTURBATIONS 27

Since the integrated divergence will only contribute a boundary term in the total energy expression we set it aside here and deﬁne our regulated energy density, Reg, by E ˜ Reg N 2ν b a 1 2 1 4γ 2 := e− r˜′ar˜′b + (˜p′) + e (˜r′) E √(2) 8 2 ( h ˜ 1 2γ 2 2 N 2ν (2) 2 + e (˜v′) +(˜u′ λr˜′) e √ h(τ ′) 2 − − 2 ) (3.16) (2) ab 1 (1) (1) 1 (2) (2) 6γ (3) (3) + N˜√ h h V V + V V + e− V V 2 a b 2 a b a b 6γ (4) (4) 4γ (5) (5) 2γ (6) (6) + e− Va Vb +6e− Va Vb + e− Va Vb

2γ (7) (7) 1 4γ (8) (8) + e− V V + e− V V a b 2 a b and corresponding total regulated energy by

(3.17) EReg := d2x Reg . {E } ZMb Evaluated with respect to either of the two maximal slicing gauges discussed above Reg and thus EReg are manifestly positive definite. E 4. Conservation of the Total Energy The ‘alternative’ energy, EAlt, defined by Eq. (G.31) has its density, Alt, given explicitly via Eqs. (G.27) and (G.30). While EAlt potentially differsE by a boundary integral from its ‘regulated’ counterpart EReg (c.f., Eqs. (3.16)–(3.17)), we shall see below that this difference actually vanishes for the class of (asymptotically-pure-gauge) perturbations that we consider here. Thus for our present purposes it will suffice to show that EAlt is conserved since this will imply the corresponding result for EReg. As discussed more completely in Appendices F and H we work in a partially gauge-fixed setting wherein the flat (2-dimensional) ‘conformal’ desensitized spatial metric, h (4.1) (2)h˜ := ab dxa dxb, √(2)h ⊗ is held fixed during the evolution through a judicious choice of the lin- ˜ c′ ∂ earized shift field X′ := N ∂xc . By contrast, at least for now, we leave the perturbative time gauge unspecified by allowing the linearized lapse function, N˜ ′, to remain arbitrary. That our total energy will be found to be conserved independently of the interior behavior of the linearized 28 VINCENT MONCRIEF AND NISHANTH GUDAPATI time gauge chosen corresponds to its essential gauge invariance. Oth- erwise one could modify its evolution by merely making a change of gauge. ∂ Alt Computing ∂t directly by means of the linearized field equations one gets: E (4.2) ˜ 2 ∂ Alt ∂ N (2) ab ′ ′ √ = b p˜ hh γ,a ∂tE ∂x ( (2)g˜ 4γ 4γ (2) ab ′ 2γ (2) ab 2γ ′ + e r˜′ ep− √ hh (ω,a + λη,a) + e v˜′ √ hh e− λ,a

2γ (2) ab 2γ ′ + e (˜u′ λr˜′) √ hh e− η − ,a 2γ (2) ab 4γ + e (˜u′ λr˜′)λ′√ hh e− (ω + λη ) − ,a ,a (2) ab (2) ab 4γ + γ′ ′ 4N˜√ hh γ + ω′ ′ N˜√ hh e− (ω + λη ) LX ,a LX ,a ,a 2γ (2) ab 2γ (2) ab + λ′ ′ Ne˜ − √ hh λ + η′ ′ Ne˜ − √ hh η LX ,a LX ,a 4γ (2) ab + Ne˜ − √ hh λ(ω,a + λη,a)

(2) ba (2) ab + 2( ′ N˜) √ hh ν′ + 2( ′ ν′)√ hh N˜ LX ,a LX ,a ′ b (2) ac 2X √ hh ν′ N˜ − ,a ,c bc a 2ν (2) ab +2Nh˜ (˜r )′e− N˜ ′ +(N˜N˜ ′ N˜ ′N˜ )τ ′√ hh c ,a ,a − ,a bc a 2ν 2N˜ ′h (˜r )′e− N˜ − c ,a 2ν ab ˜′ ′ N˜ +(N˜ ′N˜ N˜N˜ ′ )e− h ˜′ − H LX ,a − ,a Hb n √(2) ab ∂ ∂o where, since the variations of h h ∂xa ∂xb are taken to vanish (c.f., Appendices F and H), one has ⊗

√(2) ab ′ √(2) ab (4.3) hh γ,a = hh γ,a′ , √(2) ab 2γ ′ √(2) ab 2γ hh e− λ,a = hh e− λ,a′ (4.4) (2) ab 2γ 2√ hh e− γ′λ , etc. − ,a Note that the terms in the last bracket in Eq. (4.2) vanish by virtue of the linearized constraints{ leaving } only a total spatial divergence whose integral over Mb will result in pure boundary expressions. AXISYMMETRIC PERTURBATIONS 29

As we shall show below these boundary integrals vanish for the class of (asymptotically-pure-gauge) perturbations that we consider herein. 4.1. Evaluating the ‘Dynamical’ Boundary Flux Terms. Con- sider first the boundary flux contributions from Eq. (4.2) that are each linear in the perturbed momenta p˜′, r˜′, v˜′, u˜′ λr˜′ — we shall refer to these as ‘dynamical’ flux terms.{ From Eqs.− (H.73)–(H.76)} we see that the momentum factors occurring in each of these terms can, in the asymptotic regions near R = R+ and R = defined therein, be expressed as a pairing of the vector field (c.f., H.78)∞ ∂ ∂ (4.5) (2) a := N˜ 2gãb(4)Y 0 D ∂xa ,b ∂xa with a corresponding one-form taken from the list a a a a (4) 0 4γ,adx ,λ,adx , η,adx , (ω,a + λη,a)dx . Here Y is the time compo- { } (4) (4) µ ∂ nent of the gauge transforming vector field Y = Y ∂xµ introduced in this appendix. From Eq. (H.23) we see that (in the asymptotic regions defined in Appendix H where the compactly supported ‘hyper- (4) bolic gauge perturbation’ kαβ vanishes) this component is determined from integrating N˜ N˜ (4.6) (4)Y 0 ′ (2)Y a ,a ,0 → N˜ − N˜ (2) (2) a ∂ (4) a ∂ with Y = Y ∂xa = Y ∂xa given in turn by (c.f., Eqs. (H.34)– (H.35), (H.44), (H.50)–(H.56) and recall the vanishing of a0(R+) for the perturbations of interest herein established in Appendix J):

(2)Y R ∞ Rn Rn (4.7) = α(+)(t)Rn + cos(nθ), R n + Rn − Rn n=1 + X ∞ Rn Rn (4.8) (2)Y θ = α(+)(t)Rn + + sin (nθ) n + Rn Rn n=1 + X near R = R+, and by

(2) R ∞ Y ( ) n (4.9) = β − (t)R− cos(nθ), R − n n=1 X ∞ (2) θ ( ) n (4.10) Y = βn− (t)R− sin (nθ) n=1 X (+) ( ) near R = . Recall that the t-dependent coeﬃcients αn , βn− are ∞ (4) computable in terms of speciﬁed ‘sources’ (determinedn by kαβo) via Eqs. (H.57), (H.59)–(H.61). 30 VINCENT MONCRIEF AND NISHANTH GUDAPATI

Deﬁning

t (2) a (2) a (4.11) (t, R, θ) := dt′ Y (t′,R,θ) Y Zt0 we see that (4.12) t ˜ ˜ (4) 0 (4) 0 N ′ (2) a N,a Y (t, R, θ)= Y (R, θ)+ dt′ (t′,R,θ) (t, R, θ) (R, θ) t=t0 N˜ − Y N˜ Zt0 !

where (4)Y 0 (R, θ) is initial data chosen for (4)Y 0. Deﬁning t=t0

t (+) (+) (4.13) αñ (t) := dt′ αn (t′), Zt0 t ˜( ) ( ) (4.14) βn− (t) := dt′ βn− (t′), Zt0 t (4.15) c˜0(R, t) := dt′ c0(R, t′) Zt0 one arrives at explicit (Fourier representation) formulas for the (2) a by (+) (+) ( ) ( ) Y making the replacements αn αñ and βn− βñ− in Eqs. (4.7)– (4.10). → → For definiteness we shall herein eventually impose the 2+1-dimensional maximal slicing gauge condition which, taken together with suitable (homogeneous) boundary conditions for the linearized lapse function, δN˜ := N˜ ′, will imply that N˜ ′ = 0 (c.f., the discussion given in Ap- pendix K). For now however we shall retain the contributions of (a (4) 0 non-vanishing) N˜ ′ to Y so that they can be easily reinstated if alternative gauge conditions (e.g., 3 + 1-dimensional, maximal slicing, as discussed in Appendix (K)) are desired in the future. One is of course free to impose essentially arbitrary boundary conditions upon the initial data (4)Y 0 . We shall assume in the following t=t0 that these have been chosen so that, together with the boundary behavior of the linearized lapse function, one has

t ˜ (4) 0 N ′ 1 (4.16) R := Y + dt′ O , ,R t=t0 3 K t N˜ ∼ R Z 0 !,R

t ˜ (4) 0 N ′ 1 (4.17) θ := Y + dt′ O ,θ t=t0 2 K t N˜ ∼ R Z 0 !,θ

AXISYMMETRIC PERTURBATIONS 31 as R with R and θ both bounded as R R+ and with R bounded→ ∞ and KvanishingK like → K Kθ

(4.18) sin (θ) (regular factor) Kθ ∼ ×

at the axes corresponding to θ =0, π. (4) 0 For vanishing N˜ ′ these are trivial to ensure by choice of Y t=t0 but would need to be verified on a case-by-case basis for alternative time gauges. On the other hand these conditions are only sufficient for the arguments to follow and could be somewhat relaxed without disturbing the main results. The components of the vector field (2) a ∂ are given explicitly by D ∂xa

(4.19) 4 2 2 (2) R R R+ = 1 R D (r2 + a2)2 a2∆ sin2 (θ) − R2 K − ( R4 (2) R 4R2 (2) R cos θ R2 2 1 + Y + Y + 1 + (2) θ , − − R4 R − R3 R sin θ − R2 Y,R " ,R #)

and

(4.20) 2 2 2 (2) θ R R+ = 1 θ D (r2 + a2)2 a2∆ sin2 (θ) − R2 K − ( (2) R R4 (2) θ cos(θ) R2 2 Y 1 + + Y 1 + − R − R4 sin (θ) − R2 " ,θ ,θ #) 32 VINCENT MONCRIEF AND NISHANTH GUDAPATI where ∆ := r2 2Mr + a2 + Q2 (c.f., Eq. (A.4)). Evaluated in the asymptotic region− at large R via Eqs. (4.9)–(4.11), (4.14) these become

4 2 2 (2) R R R+ = 1 R D (r2 + a2)2 a2∆ sin2 (θ) − R2 K − ( 4 ∞ R+ ( ) n 1 1 β˜ − nR− − cos(nθ) − − R4 n " n=2 X 2 ∞ R+ ( ) n (4.21) 4 β˜ − R− cos(nθ) − R3 − n n=2 ! X2 2 ∞ R+ ( ) n 1 cos(θ) + 1 β˜ − nR− − sin (nθ) − R2 − n sin (θ) n=2 ! X 2 2 ( ) R R β˜ − +2 + 3 + 1 cos(θ) R2 − R2 R2 #)

and

2 2 2 (2) θ R R+ = 1 θ D (r2 + a2)2 a2∆ sin2 (θ) − R2 K − ( 2 2 2R+ ( ) R+ β˜ − sin (θ) 1 − R3 1 − R2 (4.22) 4 ∞ R+ ( ) n + 1 β˜ − nR− sin (nθ) − R4 n n=2 2 X 2 ∞ R+ ( ) n cos(θ) + 1 β˜ − R− sin (nθ) − R2 n sin (θ) n=2 ,θ)# X AXISYMMETRIC PERTURBATIONS 33 whereas for R near R+ they take the forms (c.f., Eqs. (4.7)–(4.8), (4.11), (4.13))

(4.23) 4 2 2 (2) R R R+ = 1 R D (r2 + a2)2 a2∆ sin2 (θ) − R2 K − ( R2 R2 ∞ Rn Rn Rn 1 + 1+ + α˜(+) + n + + cos(nθ) − − R2 R2 n R Rn Rn " ( n=1 + ) X 4R2 ∞ Rn Rn + α˜(+)Rn + cos(nθ) − R3 n + Rn − Rn n=1 + X 2 (+) R2 ∞ cos(θ) nα˜ Rn Rn + 1 + sin (nθ) n Rn + − R2 sin (θ) R + Rn − Rn n=1 + #) X and

(4.24) 2 2 2 (2) θ R R+ = 1 θ D (r2 + a2)2 a2∆ sin2 (θ) − R2 K − ( R2 ∞ R2n cos(θ) R2 1 + α˜(+) Rn + + sin (nθ) 1 + − − R2 n Rn sin (θ) · − R2 (n=1 ,θ X R2 ∞ Rn Rn 1+ + α˜(+)Rn + n sin (nθ) − R2 n + Rn − Rn n=1 + )) X

cos (θ) A standard trigonometric identity shows that sin(θ) sin (nθ) is expressible as a polynomial of degree n in cos (θ) and thus is regular at the cos (θ) axes θ = 0, π. From this same result it follows that sin(θ) sin (nθ) ,θ vanishes like sin (θ) at these axes. In addition it is straightforward to n Rn R+ Rn Rn + − verify that R R+ is smooth R and, in particular, has the limits R R ∀ + −

n n R R+ Rn Rn (4.25) lim + − = n R R+ R R+ → R+ − R 34 VINCENT MONCRIEF AND NISHANTH GUDAPATI and

n n R R+ Rn Rn (4.26) lim + − =0 R R+  R R+  → R R + − ,R   n =1, 2,.... ∀ From the explicit formulas for γ,λ,η,ω one finds that each of { } 2 the quantities 4γ ,λ , η , (ω + λη ) vanishes like 1 R+ as { ,R ,R ,R ,R ,R } − R2 R R+ whereas 4γ,θ,λ,θ, η,θ, (ω,θ + λη,θ) are all regular in this limit. ′ → (4){ N˜ } Assuming that Y and ˜ have been chosen so that both R and t=t0 N K (c.f., Eqs. (4.16)–(4.17)) are regular at R we see that each of the θ + Kquantities R2 (4.27) (2) aχ 1 + (regular factor) D a ∼ − R2 × as R R+ where χa =4γ,a,λ,a, η,a and (ω,a + λη,a). Thus each of the → ′ N˜p˜ Ne˜ 2γ v˜′ Ne˜ 2γ Ne˜ 4γ r˜′ (linearized momentum) factors , , (˜u′ λr˜′), √(2)g˜ √(2)g˜ √(2)g˜ − √(2)g˜ 2 (c.f., Eqs. (H.73)–(H.76)) vanishes like 1 R+ (regular factor) as − R2 × R R+. Note that in the corresponding flux terms these are each mul- → 2 tiplied by an ‘additional’ factor of N˜ = R sin (θ) 1 R+ and paired − R2 (2) ab ′ (2) ab 2γ ′ (respectively) with factors of the form √ h h γ,a , √ h h e− λ,a , etc. to yield their ultimate contributions to the flux integrals at the various boundaries. Recalling that our gauge conditions enforce the constraint that (4.28) √(2)h hab ′ =0 we see that (c.f., Eqs. (4.3) and (4.4))

√(2) ab ′ √(2) ab (4.29) h h γ,a = h h γ,a′ , √(2) ab 2γ ′ √(2) ab 2γ h h e− λ,a = h h e− λ,a′ (4.30) (2) ab 2γ 2√ h h e− γ′λ , − ,a etc.

From the explicit formulas for the (background) metric functions (C.40), (C.43), (C.45) and (C.46) the asymptotic forms for the (pure gauge) AXISYMMETRIC PERTURBATIONS 35

(2) a ∂ perturbations, (H.17)–(H.18), and the boundary conditions for Y ∂xa which yield (2) R (2) θ (4.31) Y (R+, θ)=0, Y ,R(R+, θ)=0 we find that each of γ′,λ′, η′,ω′ has a smooth limit as R R { } → + whereas their first radial derivatives γ,R′ ,λ,R′ , η,R′ ,ω,R′ are all vanishing in this limit.2 One also finds that, in the asymptotic regions near R = R and R , one has λ′,λ′ , η′, η′ ,ω′,ω′ vanishing at the + → ∞ ,θ ,θ ,θ axes θ 0, π whereas γ′ is regular in this limit with γ,θ′ 0 at θ =0, π. Given→ these results it is straightforward to verify that→ each of the ‘dynamical’ flux terms vanishes, pointwise, at the horizon boundary corresponding to R R+. One might still wonder whether the factors 1 1 → 2γ of sin2 θ and sin4 θ occurring (respectively) in the coefficients e− and 4γ e− induce some irregularity at the axes but it is not difficult to verify that such potential singularities are in fact cancelled by the rapidly vanishing angular dependences of λ,λ′,η,η′,ω,ω′ as θ 0, π. Turning to the behavior at the{ outer boundary} one finds→ that the linearized momentum factors decay, in the asymptotic region as R , according to → ∞ N˜p˜ 1 (4.32) ′ O , (2) −→ R4 g˜ Ne˜ 2γ v˜ 1 (4.33) p ′ O sin2 θ, (2) −→ R5 g˜ 2γ Ne˜ (˜u′ λr˜′) 1 (4.34) −p O sin2 θ, (2) −→ R4 g˜ Ne˜ 4γ r˜ 1 (4.35) p ′ O sin4 θ. (2) −→ R4 g˜ As noted earlier these are each multiplied by an ‘additional’ factor of 2 p N˜ = R sin θ 1 R+ but then paired with (the radial components) − R2 of terms of the form (4.29)–(4.30), etc. to determine the radial flux integrands as R . The (pure gauge)→∞ metric and wavemap perturbations, together with their needed radial derivatives, behave asymptotically as ( ) ( ) b2− + Mb1− cos(θ) 1 (4.36) γ′ + O −→ R2 R3 2Note that these results allow for a symmetrical extension of the perturbations through the background spacetime’s bifurcation 2-sphere. 36 VINCENT MONCRIEF AND NISHANTH GUDAPATI

( ) ( ) 2 b2− + Mb1− cos(θ) 1 (4.37) γ′ − + O ,R −→ R3 R4 ( ) 2 6Qab1− cos(θ) 1 (4.38) λ′ sin (θ) + O −→ R2 R3 ( ) ( ) 2 12Qab1− cos(θ) 1 (4.39) λ′ sin (θ) − + O ,R −→ R3 R4 ( ) ( ) 2 2Qb1− 1 (4.40) η′ sin (θ) + O −→ R R2 ( ) ( ) 2 2Qb1− 1 (4.41) η′ sin (θ) − + O ,R −→ R2 R3 ( ) ( ) 4 6Mab1− 1 (4.42) (ω′ + λη′) sin (θ) − + O −→ R R2 ( ) ( ) 4 6Mab1− 1 (4.43) ω′ sin (θ) + O ,R −→ R2 R3 ( )

Combining these results one ﬁnds that the ‘dynamical’ boundary ﬂux terms have the asymptotic decay properties

˜ 2 N (2) Ra ′ 1 (4.44) p˜′ √ h h γ,a O sin θ, (2)g˜ −→ R5 (4.45) p ˜ 2 N 2γ (2) Ra 2γ ′ 1 3 e (˜u′ λr˜′) √ h h e− η,a O sin θ, (2)g˜ − −→ R6 ˜ 2 2γ p N e (2) Ra 2γ ′ 1 3 (4.46) v˜′ √ h h e− λ,a O sin θ cos θ, (2)g˜ −→ R8 (4.47) p ˜ 2 4γ N e r˜′ 4γ (2) Ra ′ 1 5 e− √ h h (ω,a + λη,a) O sin θ, (2)g˜ −→ R8 (4.48) p ˜ 2 2γ N e (˜u′ λr˜′) (2) Ra 4γ 1 5 2 − λ′ √ h h e− (ω,a + λη,a) O sin θ cos θ. (2)g˜ −→ R11 p AXISYMMETRIC PERTURBATIONS 37

These rapid rates of decay, which clearly yield pointwise vanishing flux contributions at the outer boundary, reflect the fact that the perturbations are pure gauge in the asymptotic region R . Utilizing the pointwise decay rates of the perturbative→∞ quantities γ′, η′,λ′,ω′ + λη′,ω′ at the boundaries R R+ and R uncov- ered{ in this section it} is straightforward to verifyց that allր∞ of the corresponding boundary integrals potentially distinguishing EAlt from EReg actually vanish for the class of (asymptotically pure gauge) perturbations considered here. This result follows from a detailed evaluation of the asymptotic behavior (indeed rapid decay) of the coefficients of 2 2 2 the perturbative expressions (η′) +(λ′) , (ω′ + λη′) , occurring in the associated flux integrals.{ ···} Taking into account the regularity at the axes (corresponding to θ → 0, π) of the perturbative quantities γ′, η′,λ′,ω′ + λη′,ω′ developed in detail below in Section (4.4) and evaluating{ the corresponding} behavior of their coefficients in the flux expressions that potentially distinguish EAlt from EReg it is straightforward to verify that these integrals also vanish for the class of perturbations considered herein. Thus, for the class of (asymptotically pure gauge) perturbations that we consider EAlt = EReg.

4.2. Evaluating the ‘Kinematical’ Boundary Flux Terms. Con- sider next the boundary ﬂux contributions from Eq. (4.2) that are each ˜ a ∂ linear in the Lie derivative of a vector density = ∂xa taken from the list V V

∂ (4.49) ˜ := 4N˜√(2)hhacγ , VI ,c ∂xa

(2) ac 4γ ∂ (4.50) ˜ := N˜√ hh e− (ω + λη ) , VII ,c ,c ∂xa

(2) ac 2γ ∂ (4.51) ˜ := N˜√ hh e− λ , VIII ,c ∂xa

(2) ac 2γ ∂ (4.52) ˜ := N˜√ hh e− η , VIV ,c ∂xa ˜ := λ ˜ VV VII (4.53) 2Qra sin2 θ = ˜ , (r2 + a2 cos2 θ)VII

˜ c′ ∂ with respect to the linearized shift vector ﬁeld X′ := N ∂xc given, in (4) the chosen gauge, by Eq. (H.24). In the asymptotic regions where kαβ 38 VINCENT MONCRIEF AND NISHANTH GUDAPATI vanishes this expression reduces to

a′ ∂ (4) a 2 ab(4) 0 ∂ ′ ˜ ˜ X = N a = Y,0 N g˜ Y,b a (4.54) ∂x − ∂x ∂ = (2)Y a (2) a ,0 − D ∂xa where the explicit formulas for (4)Y a = (2)Y a and (2) a are given by Eqs. (4.7–4.10) and (4.19–4.24). D Recalling that, for a vector density ˜ = a ∂ , one has V V ∂xa

a c a a c (4.55) ( ′ ˜) =(X′ ) X′ LX V V ,c − ,cV it is straightforward to compute ′ ˜ , ′ ˜ and to pair these with LX VI ···LX VV their associated factors taken from the list γ′,ω′,λ′, η′ . From the regularity of these latter quantities at the horizon{ and the} readily verified vanishing of each of the radial components ( ′ ˜ )R, ( ′ ˜ )R LX VI ··· LX VV at this inner boundary it follows that the fluxn contributions of theseo Lie derivative terms are each (pointwise) vanishing at the horizon.3 Recalling Eqs. (4.36), (4.38), (4.40) and (4.42) we see that each of γ′,ω′,λ′, η′ decays of order O(1/R) or faster as R . It follows{ that the} corresponding (Lie derivative) flux integral→ expre∞ s- sions will make no contributions at the outer boundary provided that the associated radial components ( ′ ˜ )R, ( ′ ˜ )R , ( ′ ˜ )R, LX VI LX VII LX VIII ( ′ ˜ )R, ( ′ ˜ )R are all boundedn as R . A straightforward LX VIV LX VV →∞ computation of these quantitieso shows that this is indeed the case provided that (4)Y 0 and δN˜ are chosen so that the one-form components (4) 0 t δN˜ Y,a t=t0 + t dt′ ˜ and their θ-derivatives are sufficiently regular | 0 N ,a ˜ (4) 0 in this limit.R Since we shall eventually take δN = 0 and since Y t=t0 is at our discretion, this latter condition is trivial to arrange. |

4.3. Evaluating the ‘Conformal’ Boundary Flux Terms. Con- sider next the boundary ﬂux contributions from Eq. (4.2) that are each

3 (4) 0 ˜ ˜ ′ We are assuming that Y t=t0 and δN := N have been chosen so that the (4) 0 | t ′ δN˜ one-form components Y ,a t=t0 + dt ˜ and their θ derivatives are each | t0 N ,a non-singular at the horizon. R AXISYMMETRIC PERTURBATIONS 39 linear in the perturbed conformal factor ν′. These arise from the integrated divergence of the vector density

b ∂ (2) ba Q := 2( ′ N˜)(√ h h ν′ ) ∂xb LX ,a (4.56) n (2) ba b (2) ac ∂ + 2( ′ ν′)√ h h N˜ 2X′ (√ h h ν′ N˜ ) . LX ,a − ,a ,c ∂xb o One might first think to identify ν′, in the asymptotic regions where (4) (4) the ‘background perturbations’ kαβ and ℓα both vanish, with the (2) a ‘pure gauge perturbation’, Y ν,a, of the unperturbed conformal factor ν. Indeed one can, not surprisingly, verify directly that this choice combined with the complementary pure gauge perturbations (2) a (2) a Y γ,a, Y ω,a, satisfies all of the linearized constraint equations. As we shall··· see though this choice would leave an uncancelled flux contribution at the horizon boundary corresponding to R R+ and even to uncancelled, regularity violating flux terms at the symmeց - try axes corresponding to θ =0, π. The subtlety here is that the supposedly defining equation for the conformal factor ν (at the fully nonlinear level) was the decomposition (2) 2ν of the Riemannian metric g˜, viag ãb = e hab, into a conformal fac- 2ν tor e and a ‘conformal metric’ hab that was required to be flat (c.f., the discussion in Appendix F). But, thanks to a well-known conformal λ 2λ identity, valid in 2-dimensions, any metric of the form hab = e hab conformal to a flat metric hab will still be flat if and only if the function λ is harmonic (with respect to hab or, equivalently, to any metric conformal thereto). In other words the decomposition recalled above does not uniquely determine ν (and therefore also hab) at the nonlinear level and, of course, therefore also at the corresponding linearized level. Indeed, as one can easily see from the explicit form of the linearized Hamiltonian constraint (G.7), ν′ is only determined, without further information, up to a harmonic function of the metricg ãb (or, equivalently, of any metric conformal thereto). As a special case of the above recall that a pure gauge transformation of the flat-metric hab generated by an analytic change of coordinates (i.e., coordinates satisfying the Cauchy Riemann equations) automatically preserves hab up to a conformal factor of the above type (i.e., a factor e2λ with λ in fact harmonic). But such a conformal transformation to hab can, by convention, be absorbed unambiguously into an inhomogeneous transformation of the logarithm ν occurring in the 2ν ‘unified’ expression for the ‘physical’ 2-metricg ãb = e hab wherein hab, by conventional fiat, remains fixed while ν picks up an (additive) non-tensorial complement. 40 VINCENT MONCRIEF AND NISHANTH GUDAPATI

At the linearized level, wherein a pure gauge perturbation of (2)g˜ (2) has the (unambiguous) form (2) g˜, with L Y p (2) (2) a (2)p (2) a 2ν √(2) (2)Y g˜ = Y g˜ = Y e h L ,a ,a (4.57) p 2ν (2)pa (2) 2ν (2) a (2) = e Y √ h +2e Y ν,a√ h, ,a but where (by the convention of holding √(2)h ﬁxed) we regard this as a (pure gauge) variation of (2)g˜ of the form

(2) p(2) 2ν√(2) (4.58) (2)Y g˜ = δ g˜ =2e h δν L pure gauge p p it follows that this pure gauge perturbation, ν , of ν takes the ′ gauge form |

(2) a 1 1 (2) (2) a (4.59) ν′ = δν = Y ν,a + ∂a √ h Y . |gauge |pure gauge 2 √(2)h In the asymptotic regions near R+ and where (as discussed in Ap- (2) ∞ pendix H) Y is a conformal Killing ﬁeld of the ﬂat-metric hab it is straightforward to verify that the supplementary, ‘correction’ term

1 1 (2) (2) a 1 (2) (2) a (4.60) ∂a √ h Y = a(h) Y 2 √(2)h 2 ∇ is indeed harmonic with respect to hab, i.e., to check that 1 (4.61) (2) (h)(2) c(h) (2) (h)(2)Y a =0. ∇c ∇ 2 ∇a It follows therefrom that satisfaction of the linearized field equations, in particular the (linearized) Hamiltonian constraint, is not disturbed by the inclusion of this correction to ν′. b ∂ The integrals with respect to θ of the radial component of Q ∂xb , namely 2 R θ R+ Q =2ν′ X′ R sin θ 1+ ,θ R2 2 R R+ X′ cos θ 1 − − R2 (4.62) 2 R R+ +2ν′ X′ sin θ R 1+ ,R R2 2 θ 2 R+ + X′ cos θ R 1 , − R2 AXISYMMETRIC PERTURBATIONS 41 evaluated in the limits R and R R+ yield the potential flux contributions at these boundariesր ∞ for theց given, specific (‘corrected’, pure gauge) choice for ν′

(2) a 1 (2) (2) a (4.63) ν′ Y ν + (h) Y . → ,a 2 ∇a By using only the basic Green’s function asymptotics for the vector fields (2)Y and (2) characterized in Appendix H one finds that QR Y 1 vanishes (pointwise) at least of order O R as R . Using however the more immediately detailed Fourier expansionր∞ formulas for these quantities (c.f., Eqs. (4.7)–(4.11)) one finds that, thanks to a leading ( ) order cancellation of some terms involving the Fourier coefficient β1− , 1 the actual rate of decay is O R2 . By either reckoning the corrected, pure gauge perturbation, ν = (2)Y aν + 1 (2) (h)(2)Y a, ‘contributes’ ′ ,a 2 a pointwise vanishing flux at spatial infinity. ∇ At the other limit, on the other hand, the boundary flux angular integrand reduces to R Q 4R+ sin θ ν,R′ R R+ −−−−→ց (4.64) 4 (2) R (2) R R+ 4 Y ,0 + 2 2 2 Y . × (r + a ) R+ R+ R R+ ց Exploiting Eq. (H.30) one can show that

(4.65) (2) R 2 ∂ (2) a 1 (2) (2) a ∂ Y (2) R R+ Y ν,a + a(h) Y + Y 1 2 ∂R 2 ∇ −−−−→R R+ ∂R − R − R ց 2r(r2 + a2) a2 sin2 θ(r M) − − × (r2 + a2)2 a2∆ sin2 θ " − # (2) θ 2 2 2 2 2 Y cos θ (r + a ) 2a∆ sin θ (2) R + − 2 + Y,R . sin θ (r2 + a2)2 a2∆ sin θ )) R R+ − ց 2 ∂r R+ Recalling that ∂R = 1 R2 and employing Eq. (H.31) to express (in −(2) θ the asymptotic regions) Y,R via cos θ 1 cos θ (2)Y R (4.66) (2)Y θ ,R sin θ −→ − R sin θ R ,θ! we can exploit the chosen boundary condition (c.f., Appendix H), (4.67) (2)Y R 0, R R+ ց −→

42 VINCENT MONCRIEF AND NISHANTH GUDAPATI to deduce that

(2) a 1 (2) (2) a ν′ = Y ν,a + a(h) Y ,R R R+ ց 2 ∇ ,R R R+ ց ∂ (2)Y R = R ∂R ( R ,R) (4.68) R R+ ց (2) θ ∂ (2) θ = Y,θR = Y,R R R+ ∂θ ց R R+ ց 1 (2)Y R = =0 − R R ,θθ! R R+ ց where we have again exploited Eqs. (H.30) and (H.31) to reexpress derivatives of (2)Y R in terms of those of (2)Y θ and vice versa and appealed to the boundary condition (4.67) in the final step. Recalling Eq. (4.64) we thus see that the corresponding ‘conformal’ boundary flux integrand QR vanishes at the horizon boundary. R R+ ց To give the result (4.68) a precise geometrical interpretation recall that, in our notation (c.f. Eqs. (2.11)–(2.12) and (C.13), the ‘spatial’ Riemannian metric induced on a t = const. hypersurface is given by i j 2γ a b gij dx dx = e− gãb dx dx (4.69) ⊗ ⊗ + e2γ (dϕ + β dxa) (dϕ + β dxb) a ⊗ b where, in Weyl-Papapetrou coordinates xa = R, θ , { } { } (4.70)g ˜ dxa dxb = e2ν(dR dR + R2dθ dθ). ab ⊗ ⊗ ⊗ A It is straightforward to evaluate the first fundamental form, µAB dx dxB, and second fundamental form, λ dxA dxB, induced thereby⊗ AB ⊗ upon a (topologically spherical) surface R = R0 = const. and to AB calculate the mean curvature, trµ λ := µ λAB of the latter (where xA = θ,ϕ ). The result is { } { } γ ν 1 ∂ ν (4.71) tr λ = e − (Re ) µ − Reν ∂R which of course vanishes at the event horizon, R R+, of a Kerr- Newman black hole, the latter being a minimal surfaceց . Linearizing (4.71) about this background one finds that

γ ν ∂ (4.72) (trµ λ)′ e − ν′ . −−−−→R R+ − ∂R ց R=R+

AXISYMMETRIC PERTURBATIONS 43

Thus the boundary condition ν′ = 0 corresponds precisely to ,R R=R+ preserving minimality of the surface R = R at the linearized level. +

4.4. Axis Regularity and Evaluation of Flux Terms at the ‘Ar- tificial Boundaries’. The sections above have dealt with the evaluation of (potential) energy flux contributions at the actual boundaries corresponding to R and R R+ and established the pointwise vanishing of these fluxր ∞ expressionsց for the boundary conditions chosen. But the full flux integral formula (4.2) also includes a potential contribution from the ‘artificial boundaries’ provided by the axes of symmetry corresponding to θ = 0, π. Needless to say the evaluation of these potentially energy violating flux contributions hinges upon the regularity of the various fields involved at these axes of symmetry. Since, by assumption, we begin with a globally regular solution to the linearized field equations expressed in (say) a ‘hyperbolic gauge’ (c.f., Appendix B) and transform this solution to the desired gauge with an everywhere smooth gauge transformation (c.f.,Appendix H) the regularity of the resulting perturbations (as smooth tensorial fields on the spacetime manifold) is not in question. But is this smoothness and its implicit axial regularity sufficient to ensure the vanishing of the potential flux contributions? In this section we shall verify that this is indeed the case. An especially useful resource in this regard is the article [68] by Rinne and Stewart which derives the natural regularity conditions satisfied (at an axis of symmetry) by various smooth tensor fields (including scalar fields, vector fields, one forms and symmetric, second rank tensor fields) on a smooth, axi-symmetric spacetime under the assumption that the various ‘perturbations’ are themselves axi-symmetric. Exploiting their results in conjunction with our linearized field equations it is straightforward to evaluate the various ‘dynamical’ boundary flux terms and establish their (pointwise) vanishing at the symmetry axes to the following orders:

˜ Np˜′ (2) θa ′ 2 (4.73) N˜ √ hh γ,a O(sin θ) µ(2)g˜ ! → ˜ 4γ Ne r˜′ 4γ (2) θa ′ 2 (4.74) N˜ e− √ hh (ω,a + λη,a) O(sin θ) µ(2)g˜ ! → ˜ 2γ Ne v˜′ (2) θa 2γ ′ 2 (4.75) N˜ √ hh e− λ,a O(sin θ) µ(2)g˜ ! → 44 VINCENT MONCRIEF AND NISHANTH GUDAPATI

˜ 2γ Ne (˜u′ λr˜′) (2) θa 2γ ′ 2 (4.76) N˜ − √ hh e− η,a O(sin θ) µ(2)g˜ ! → ˜ 2γ Ne (˜u′ λr˜′) (2) θa 4γ 4 (4.77) N˜ − λ′√ hh e− (ω,a + λη,a) O(sin θ) µ(2)g˜ ! → Thus these versions ‘dynamical’ flux terms provide no (energy violating) contributions at the axes of symmetry, θ =0, π. Turning to the ‘kinematical’ flux terms involving the Lie derivatives of the vector densities (4.49)–(4.53) one finds, in the analogous way, that each of the factors X′ ˜I , , X′ ˜V has a regular (but, in general non-vanishing) limit atL theV axes··· of symmetryL V corresponding to θ =0, π. On the other hand, as discussed fully in Appendix E, each of the multiplicative factors ω′,λ′, η′ is, for the class of perturbations considered herein, required to{ vanish} on these axes of symmetry. In fact, as smooth scalar fields, they must vanish at least of order O(sin2 θ) as θ 0, π. Thus the corresponding energy flux terms vanish (pointwise) at→ these artificial boundaries. The remaining factor, γ′, however has a smooth (but in general, non-vanishing) limit as θ 0, π. The corresponding flux term has the limiting values →

θ 2 R R+ ′ ′ ˜ ′ ′ (4.78) γ X , I 4γ X cos θ 1 2 L V θ 0,π → ( − R ,R!) → θ=0,π

at the respective axes θ = 0, π over which it is to be integrated from R+ to . Fortunately,∞ however, this (in general non-vanishing) net ﬂux contribution combines naturally with the remaining ‘conformal’ ﬂux contribution comprised of the integrals (over the two axes) of (c.f. Eq. (4.56))

2 θ R R+ Qθ 0,π 2ν,R′ X′ cos θ 1 → → − R2 θ=0,π 2 R R+ = 2ν′X′ cos θ 1 (4.79) − R2 ( ,R 2 R R+ ′ ′ 2ν X cos θ 1 2 . − − R ,R) θ=0,π

The integrals with respect to R of the total derivative terms (evaluated at θ =0, π) from R to are readily shown to vanish by virtue of the + ∞ AXISYMMETRIC PERTURBATIONS 45

R limiting behavior of the factor 2ν′X′ along the axes, namely

R 1 (4.80) 2ν′X′ O θ=0,π −−−→R R2 ր∞ with this same quantity vanishing in the limit as R R+. In deriving this result one needs to exploit Eq. (H.30) togetherց with L’Hospital’s rule to show that, in the asymptotic regions along the axes, one has

(2) a 1 (2) (2) a (4.81) ν′ Y ν + (h) Y −−−→ ,a 2 ∇a (2) R 2 (2) R 2 Y ,R Y −−−→θ 0,π − R → R2 2r (4.82) + (2)Y R 1 + . − R2 r2 + a2 θ=0,π

R The vanishing at R+ follows from the vanishing of X′ θ=0,π there together with the regularity of ν′ in this limit. Note also the additional R2 factor of 1 + in the resulting ‘end point’ expression. − R2 It follows from the above that the full integrated ﬂux expression will vanish provided that

(4.83) ν′ =2γ′ along the axes of symmetry. One can, however, again appeal to the Rinne/Stewart results [68] to verify that (expressed in our notation)

(4.84) ν′ = 2γ′ |θ=0,π |θ=0,π for any regular (axi-symmetric) metric perturbation. This equivalence can also be checked explicitly in the asymptotic regions (along the axes) where the perturbations are pure gauge. It then follows that we have proven: Theorem 2. For the class of axisymmetric, asymptotically-pure-gauge Kerr-Newman perturbations considered herein (c.f., Appendices H and I) the positive-definite energy functional defined by Eqs. (3.16)–(3.17) is strictly conserved when the Weyl-Papapetrou and 2+1-dimensional maximal slicing gauge conditions are imposed. Remarks: It is somewhat curious to note that the ultimate vanishing of the net (potentially energy conservation violating) flux terms along the artificial boundaries provided by the axes of symmetry is obtained only after the ‘integration by-parts’ procedure outlined above is carried out. Note also that the additional boundary flux terms from Eq. (4.2) that are linear in the perturbed lapse function N˜ ′, vanish identically in 46 VINCENT MONCRIEF AND NISHANTH GUDAPATI our (2+1 –dimensional maximal slicing) gauge for which N˜ ′ 0. Such terms would need to be considered, however, in alternative gauges≡ for which N˜ ′ is non-vanishing. Perhaps the most interesting such choice is the 3+1 –dimensional maximal slicing gauge which is discussed in some detail in Appendix K. In particular we show therein that these additional (potentially energy conservation violating) boundary flux terms do indeed vanish as desired. While we do not actually prove the existence of this gauge for our problem (the necessary elliptic theory being rather technically involved) we thus nevertheless show that, if this gauge does indeed exist (as is most plausible), then our energy functional continues to be conserved upon employing it. It is clear from the form of Eq. (4.2) though that conservation of the energy depends only upon N˜ ′ through the vanishing of its (potential) boundary flux contributions and not upon the behavior of this quantity in the DOC’s ‘interior’. Thus any choice of linearized time guage which secures the vanishing of these boundary flux terms would yield a corresponding conservation result. This is the essential gauge invariance of our energy expression alluded to previously.

5. Summary, Concluding Remarks, and Outlook The mathematical problem of stability of black hole spacetimes is the subject of a long standing research program that dates back to the 1960s. Historically, an essential first step was to study the stability of such spacetimes with respect to linear scalar wave, Maxwell and linearized Einstein perturbations. To establish the stability of such black hole ‘backgrounds’ it is necessary to verify the boundedness and decay of the perturbations. Arguably, the most important obstacle to controlling the perturbations of rotating black holes is the fact that the energy of even linear waves propagating in such a spacetime is not necessarily positive- definite due to the ergo-region that always surrounds a black hole with non-vanishing angular momentum. This issue, which has both technical and physical ramifications, limits the immediate use of standard techniques for proving the decay of waves. From a mathematical perspective one should recall that the energy of the waves being not necessarily positive-definite is a consequence of the fact that the Killing vector ∂t is not globally timelike throughout a (rotating) black hole’s DOC, becoming instead spacelike within its ergo-region. An example of this phenomenon can be seen in the linear, scalar wave perturbations of Kerr black holes which fail to admit a conserved and positive-definite energy. In the special case of axially AXISYMMETRIC PERTURBATIONS 47 symmetric scalar wave perturbations, however, this problem evaporates since the troublesome, indefinite term in the energy density actually vanishes but the problem reappears for both axially symmetric Maxwell and linearized Einstein waves.4 Indeed, the lack of a positive-definite energy and the related so- called ‘super-radiance effect’ could in principle allow the perturbations to blow up exponentially, even in the axisymmetric case [63]. A common technique to exclude this possibility is to introduce a ‘blended’ vector field Tχ such as

(5.1) Tχ = ∂t + χ∂ϕ where χ is a suitable ‘cutoff function’ chosen so that Tχ is globally timelike and the corresponding energy is positive-definite. However, since this energy is not in general conserved, suitable Morawetz-type spacetime integral estimates are needed to establish its boundedness and for the cases of Maxwell and linearized Einstein waves propagating on Kerr backgrounds these techniques seem to be currently limited to the treatment of small, subextremal angular momentum, a M and little is known about the stability of Kerr or Kerr-Newman| |≪ spacetimes with respect to Maxwell and linearized Einstein perturbations in the case of arbitrary (subextremal) angular momentum, a < M. In this work, using Hamiltonian methods, we establish| | the existence of a conserved and positive-definite total energy for the fully coupled, axially symmetric Einstein-Maxwell perturbations of Kerr- Newman spacetimes for the entire subextremal range ( a , Q < M,a2+ Q2 < M 2). Our proof of energy conservation has necessitated| | | | a demonstration that a plethora of (potentially conservation violating) boundary flux terms actually all vanish. This argument was quite intricate in view of the elliptic nature of our chosen (Weyl-Papapetrou) gauge conditions which, in turn, were needed for the employment of the famous Carter-Robinson identities in their traditional form. These identities were needed to transform our energy expression into its desired positive-definite form. Our use of the Carter-Robinson identities exploits, of course, the wave map structure resulting, in a well-known way, from the dimensional reduction of the Einstein-Maxwell equations with one rotational isometry. The general methods developed herein can be used to study the stability of a variety of black hole spacetimes which exhibit analogous wave map structure. In [37] for example, it was shown that

4Indeed, as shown in Section 2, the conventional local energy density for axisymmetric Maxwell ﬁelds can be negative inside the ergo-region. 48 VINCENT MONCRIEF AND NISHANTH GUDAPATI

Lorentzian Einstein manifolds (i.e., those satisfying the Einstein equations with a non-vanishing cosmological constant) with one rotational isometry admit a Lagrangian dimensional reduction to the (2+1-dimensional) Einstein equations coupled to a ‘modified’ wave map system wherein the traditional wave map is shifted by a term in the cosmological constant. A crucial observation in this work was that the cosmological constant effectively decouples in such a way that it acts as a ‘source term’ for the wave map without destroying its essential geometric structure. Another application of the ideas developed herein is that one can use them to derive, for the axisymmetric, purely Maxwellian perturbations of a Kerr spacetime, a conserved, positive-definite energy functional expressible, albeit nonlocally, in terms of the Newman-Penrose scalars for the Maxwell field [40]. By contrast the conventional energy expression for these quantities, while local, fails to have the corresponding positivity. A first step towards extending this result to deal with the gravitational perturbations of Kerr black holes is presented in Appen- dix L wherein the Weyl tensor for vacuum axisymmetric spacetimes is expressed in terms of the wave map and 2+1-dimensional metric variables. For this special case of (axisymmetric, gravitational) Kerr perturbations one of us (N.G.) has shown how to correlate positive- definiteness of the perturbative energy to the negativity of the curvature of the corresponding wave map target space (hyperbolic 2-space) [39]. This argument is naturally covariant with respect to the target space geometry. As is well-known, for sufficiently smooth but non-stationary solutions to the linearized equations for a stationary background, one can derive a sequence of new solutions to the same equations by sequentially Lie differentiating a given solution with respect to the (asymptotically timelike) Killing field of the background. In standard coordinates adapted to the stationarity of the background, wherein the relevant Killing field, ζ, takes the form ζ = ∂t, this simply amounts to time differentiating the chosen, linearized solution as many times as its smoothness allows. At each stage of this procedure one can apply the linearized field equations themselves to reexpress time derivatives in terms of spatial ones, thus generating a family of solutions to the linearized equations built from sequentially higher order spatial derivatives of the initial one. For the Kerr-Newman problem in particular one can thus derive a sequence of higher order (conserved, positive definite) energy expressions which, combined with standard Sobolev inequalities, could, in principle, be exploited to derive corresponding uniform bounds on the perturbations. AXISYMMETRIC PERTURBATIONS 49

A well-known complication in this procedure, however, is the sequential occurrence, in each of these higher order energy expressions, of certain ‘weight factors’ arising from the background spacetime’s (2+1- dimensional) lapse function, N˜, R2 (5.2) N˜ = R sin θ 1 + − R2 which vanishes at the black hole’s horizon (R R+) and at the axes of symmetry (θ 0, π) and which blows upց (linearly) at spatial infinity (R ).→ One can see this phenomenon occurring already at the lowestր order ∞ wherein the formula (3.16) for Reg has an overall, E multiplicative factor of N˜. New such factors arise from each sequential time differentiation of the chosen perturbation when one applies the linearized field equations to replace time derivatives with spatial ones. Fortunately, however, the associated, so-called redshift effect arising from the vanishing of N˜ at the black hole’s horizon is a familiar one and has been analyzed in other, ‘model’ stability problems. Even so the use of Sobolev inequalities for the extraction of optimal uniform bounds on the perturbations from the higher order energy expressions is a technically intricate problem which we shall not pursue here. It is worth remarking though that, since the particular class of perturbations that we consider is, by construction, pure gauge in the asymptotic regions near the horizon and ‘near’ infinity, not to mention constructively regular at the axes of symmetry, the behavior of these perturbations in these asymptotic regions (and at the axes of symmetry) is not expected to be problematic. On the other hand the natural longer range aim of appli- cability of our (higher order) energies would encompass the treatment of more general classes of perturbative solutions and thus necessitate a more detailed analysis of this redshift effect in the asymptotic region near the horizon as well as one of the behavior at infinity and near the axes of symmetry. An interesting first step in this direction would be to carry out the corresponding analysis for the purely Maxwellian perturbations of the Kerr backgrounds considered in Section 2. Another potentially interesting application of our approach would be to the perturbations of (arbitrarily rapidly rotating) Kerr-Newman-de Sitter black holes arising through the inclusion of a positive cosmological constant Λ in the Einstein-Maxwell equations. As we have already mentioned in the Introduction, a fortuitous feature of the Carter- Robinson identity that plays such a crucial role in our program but which is normally applied to purely electrovacuum problems (i.e., those having Λ = 0), is that it only generates, thanks to a favorable sign in one of its terms that vanishes for electrovacuum backgrounds, a new 50 VINCENT MONCRIEF AND NISHANTH GUDAPATI term of positive sign in the presence of a positive cosmological constant. This feature (of the Carter-Robinson identity) has already been exploited by one of us (N.G.) to extend the arguments of Section 2 above to the treatment of the purely Maxwellian perturbations of Kerr-de Sitter black hole backgrounds [38]. As we have also alluded to in the Introduction there is the interesting potential of applying our approach to the analysis of stability of black holes in higher than 4 spacetime dimensions. The most significant open question in this regard would seem to be the stability of the famous 5-dimensional Myers-Perry rotating black hole solution in [62] and its (still not explicitly known) electrovacuum generalization [41]. For perturbations preserving the T 2, ‘axial’ isometry of such an axially symmetric background (c.f., [60] and [44]), one can apply well- known Kaluza-Klein reduction techniques to reduce the field equations to those of a wave map coupled to a 2+1-dimensional Lorentzian metric that closely resembles the system we have already treated [49, 52]. Fur- thermore the needed Carter-Robinson type identities for these (higher dimensional, reduced) field equations have already been derived and systematically applied to the development of corresponding black hole uniqueness theorems [43, 47, 46, 21]. In addition, the relevant linearization stability (LS) ‘technology’ can be readily extended to the higher dimensional setting of interest so that one should be able to generalize the arguments given herein to the treatment of such higher dimensional black holes. An attractive feature of the LS ‘machinery’ alluded to above is that, being essentially spacetime covariant in nature, it lends itself to the treatment of alternative slicings of the background such as those foli- ated by hypersurfaces that either intersect the future horizon or future null infinity, Scri+, or both instead of being ‘locked down’ at the bifurcation 2-sphere and at spacelike infinity, i0, as ours were required to do. Such alternative slices are not true Cauchy surfaces for the full DOC of a Kerr-Newman black hole but perturbative data given on them does uniquely control the evolution of such data to their causal futures. Fur- thermore the corresponding energy fluxes at the future horizon and at Scri+ are expected to have good signs (for the axisymmetric perturbations to which our formalism naturally applies) and thus to yield decaying (as opposed to strictly conserved) energy expressions (and their higher order generalizations). AXISYMMETRIC PERTURBATIONS 51

Acknowlegements This article is the outcome of several years of joint work and we would like to take this opportunity to express our gratitude to colleagues and institutions that provided conducive conditions to complete this work. We thank IHES for the gracious hospitality extended to both of us during the fall of 2016. N. Gudapati gratefully acknowledges the support of Deutsche Forschungs- gemeinschaft (DFG) Fellowships GU 1513/1-1 and GU 1513/2-1, hosted by the Department of Mathematics, Yale University and the Albert Einstein Institute (AEI, Golm) respectively. N. Gudapati also acknowledges the support from the Gordon and Betty Moore Foundation and the John Templeton Foundation through the Black Hole Initiative of Harvard University, during his postdoctoral stay at the Center of Math- ematical Sciences and Applications (CMSA). Finally, N.Gudapati expresses special thanks to Igor Frenkel for his postdoctoral stay at Yale University in the academic year 2017-2018 and Hermann Nicolai for hosting him at Albert Einstein Institute in the summer of 2016 and the fall of 2019.

Appendix A. Explicit Representations of Kerr-Newman Spacetimes Several diﬀerent coordinate systems for the Kerr-Newman, black hole spacetimes are employed in the present paper. We give these coordinate expressions here together with the explicit transformations connecting them. Except for the elementary degeneracies of the familiar angular coordinates for topological 2-spheres, each of these covers the domain of outer communications of the corresponding black hole in a non- singular way. They each, however, break down at the black hole’s event horizon which would necessitate a further transformation to be properly covered. We shall work throughout in ‘geometrical’ units for which Newton’s constant G and the speed of light c are both set to unity. Each Kerr-Newman black hole is characterized by three parameters, (M, a, Q), where M designates the mass, Q its electric charge and where a determines its angular momentum , along its axis of rotational symmetry, through = aM. These areS subject to the inequalities M > 0 and M 2 a2 +SQ2 with M 2 = a2 + Q2 corresponding to the extremal case. Solutions≥ violating either of these do not correspond to black holes. 52 VINCENT MONCRIEF AND NISHANTH GUDAPATI

In Boyer-Lindquist coordinates the line element and vector potential are given by ∆ a2 sin2 θ 2a sin2 θ(r2 + a2 ∆) ds2 = − dt2 − dtdϕ − Σ − Σ (r2 + a2)2 ∆a2 sin2 θ Σ (A.1) + − sin2 θdϕ2 + dr2 + Σdθ2 Σ ∆ Qr (A.2) A = − [dt a sin2 θdϕ] Σ − where (A.3) Σ := r2 + a2 cos2 θ and (A.4) ∆ := r2 2Mr + a2 + Q2 − The domain of the outer communications (or black hole ‘exterior’) is the region for which t R, ∈ (A.5) r>r := M + M 2 (a2 + Q2) + − and where the angles θ,ϕ , withpθ [0, π] and ϕ [0, 2π), label the points of topological 2-spheres{ } having∈t = constant∈ and r = constant. The black hole’s event horizon (not properly covered by these coordinates) lies at the limiting coordinate radius r = r+. When a = 0 the spacetime has precisely two independent Killing fields, 6 ∂ ∂ (A.6) ζ = and ψ = , ∂t ∂ϕ which correspond to its stationarity and axial symmetry whereas the special cases (a = 0, Q = 0) and (a = 0, Q = 0) yield the Reissner- Nordström and Schwarzschild6 solutions (respectively) which, each being spherically symmetric, admit two additional, rotational Killing fields. When a = 0 the Killing field ζ, which is timelike at large radius, becomes spacelike6 inside the so-called ‘ergo-region’ characterized by (A.7) r>r , ∆ a2 sin2 θ< 0. + − The presence of this region in these rotating cases causes serious difficulties for the task of finding positive energy expressions for the gravitational and electromagnetic perturbations. The main aim of this article is to construct such an energy for axisymmetric perturbations and to analyze its implications for the black hole stability problem in linear approximation. AXISYMMETRIC PERTURBATIONS 53

A transformation of the radial coordinate given by 1 (A.8) R = r M + r2 2Mr +(a2 + Q2) 2 − − with inverse p (M 2 a2 Q2) (A.9) r = R + M + − − 4R combined with the introduction of ‘isothermal’ coordinates defined via (A.10) ρ = R sin θ, z = R cos θ, puts the line element into Weyl-Papapetrou form (A.11) Σ (r2 + a2)2 a2∆ sin2 θ ds2 = ∆dt2 + − (dρ2 + dz2) (r2 + a2)2 a2∆ sin2 θ − R2 − ! 2 2 2 sin θ 2 2 2 2 2 a(2Mr Q )dt + (r + a ) a ∆ sin θ dϕ − 2 Σ − " − (r2 + a2)2 a2∆ sin θ !# − where ρ z (A.12) R = ρ2 + z2, sin θ = , cos θ = ρ2 + z2 ρ2 + z2 p and p p (M 2 a2 Q2) (A.13) r = ρ2 + z2 + M + − − 4 ρ2 + z2 p In these coordinates the domain of outer communicationsp corresponds to 1 1 R = ρ2 + z2 > (r M)= M 2 (a2 + Q2) (A.14) 2 + − 2 − :=pR 0. p + ≥ Note that R+ = 0 only in the extremal case. The Carter [15] and Robinson [69] identities, which play a crucial role in the present paper, are traditionally expressed in alternative variations of Weyl-Papapetrou coordinates in which the event horizon at r = r+ is mapped to an interval (or ‘cut’) along the symmetry axis. Recalling that solutions of the Cauchy-Riemann equations preserve the ‘isothermal’ form of the Riemannian 2-metric dρ2 +dz2 one easily shows that the transformation defined by (conjugate harmonic functions) (M 2 a2 Q2)ρ (A.15)ρ ¯ = ρ − − − 4(ρ2 + z2) 54 VINCENT MONCRIEF AND NISHANTH GUDAPATI and (M 2 a2 Q2)z (A.16)z ¯ = z + − − 4(ρ2 + z2) induces the conformal mapping C(ρ2 z2) C2 (A.17) dρ¯2 + dz¯2 = 1+ − + (dρ2 + dz2) 2(ρ2 + z2)2 16(ρ2 + z2)2 × where C = M 2 Q2 a2. The inverse transformation can be readily derived by exploiting− − the identity (A.18) C2 1 C (ρ2+z2)+ = (¯ρ2 +¯z2)+ (¯ρ2 +¯z2)2 +2C (¯ρ2 z¯2)+ 16(ρ2 + z2) 2 − 2 ( s ) to solve a quadratic equation for ρ2 + z2 in terms ofρ ¯ andz ¯. It is easily verified (for the non-degenerate cases having C > 0) that the horizon ‘semi-circle’ defined by 1 (A.19) ρ2 + z2 = R2 = M 2 (a2 + Q2) > 0 + 4 − gets mapped to a ‘cut’ on thez ¯ axis given by (A.20)ρ ¯ =0, z¯ M 2 (a2 + Q2), M 2 (a2 + Q2) . ∈ − − − For the degenerate casesh p (having C = 0) transformationp (A.15–A.16)i reduces to the identity and the horizon, in these coordinates, ‘collapses’ to a point. Finally, setting c := M 2 (a2 + Q2), consider the transformation defined by − p ρ¯ = (λ2 c2)1/2(1 µ2)1/2 − − (A.21) z¯ = µλ where c<λ< , 1 µ 1. It is readily verified that ∞ − ≤ ≤ dλ2 dµ2 (A.22) dρ¯2 + dz¯2 =(λ2 c2µ2) + − λ2 c2 1 µ2 − − In these coordinates the two symmetry axis components correspond to µ = 1 whereas the horizon occurs at λ c. The transformation in (A.21)± is readily inverted through the useց of the identity c2z¯2 (A.23) λ2 + = c2 +ρ ¯2 +¯z2 λ2 and the λ,µ coordinates play a key role in the Robinson identity presented{ in [69].} AXISYMMETRIC PERTURBATIONS 55

Appendix B. The Global Cauchy Problem for the Linearized Einstein-Maxwell Equations The Einstein-Maxwell equations, in the absence of a charged current source, are expressible, in their most general 4-dimensional form, as (B.1) 1 (4)Ein((4)g) := (4)Ric((4)g) (4)g (4)R((4)g) αβ − 2 αβ (4) (4) (4) =8π T ( g, F ) αβ 1 =2 (4) F (4)F (4)gµν (4)g (4)F (4)F µν , αµ βν − 4 αβ µν (B.2) (4) α (4) (4) αβ δ(4) F := F =0, g · ∇β (4) (4) (4) (4) d F := Fαβ,γ + Fβγ,α + Fγα,β (B.3) αβγ =0 where (4)g = (4)g dxµ dxν is the spacetime-metric, (4)Ric((4)g) and µν ⊗ (4)R((4)g) are its associated Ricci tensor and scalar curvature, (4)F = (4) µ ν (4) Fµν dx dx is the electromagnetic 2-form field and where α (or, equivalently⊗ ; α) designates covariant differentiation with respect∇ to (4)g. In the above and throughout we have set Newton’s constant of gravitation, G, and the speed of light, c, equal to unity by choice of units. We shall assume here and throughout that the field tensor (4)F is (4) (4) µ (4) (4) derived from a ‘vector potential’ A = Aµ dx such that F = d A or, in coordinates,

(4) (4) Fµν = [d A]µν (B.4) = (4)A (4)A ν,µ − µ,ν so that Eq. (B.3) is satisfied identically. Henceforth we regard (4)F as expressed, as above, in terms of (4)A and regard the pair (4)g, (4)A as the ‘fundamental fields’ upon which the field equations are{ imposed.} Designating the first variations (δ (4)g, δ (4)A) of ((4)g, (4)A) by

(4) (4) (4) µ ν (4) µ (B.5) ( h, A′)=( h dx dx , A′ dx ) µν ⊗ µ we can express the corresponding linearized equations as

(4) (4) (4) (4) (4) (4) (4) (4) (B.6) D Ein( g) h =8πD T ( g, A) ( h, A′), · · 56 VINCENT MONCRIEF AND NISHANTH GUDAPATI and (4) (4) (4) (B.7) D(δ(4) F ) ( h, A′)=0 g · · where D Ein((4)g) (4)h · αβ 1 (4)¯ ;µ (4)¯ ;µ (4)¯ ;µ = hαµ;β + hβµ;α hαβ;µ (B.8) 2 − n(4)g (4)h¯ ;µν (4)R((4)g) (4)h¯ − αβ µν − αβ (4) (4) (4) µν (4) + gαβ Ric( g) h¯µν and o (4) (4) (4) D(δ(4) F ) ( h, A′) g · · α 1 = (4)h (4)F µ;ν (4) F βγ((4)h (4)h ) − µν α − 2 αβ;γ − αγ;β (B.9) ;ν 1 (4)F β (4)h (4)g (4)hγ − α νβ − 2 νβ γ (4) ;µ (4) ;ν (4) (4) ν (4) ( A′ ) +( A′ ) + Ric( g) A′ − α ;µ ν ;α α ν (4) (4) (4) (4) (4) with D T ( g, A) ( h, A′) readily computable algebraically in terms of (4)h and · (4) (4) (4) D F ( A) A′ (B.10) · µν (4) (4) = ∂µ A′ ∂ν A′ . ν − µ (4) (4) µ ν In the above h¯ = h¯µν dx dx designates the ‘trace-reversed’ metric perturbation deﬁned by ⊗

(4) (4) 1 (4) (4) (4) αβ h¯µν := hµν gµν hαβ g (B.11) − 2 1 = (4)h (4)g (4)h γ µν − 2 µν γ which is readily inverted to yield 1 (B.12) (4)h = (4)h¯ (4)g (4)h¯ γ µν µν − 2 µν γ with (4)h γ := (4)h (4)gµν = (4)h¯ γ := (4)h¯ (4)gµν. γ µν − γ − µν As is well known, when the background field equations (B.1–B.3) are satisfied the corresponding linearized equations (B.6–B.7) are invariant with respect to an Abelian group of gauge transformations generated by pairs of the form (4)Λ, (4)Y where (4)Λ is a scalar field and (4) (4) µ ∂ { } Y = Y ∂xµ a vector field on the given background spacetime. The AXISYMMETRIC PERTURBATIONS 57

(4) (4) fundamental linearized variables ( h, A′) undergo the gauge transformations (4) (4) (4) (4) (B.13) A′ A′ + ∂ Λ+( (4) A) , µ → µ µ L Y µ (4) (4) (4) (4) (4) (B.14) F ′ := ∂ A′ ∂ A′ F ′ +( (4) F ) , µν µ ν − ν µ → µν L Y µν (4) (4) (4) (4) (4) (4) hµν hµν + µ Yν + ν Yµ (B.15) → ∇ ∇ (4) (4) = h +( (4) g) µν L Y µν (4) where (4)Y designates Lie diﬀerentiation with respect to Y and (4)L (4) (4) ν where Yµ := gµν Y is the latter’s covariant form. One can exploit this gauge invariance to impose the electromagnetic ‘Lorenz’ and gravitational ‘harmonic’ (or de Donder) gauge conditions given (respectively) by (4)¯ ;ν (B.16) hµν =0 and

(4) ;ν (B.17) A′ν =0. This is accomplished by solving the inhomogeneous wave equations ν (B.18) (4)Y ;ν + (4)Ric((4)g) (4)Y = (4)h¯ ;ν µ;ν µ ν − µν and

(4) ;µ (4) ;µ (4) ;µ (B.19) Λ = A′ ( (4) A) ;µ − µ − L Y µ (4) µ (4) for Yµ dx and Λ respectively. Theorems guaranteeing the global existence and uniqueness of solutions to the corresponding linear Cauchy problems, formulated on a globally hyperbolic spacetime, are proven in [67]. The solutions to (B.18) and (B.19) are, of course, only unique up to the addition of arbitrary solutions to the corresponding homogeneous equations. When the foregoing gauge conditions are imposed, the linearized ﬁeld equations (B.6) and (B.7) reduce to the manifestly hyperbolic, coupled system (B.20) 1 ρµ (4)h¯ ;µ + (4)Riem((4)g) (4)h¯ 2 − αβ;µ α β ρµ n (4) (4) ρ (4)¯ (4) (4) ρµ (4)¯ + Ric( g) β hαρ + Riem( g) β α hρµ ρ µν + (4)Ric((4)g) (4)h¯ (4)R((4)g) (4)h¯ + (4)g (4)Ric((4)g) (4)h¯ α βρ − αβ αβ µν (4) (4) (4) (4) (4) =8π D T ( g, A) ( h, A′) , o · αβ 58 VINCENT MONCRIEF AND NISHANTH GUDAPATI and (B.21) (4) ;ν (4) (4) ν (4) A′ + Ric( g) A′ − µ;ν µ ν 1 ;ν (4)h (4)F α;β (4)F β (4)h (4)g (4)hγ − αβ µ − µ νβ − 2 νβ γ 1 (4)F βγ (4)h (4)h − 2 µβ;γ − µγ;β =0 where [(4)Riem((4)g)]α ∂ dxβ dxγ dxδ is the Riemann curvature βγδ ∂xα ⊗ ⊗ ⊗ tensor of (4)g. To ensure satisfaction of the gauge conditions however one must restrict the choice of allowed Cauchy data for the above system accordingly. If Σ is a Cauchy hypersurface of the background spacetime (assumed here to be globally hyperbolic and time orientable) then one must impose

(4) ;ν α (4) ;ν (B.22) A′ν =0, n A′ν =0 Σ ;α Σ (4)¯ ;ν α (4)¯ ;ν (B.23) hµν =0, n hµν =0 Σ ;α Σ α ∂ where n ∂xα is the unit, future pointing normal field to Σ. To show that Eqs. (B.22–B.23) are both necessary and sufficient for the implementation and preservation of the gauge conditions we first (4) ;ν (4)¯ ;ν derive wave equations satisfied by the quantities A′ν and hµν . These are most easily obtained by computing the first variations of the identities ;µ (B.24) (4)F ;ν 0, (4)Ein((4)g) ;ν 0. µν ≡ µν ≡ Reducing the resultant variational identities through imposition of the gauge fixed field equations (B.20) and (B.21) leads directly to (4) ;ν ;µ (4) (4) α;β A′ν ;µ = hαβ K (B.25) 1 ;ν + (4)Kβ (4)h (4)g (4)h γ , νβ − 2 νβ γ and ;β β (B.26) (4)h¯ ;µ + (4)Ric((4)g) (4)h¯ ;µ =0 αµ ;β α βµ where

(4) (4) ;ν (B.27) Kµ := Fµν AXISYMMETRIC PERTURBATIONS 59 which of course vanishes when the background (Maxwell) ﬁeld equations are satisﬁed. In deriving Eq. (B.26) we have exploited the fact that

(4) ;ν (4) (4) (4) (4) (B.28) D T ( g, A) ( h, A′)=0 µν · when the background and the linearized (Maxwell) ﬁeld equations are satisﬁed. Under these circumstances we thus arrive at the homogeneous wave equations

(4) ;ν ;µ (B.29) A′ν ;µ =0, and ;β β (B.30) (4)h¯ ;µ + (4)Ric((4)g) (4)h¯ ;µ =0 αµ ;β α βµ (4) ;ν (4)¯ ;µ satisfied by the gauge fixing quantities A′ν and hαµ . By standard (4) ;ν (4)¯ ;µ results [67] one concludes that both A′ν and hαµ vanish throughout the globally hyperbolic, background spacetime with Cauchy surface Σ if and only if conditions (B.22–B.23) are satisfied on Σ. While it may seem that we have thus reduced the linearized field equations to a purely hyperbolic problem this conclusion is slightly misleading for the following reason. By combining the gauge fixed, linearized Maxwell equation (B.21) with the constraint upon the gauge fixing initial conditions (B.22) one arrives at

(4) ;ν µ µ (B.31) A′ν n = n Kµ′ =0 ;µ Σ Σ where the latter equality is precisely the usual, elliptic constraint upon linearized Maxwell initial data expressed in 4-dimensional notation (with Kµ′ the first variation of Kµ given explicitly by Eq. (B.9)). In a similar way, by combining the gauge fixed, linearized Einstein equation (B.20) with the gauge fixing initial condition (B.23) one arrives at (B.32) (4) (4) (4) (4) (4) (4) (4) (4) µ D Ein( g) h 8π D T ( g, A) ( h, A′) n · µν − · µν Σ n 1 o = (4)C nµ + (4)C nµ n (4)C ;µ 2 ν;µ µ;ν − ν µ Σ =0 where (4) (4)¯ ;β (B.33) Cα := hαβ 60 VINCENT MONCRIEF AND NISHANTH GUDAPATI and where the final equality follows from the imposition of the gauge fixing initial data constraints (B.23). But the resulting equation is precisely the usual, elliptic constraint upon the linearized Einstein initial data expressed in 4-dimensional form. Since we have already shown that the gauge conditions are preserved in time by the gauge fixed field equations it follows that the (linearized) Einstein-Maxwell constraint equations (B.34) (4) (4) (4) (4) (4) (4) (4) (4) µ D Ein( g) h 8π D T ( g, A) ( h, A′) n · µν − · µν n=0 o and µ (B.35) Kµ′ n =0 µ hold on an arbitrary Cauchy surface (with unit normal field n ∂µ) and not merely on the ‘initial’ one. The results given in this Appendix are, of course, simply a linearized version of the local existence and uniqueness theorem for the fully nonlinear Einstein-Maxwell equations proven by Choquet-Bruhat in Ref. [18]. But in view of the linear character of our field equations one can adapt arguments of the type presented in [67] to establish the global extendibility of solutions to the full, maximal Cauchy development of a chosen initial data surface. Thus, in particular, solutions generated from appropriate initial data will automatically extend to the full domain of outer communications of a background black hole solution that we choose to perturb. A well known, important feature of the hyperbolic form of the perturbation equations is that it guarantees the strictly causal propagation of the corresponding solutions. For a Kerr-Newman background, for example, this ensures that Cauchy data having ‘initially’ compact support, bounded away from the horizon and from spatial infinity, will retain this property for all finite Boyer-Lindquist time, t. This reflects the fact that Boyer-Lindquist time slices are ‘locked down’ at i0 (spacelike infinity) and at the bifurcation 2-sphere lying in the horizon. For the (causally propagating) purely Maxwellian perturbations of the Kerr spacetime analyzed in Section 2 it follows that, for such compactly supported initial data, the potential energy flux contributions at spatial infinity and the horizon, arising from the ‘continuity’ equation (2.45), will vanish identically. This leaves only the possibility of a non-vanishing energy flux at the ‘artificial’ boundary provided by the axes of symmetry at θ =0, π. To verify that these also vanish for AXISYMMETRIC PERTURBATIONS 61

θ regular perturbations one needs to evaluate reg, (c.f., Eq. (2.43)) at these axes. J From the discussion in Appendix E we know that the perturbative quantities, λ′ and η′ both vanish along the axes of symmetry, a fact that results from our demand that the electric and magnetic charges of the ‘background’ spacetime remain unperturbed. It then follows from the smoothness criteria developed in Ref. [68] that each of these functions vanishes sin2 θ at these axes. From Eqs. (2.21), (2.22) ∼ 2 and (2.37) it then follows that each ofu ˜′ andv ˜′ also vanishes sin θ at the axes and thus, after a straightforward calculation, that∼ θ Jreg vanishes sin2 θ as well. Consequently the energy HReg deﬁned via Eqs. (2.35)–(2.38)∼ is strictly conserved for this class of (spatially compactly supported) perturbations. A more comprehensive treatment would allow the Maxwellian perturbations to lie in suitable (weighted) Sobolev spaces and appeal to their (presumed) dense ﬁlling by the compactly supported solutions to establish the corresponding energy conservation result. While this would seemingly be straightforward to carry out, we shall not pursue it further here.

Appendix C. The Reduced Hamiltonian Formalism for Axi-Symmetric Spacetimes This article deals primarily with the linearized Einstein-Maxwell equations restricted to the domain of the outer communications, V, of a charged (if Q = 0), rotating (if a = 0) Kerr-Newman black hole. The coordinate6 systems discussed in Appendix6 A cover such domains and are adapted to the stationarity and axial symmetry of the Kerr-Newman solutions. Each such domain is a product of the form 3 V = R (R Bb) where Bb is a closed ball (or exceptionally a point) of coordinate× radius\ b 0. In the spatially cylindrical (Weyl-Papapetrou) coordinates t,ρ,z,ϕ≥ introduced in that appendix, for example, { } M 2 a2 Q2 (C.1) B = (ρ,z,ϕ) ρ2 + z2 b2 = − − 0 b ≤ 4 ≥ and the corresponding Kerr-Newman spacetime (restricted to V ) ad- ∂ mits ψ = ∂ϕ as a spacelike Killing field. Since we shall only consider perturbations that preserve this axial Killing symmetry it will be natural to pass to the corresponding quotient space V/U(1) (where U(1) is the circle action generated by ψ) and to formulate the linearized equations thereon. Since points on the symmetry axis are invariant under this group action (since ψ vanishes there) the resulting quotient space is a manifold with boundary of the 62 VINCENT MONCRIEF AND NISHANTH GUDAPATI form (C.2) V/U(1) = R M × b where Mb is the half-plane (ρ, z) ρ 0, z R with the half-disk 2 2 { 2 | ≥ ∈ } Db = (ρ, z) ρ 0, ρ + z b > 0 or point (ρ = z = 0) removed. { | ≥ ≤ } 2 2 The boundary points of Mb (i.e., those on the z-axis with z > b 0) correspond to those on the spacetime’s axis of symmetry. In this≥ appendix we shall focus on deriving the requisite linearized field equations at interior points of the quotient space R Mb (i.e. points in the complement of the boundary), keeping in mind× that certain geometrically natural regularity conditions will need to be imposed on the linearized fields, not only at the boundary axis but also at the background black 2 2 2 1 2 2 2 hole’s event horizon (corresponding to ρ +z b = 4 (M a Q ) 0) and, asymptotically, at ‘infinity’. Such regularityց conditions− − will be≥ necessary to ensure that linearized solutions on R Mb can be ‘lifted’ to yield sufficiently smooth and asympotically acceptable× perturbations on V. In coordinates xµ = t, xa, x3 of the aforementioned type for the { } 3 { } 0 a 1 2 4-manifold V = R (R Bb), where x = t, x = x , x = ρ, z and x3 = ϕ, we begin× by\ expressing the spacetime{ } { line element} { in} Arnowitt, Deser and Misner (ADM) form [53]:

2 (4) µ ν ds = gµνdx dx (C.3) = N 2dt2 + g (dxi + N idt)(dxj + N jdt) − ij where µ, ν, . . . range over 0, 1, 2, 3 while i, j, . . . range over 1, 2, 3 . For (4) (4) µ{ ν } { } the metric g = gµν dx dx to be properly Lorentzian it is essential that the ‘lapse function’ N⊗be nowhere vanishing and that the induced (3) i j metric, g = gijdx dx , and t = constant hypersurfaces be Riemann- ian. To avoid confusion⊗ with the lapse function N we shall designate the i ∂ ‘shift vector ﬁeld’, N ∂xi , in coordinate free notation, by X. When the spacetime V, (4)g admits an electomagnetic ﬁeld (4)F = (4)F dxµ dxν µν ∧ that is globally derivable from a ‘vector potential’ (4)A = (4)A dxµ then µ we have (C.4) (4)F = (4)A (4)A µν ν,µ − µ,ν and introduce an ADM parameterization for (4)A by setting

(4) i (C.5) A = A0dt + Aidx . Let Ω V be an arbitrary compact domain in V with (at least piecewise)⊂ smooth boundary ∂Ω. The Einstein-Maxwell equations (at AXISYMMETRIC PERTURBATIONS 63 interior points of Ω) follow from the ADM variational principle

ij i δgij IΩ = δπ IΩ = δAi IΩ = δ IΩ (C.6) E i = δN IΩ = δN IΩ = δA0 IΩ =0 (subject to suitable boundary conditions on the variations of the ﬁelds on ∂Ω) with

(C.7) I := d4x πijg + A i N N i A i Ω ij,t iE,t − H − Hi − 0E,i ZΩ where

1 ij 1 i 2 (3) 1 gij i j i j (C.8) = π πij (πi) µ(3)g R + ( + ), H µ(3) − 2 − 2 µ(3) E E B B g g j j k (C.9) i = 2πi j ǫijk H − | − E B with 1 (C.10) i = ǫijk(A A ). B 2 k,j − j,k (3) Here µ(3) and R are the volume element µ(3) = det g and g g | ij| (3) i j scalar curvature of the Riemannian metric g = gijdxp dx , designates covariant differentiation with respect to this metric⊗ and| spatial indices i, j, . . . are raised and lowered using (3)g and its inverse, (3) 1 ij ∂ ∂ g− = g ∂xi ∂xj . The (contravariant) symmetric tensor density (3) ij ∂ ⊗ ∂ (3) π = π ∂xi ∂xj is the momentum canonically conjugate to g ⊗ (3) i ∂ whereas the vector density = ∂xi is (up to sign) that conju- (3) i E E ijk gate to A = Aidx . The Levi-Civita symbols ǫijk and ǫ are covariant and contravariant, completely antisymmetric tensor densities (such 1 ijk 123 that µ(3)gǫijk and µ ǫ are tensor fields) satisfying ǫ123 = ǫ = 1. (3)g A derivation of this action principle from its (perhaps more familiar) Lagrangian form is presented in Chapter 21 of the text “Gravitation” by Misner, Thorne and Wheeler (MTW) [53]. Our notation differs somewhat from theirs in that we have absorbed a factor of 2 into the (4) (4) (3) i (3) i ∂ (3) i ∂ symbols A, F, A0, A = Aidx , = ∂xi and = ∂xi in order to simplify the forms of the electromagneticE E HamiltonB equationB s. ijk In addition we write µ(3)g for their √g and use ǫ and ǫijk instead of [ijk] to designate the Levi-Civita tensor densities. To recover the i i ∂ i ∂ expressions of MTW one should replace our A0, Aidx , ∂xi and ∂xi i i ∂ i ∂ E B by 2A0, 2Aidx , 2 ∂xi and 2 ∂xi respectively, write √g in place of our E B ijk µ(3)g and substitute [ijk] for our ǫ and ǫijk. Now restrict attention to those Lorentizan metrics on V which have ∂ the circle action generated by ψ = ∂ϕ as a (spacelike) isometry group 64 VINCENT MONCRIEF AND NISHANTH GUDAPATI and impose the corresponding (U(1)) invariance on (4)A by demanding that

(4) ∂ (4) (C.11) ∂ g = gµν =0 L ∂ϕ µν ∂ϕ and

(4) ∂ (4) (C.12) ∂ A = Aµ =0. L ∂ϕ µ ∂ϕ One can now express the field equations alluded to above entirely in terms of fields induced on the quotient space R Mb. To this end it is convenient to reparametrize the (U(1)-invariant)× Lorentzian metric (4)g on V by setting (C.13) 2 (4) µ ν 2γ 2 2 a a b b ds = g dx dx = e− N˜ dt +˜g (dx + N˜ dt)(dx + N˜ dt) µν − ab + e2γ dϕ + β dt + β dxn a 2 o { 0 a } and, correspondingly, to write (4) µ a (C.14) Aµdx = A0dt + Aadx + A3dϕ for the (U(1)-invariant) vector potential. Here, a, b, . . . range only over 1, 2 the indices for coordinates for Mb. Abusing notation slightly {we shall} employ the same symbols to designate the fields induced, in Kaluza-Klein fashion, on the quotient space. At interior points of the quotient space (i.e., on the complement of the symmetry axis) we may regard (C.15) dσ2 := N˜ 2dt2 +˜g (dxa + N˜ adt)(dxb + N˜ bdt) − ab as the ADM form ofn the line element for an induced, 2+1-dimensional,o 2γ (4) ∂ ∂ (4) ∂ Lorentz metric and view e = g ∂ϕ , ∂ϕ and A3 = A, ∂ϕ as a a induced functions and β0dt + βadx and A0dt + Aadx asD inducedE one- forms on (interior points of) the quotient space R Mb. Note however that since e2γ must vanish at boundary points of× this quotient (which corresponds to points on the symmetry axis in V ), the function γ must entail a logarithmic singularity in this limit and, accordingly, N˜ and gãb must incorporate a singular (vanishing at the boundary) conformal 2γ factor to cancel the singularity coming from e− . While one could explicitly remove these singularities from the base fields by a change of parametrization the elegant form of the projected field equations (at interior points of R M) would thereby be disturbed. To avoid this we shall retain the notation× introduced above, keeping in mind that certain fields induced on the quotient must exhibit well-defined singular AXISYMMETRIC PERTURBATIONS 65 behaviors at the boundary in order to ‘lift’ naturally to yield smooth fields on V. The background Kerr-Newman solutions of course automatically exhibit this (geometrically natural) singular behavior when parametrized as above (c.f., Eqs. (A.11–A.13) of Appendix A) and we shall need to impose suitable regularity conditions on their perturbations in order that such perturbations lift smoothly back to V. For the moment however we shall focus on transforming the projected field equations at interior points of the quotient and postpone the discussion of the regularity conditions needed at the boundary until later. Letting represent an arbitrary compact domain in Mb, disjoint D a ab from the boundary, define momenta p,˜ e˜ , π˜ conjugate to γ, βa, gãb by setting { } 3 ij 3 ab a d x π gij,t = d x π˜ gãb,t +˜e βa,t +˜pγ,t S1 S1 (C.16) ZD× ZD× 2 ab a =2π d x π˜ gãb,t +˜e βa,t +˜pγ,t . ZD This leads, together with (C.3) and (C.13) to relations such as ab 2γ ab 2γ 2γ (C.17)π ˜ = e− π , gab = e− gãb + e βaβb, etc. which can be read off from the above defining expression. To incorporate the electromagnetic terms introduce also the definitions (C.18) fã = (˜ea aA ), F 3 = ( 3 + β a) −E 3 − E aE (C.19) C = (A β A ), C = (A β A ) a − a − a 3 0 − 0 − 0 3 and reexpress the ADM action in terms of the new variables. The result (modulo an inessential boundary term) is expressible (on domains of the form Ω = [t , t ] S1) as 0 1 × D × t1 2 ab a I˜Ω := dt d x π˜ gãb,t +˜pγ,t + f˜ βa,t t0 (C.20) Z ZD n +F 3A + aC + β fã + C a N˜ ˜ N˜ a ˜ 3,t E a,t 0 ,a 0E,a − H − Ha =(I /2π) (boundary term) o Ω − where ˜ 1 ab a 2 1 2 1 4γ ã a ˜b b = π˜ πãb (˜π a) + p˜ + e− gãb(f + A3)(f + A3) H µ(2) − 8 2 E E g˜ (2) ab 1 4γ ac bd + µ(2) R˜ +2˜g γ γ + e g˜ g˜ (β β )(β β ) g˜ − ,a ,b 4 a,b − b,a c,d − d,c 1 2γ 3 2 2γ ab 2 + e (F ) + e ǫ (Ca,b A3βa,b) 2µ(2) − g˜ h 66 VINCENT MONCRIEF AND NISHANTH GUDAPATI

(C.21) 2γ a b ac bd +e− g˜ ( + ǫ A ǫ A ) , ab E E 3,c 3,d (C.22) i ˜ = 2 (2)˜ π˜b +˜pγ + F 3A + f˜b(β β )+ b(C C ). Ha − ∇b a ,a 3,a b,a − a,b E b,a − a,b In these formulas indices a, b, . . . are raised and lowered using the Rie- (2) 2 b (2) manian 2-metric g˜ =g ãb dx dx , R˜ is the scalar curvature of this (2)˜ ⊗ metric, a its covariant derivative operator and µ(2)g˜ its volume ele- ∇ ab ment (µ(2)g˜ := det gãb ). In addition ǫ is the antisymmetric tensor ab| | 12 density (such that ǫ /µ(2) is a tensor) satisfying ǫ = 1. p g˜ The constraint equations are obtained by varying I˜Ω with respect to a N,˜ N˜ , β0 and C0 and are thus given by (C.23) ˜ = ˜ = fã = a =0. H Ha ,a E,a The evolution equations are obtained by varying I˜Ω with respect to ab a 3 a the canonical variables gãb, π˜ , γ, p,˜ βa, f˜ , A3, F ,Ca, . There are { E } a neither constraints nor evolution equations for the quantities N,˜ N˜ , β0 and C0 which must be fixed (either explicitly or implicitly) by a choice of gauge. At fixed t the constraint equations fã = 0 and a = 0 may, on the ,a E,a topologically trivial space Mb, be solved in generality by setting ã ab (C.24) f = ǫ ω,b (C.25) a = ǫabη E ,b where ω and η are uniquely determined up to additive constants (that a ã can vary with t). The Hamilton equations for ,t and f,t may be ma- nipulated to yield E (C.26) ˜ 2γ Ne ab b a η,t = ǫ (Ca,b A3βa,b)+ ǫabN˜ + f(t) µ(2)g˜ − E (C.27) ˜ 2γ ˜ 4γ Ne ab Ne ab b ã ω,t = A3ǫ (A3βa,b Ca,b)+ ǫ βa,b + ǫabN˜ f + k(t) µ(2)g˜ − µ(2)g˜ where f(t) and k(t) are certain undetermined functions of t which arise from passing from the equations for a = (ǫabη ) and fã = (ǫabω ) E,t ,t ,b ,t ,t ,b to those for η,t and ω,t. Since, however, ω and η are only determined by (C.24) and (C.25) up to arbitrary additive functions of t we may smoothly resolve the ambiguity in their definitions (up to additive, true constants) by demanding that f(t)= k(t) = 0. AXISYMMETRIC PERTURBATIONS 67

Defining ab ab (C.28)r ˜ = ǫ βa,b, u˜ = ǫ Ca,b we therefore fix the equations of motion for the ‘twist potentials’ η and ω to be ˜ 2γ Ne b (C.29) η,t = (˜u A3r˜)+ N˜ η,b µ(2)g˜ − ˜ 2γ ˜ 4γ Ne Ne a (C.30) ω,t = A3(A3r˜ u˜)+ r˜ + N˜ ω,a µ(2)g˜ − µ(2)g˜ These equations, together with all the remaining evolution and constraint equations, may be derived from the reduced action J˜Ω obtained from I˜Ω by substituting the expressions (C.24), (C.25) and (C.28) and discarding an inessential boundary term. Upon defining 3 (C.31) λ = A3, v˜ = F we get, for the reduced action, (C.32) t1 2 ab a J˜Ω = dt d x π˜ gãb,t +˜pγ,t +˜rω,t +˜uη,t +˜vλ,t N˜ ˜ N˜ ã t0 − H − H Z ZD n o where ˜ and ˜ now take the forms H Ha (C.33) ˜ 1 ab a 2 1 2 1 4γ 2 1 2γ 2 2 = π˜ πãb (˜π a) + (˜p) + e (˜r) + e v˜ +(˜u λr˜) H µ(2) − 8 2 2 − g˜ (2) ab 1 2γ ab 1 4γ ab + µ(2) R˜ +2˜g γ γ + e− g˜ (η η + λ λ )+ e− g˜ (ω + λη )(ω + λη ) , g˜ − ,a ,b 2 ,a ,b ,a ,b 2 ,a ,a ,b ,b (C.34) ˜ = 2 (2)˜ π˜b +˜pγ +˜rω +˜vλ +˜uη Ha − ∇b a ,a ,a ,a ,a a Variation of J˜Ω with respect to N˜ and N˜ yields the remaining constraints ˜ = 0 and ã = 0 whereas variation with respect to the H Hab canonical pairs (˜gab, π˜ ), (γ, p˜), (ω, r˜), (η, u˜), (λ, ˜v) yields the Hamil- tonian evolution equations for the reduced system. It is well-known, though perhaps less evident in the present Hamiltonian setting, that this set of reduced field equations is (at interior points of R Mb) equivalent to the 2+1-dimensional Einstein equations (for the Lore× ntz metric given in (C.15)) minimally coupled to a wave map defined by the four scalar fields γ,ω,λ,η . The naturally occurring target space for this wave map (whose{ metric} can be read off from the expression 68 VINCENT MONCRIEF AND NISHANTH GUDAPATI

(C.33) for ˜) is the Riemannian 4-manifold (R4,dk2) with line element H 2 2 2γ 2 2 4γ 2 (C.35) dk = 4(dγ) + e− (dη + dλ )+ e− (dω + λdη) which can be recognized as a (global) coordinate representation of complex hyperbolic space. If the Maxwell ﬁeld is ‘turned oﬀ’ so that only vacuum spacetimes are considered then

2 2 4γ 2 (C.36) dk 4(dγ) + e− (dω) −→ which, defined over R2, is nothing but a coordinate representation for real hyperbolic space. Some background on this 4-dimensional target space and its 8-dimensional isometry group U(2, 1) is given in Ref. [59] and in further references cited therein, andS will not be included here. In particular though Eq. (2.60) of this reference lists, explicitly, a set of eight (locally) conserved quantities that one builds appealing to Noether’s theorem from the eight independent Killing fields of the target metric. To reconstruct an Einstein-Maxwell field on V from a solution to the reduced field equations on R Mb one needs to reconstruct the a × a one forms β0dt + βadx and C0dt + Cadx of which only the ‘transverse ab ab projections’r ˜ = ǫ βa,b andu ˜ = ǫ Ca,b directly survive (as momenta conjugate to the wave map variables ω and η) in the reduced formulation. The time components, β0 and C0, of the one-forms are essentially gauge variables and can be chosen arbitrarily together with initial data for βa and Ca compatible with (C.28). To recover βa and Ca one inte- grates the Hamiltonian equations for these quantities, which, expressed in terms of wave map variables, take the form (C.37) ˜ b N 4γ bc βa,t = β0,a + N˜ ǫabr˜ + e− gãbǫ (ω,c + λη,c) µ(2)g˜ (C.38) ˜ 2γ ˜ 4γ Ne− bc b Ne− bc Ca,t = C0,a + gãbǫ η,c + N˜ ǫabu˜ + gãbλ ǫ (ω,c + λη,c) µ(2)g˜ µ(2)g˜ Upon reverting to the original notation one finds that Eqs. (C.37) and (C.38) are indeed equivalent to the original Hamilton equations for these fields (derivable from the action I˜Ω) and that they guarantee preservation of the defining equations given in (C.28). The remaining Hamiltonian evolution and constraint equations also revert to their original forms. AXISYMMETRIC PERTURBATIONS 69

Needless to say all of the above equations are automatically satisfied by the Kerr-Newman fields. Our main aim is to study linear perturbations of these ‘backgrounds’ and, in particular, to do so within the reduced Hamiltonian framework sketched above. To this end however it is first necessary to compute the twist potentials ω and η for these Kerr- Newman backgrounds since these potentials cannot be simply read off the explicit formulas for (4)g and (4)A. From the formulas given in Appendix A one sees immediately that, in the chosen coordinate systems, βa = Aa = 0 from which it follows, via the definitions (C.19) and (C.28) that Ca =0, r˜ =0andu ˜ = 0. Noting ˜ a ∂ also that the (2+1-dimensional) shift vector field X = N ∂xa vanishes as well one sees, from Eqs. (C.29) and (C.30) that ω,t = η,t = 0, as one should have expected for a stationary solution. From the Hamilton ab equations forg ãb,γ and λ it also follows thatπ ˜ =p ˜ =v ˜ = 0 for these (stationary) Kerr-Newman backgrounds. Reading off the (Boyer-Lindquist) coordinate expressions a(2Mr Q2) (C.39) β0 = − − , (r2 + a2)2 a2∆ sin2 θ − 2Qra sin2 θ (C.40) λ = , r2 + a2 cos2 θ 2Qr(r2 + a2) (C.41) C0 = , (r2 + a2)2 a2∆ sin2 θ − ˜ 1/2 (C.42) N = ∆ sin θ, sin2 θ (C.43) e2γ = (r2 + a2)2 a2∆ sin2 θ r2 + a2 cos2 θ − and

1 a b 1/2 2 1/2 2 (C.44) gãbdx dx = ∆− dr + ∆ dθ µ(2) g˜ where ∆ = r2 2Mr + a2 + Q2 and substituting these expressions into Eqs. (C.37)− and (C.38) one arrives at a system of first order linear equations for the unknowns ω and η. The integrability conditions for this system are readily verified and the system integrated to yield (with a particularly simple choice for the arbitrary additive constants) (C.45) 4Q(a2 + r2)cos(θ) η = − a2 +2r2 + a2 cos(2θ) 70 VINCENT MONCRIEF AND NISHANTH GUDAPATI and

(C.46) ω = aM cos(θ)(5 cos(2θ)) − 4a3 cos(θ) sin4 (θ)[a2M +2r(Q2 + Mr)+ a2M cos(2θ)] + (a2 +2r2 + a2 cos(2θ))2 Note that these yield

(C.47) η(r, 0) η(r, π)= 4Q − − and

(C.48) ω(r, 0) ω(r, π)=8aM − for the (unambiguous) diﬀerences of these functions on the upper and lower symmetry axes (which thread through ‘wormholes’ in the analytically extended black hole spacetimes and are actually disjoint). Though one can readily derive the reduced ﬁeld equations by variation of the reduced action J˜Ω (c.f. Eqs. (C.32)–(C.34)) we present them here explicitly to lay the groundwork for their linearization. The evolu- ab tion equations for the canonical pairs (γ, p˜), (ω, r˜), (η, u˜), (λ, ˜v), (˜gab, π˜ are given by: (C.49) N˜p˜ γ,t = + X γ, 4µ(2)g˜ L

(C.50) ˜ ˜ 2N 4γ 2 N 2γ 2 2 p˜,t = − e (˜r) e v˜ +(˜u λr˜) ( µ(2)g˜ − µ(2)g˜ − ˜ ab ˜ 2γ ab + 4(Nµ(2)g˜ g˜ γ,a),b + Nµ(2)g˜ e− g˜ (η,aη,b + λ,aλ,b)

˜ 4γ ab + 2Nµ(2)g˜ e− g˜ (ω,a + λη,a)(ω,b + λη,b)+ X p˜ , L )

(C.51) Ne˜ 4γ Ne˜ 2γ ω,t = r˜ + λ(λr˜ u˜)+ X ω, µ(2)g˜ µ(2)g˜ − L AXISYMMETRIC PERTURBATIONS 71

(C.52) ˜ 4γ ab r˜,t = Nµ(2)g˜ e− g˜ (ω,a + λη,a) + X r˜ , ,b L (C.53) Ne˜ 2γ η,t = (˜u λr˜)+ X η, µ(2)g˜ − L

(C.54) ˜ 2γ ab ˜ 4γ ab u˜,t = (Nµ(2)g˜ e− g˜ η,a),b + Nµ(2)g˜ e− g˜ λ(ω,a + λη,a) + X u˜ , ,b L (C.55) Ne˜ 2γ v˜ λ,t = + X λ, µ(2)g˜ L

(C.56) ˜ N 2γ ˜ 2γ ab v˜,t = e r˜(˜u λr˜)+(Nµ(2)g˜ e− g˜ λ,a),b (µ(2)g˜ −

˜ 4γ ab Nµ(2)g˜ e− g˜ (ω,a + λη,a)η,b + X v˜ , − L )

(C.57) ˜ 2N cd (2) g˜ab,t = (˜gacg˜bd g˜abg˜cd)˜π +( X g˜)ab, µ(2)g˜ − L

(C.58) ˜ ab 2N ac bd ab c ab π˜,t = − [˜π π˜ g˜cd π˜ π˜c]+( X π˜) ( µ(2)g˜ − L ˜ 1 N ab cd c 2 ab ab c ˜ ˜ | + g˜ π˜ π˜cd (˜πc) + µ(2)g˜(N | g˜ N c) 2 µ(2)g˜ − − | 1 N˜ 1 1 1 + g˜ab (˜p)2 + e4γ (˜r)2 + e2γ v˜2 +(˜u λr˜)2 2 µ(2) 8 2 2 − g˜ 72 VINCENT MONCRIEF AND NISHANTH GUDAPATI

ac bd 1 ab cd 1 2γ + Nµ˜ (2) g˜ g˜ g˜ g˜ 2γ γ + e− (η η + λ λ ) g˜ − 2 ,c ,d 2 ,c ,d ,c ,d 1 4γ + e− (ω + λη )(ω + λη ) 2 ,c ,c ,d ,d ) whereas the constraints are now simply (C.59) ˜ = 0 and ˜ =0 H Ha with ˜ and ã defined by Eqs. (C.33) and (C.34). In the above formu- H H ˜ a ∂ las the Lie derivatives with respect to X = N ∂xa of scalars (γ,ω,η,λ) are simply their directional derivatives with, for example, (C.60) γ = N˜ aγ , LX ,a whereas those of the scalar densities (˜p, r,˜ u,˜ v˜) are (C.61) p˜ =(N˜ ap˜) , etc. LX ,a (2) a b (2) while those for the tensor g˜ =g ãbdx dx and tensor density π˜ := πãb ∂ ∂ are ⊗ ∂xa ⊗ ∂xb (2) c c c ( X g˜)ab = N˜ gãb,c + N˜ g˜cb + N˜ gãc (C.62) L ,a ,b = Nã b + N˜b a | | and (C.63) ( (2)π˜)ab =(N˜ cπãb) N˜ aπ˜cb N˜ b πãc LX ,c − ,c − ,c respectively. The two dimensional indices a, b, . . . are raised and low- (2) (2) 1 ab ∂ ∂ ered using g˜ and g˜− :=g ˜ a b whereas covariant differentia- ∂x ⊗ ∂x tion with respect to (2)g˜ is designated by a vertical bar. The last two of equations (C.49)–(C.58) together with the constraints (C.59) comprise the 2+1-dimensional Einstein equations with a wave map source whereas the first eight of these equations are the corresponding (curved space) wave map equations in Hamiltonian form. The Kerr-Newman solutions given explicitly in Appendix A are of course stationary and have vanishing (2+1-dimensional) shift, X = ˜ a ∂ N ∂xa = 0. It follows immediately from Eqs. (C.49)–(C.58) that all of the canonical momenta vanish, i.e., that (C.64)p ˜ =r ˜ =u ˜ =v ˜ =π ãb =0 and therefore that the evolution equations reduce to (C.65)

˜ ab ˜ 2γ ab 4(Nµ(2)g˜ g˜ γ,a),b + Nµ(2)g˜ e− g˜ (η,aη,b + λ,aλ,b) n AXISYMMETRIC PERTURBATIONS 73

˜ 4γ ab +2Nµ(2)g˜ e− g˜ (ω,a + λη,a)(ω,b + λη,b) =0, o (C.66) ˜ 4γ ab Nµ(2)g˜ e− g˜ (ω,a + λη,a) =0, ,b (C.67) ˜ 2γ ab ˜ 4γ ab (Nµ(2)g˜ e− g˜ η,a),b + Nµ(2)g˜ e− g˜ λ(ω,a + λη,a) =0, ,b (C.68) 2γ ab 4γ ab (Nµ˜ (2) e− g˜ λ ) Nµ˜ (2) e− g˜ (ω + λη )η =0, g˜ ,a ,b − g˜ ,a ,a ,b n o (C.69)

ab ab c ac bd 1 ab cd ˜ ˜ | ˜ µ(2)g˜(N | g˜ N c)+ Nµ(2)g˜ g˜ g˜ g˜ g˜ − | − 2 × 1 2γ 1 4γ 2γ γ + e− (η η + λ λ )+ e− (ω + λη )(ω + λη ) =0, ,c ,d 2 ,c ,d ,c ,d 2 ,c ,c ,d ,d whereas the Hamiltonian constraint, ˜ = 0, takes the form H

(2) ab 1 2γ ab µ(2) R˜ +2˜g γ γ + e− g˜ (η η + λ λ ) g˜ − ,a ,b 2 ,a ,b ,a ,b (C.70) 1 4γ ab + e− g˜ (ω + λη )(ω + λη ) =0 2 ,a ,a ,b ,b while the momentum constraint, ˜a = 0, is satisﬁed identically. Note especially that the trace ofH Eq. (C.69) results in the formula

c ˜ | (C.71) N c =0. |

This fact that the (2+1-dimensional) lapse for Kerr-Newman solutions is harmonic will play an important role in our treatment of the linearized equations. 74 VINCENT MONCRIEF AND NISHANTH GUDAPATI

Appendix D. Covariance and regularity of the fundamental wavemap fields It is clear from their definitions in terms of the axial Killing field, ψ = ψµ ∂ ∂ , that the wavemap variables ∂xµ → ∂ϕ (D.1) e2γ := (4)g ψµψν (4)g µν → ϕϕ and (D.2) λ := ψµ (4)A A = A µ → 3 ϕ both transform as spacetime scalars5. On the other hand the covariance properties of the complementary variables, ω and η, are not immediately evident from our (reduced Hamiltonian framework) introduction of these objects in Appendix C. As we shall show herein however, all the wavemap fields do indeed transform as spacetime scalars. It will then follow that their corresponding first variations, γ′,λ′,ω′, η′ , undergo linearized gauge transformations of the familiar{ form }

(D.3) γ′ γ′ + (4) γ λ′ λ′ + (4) λ, → L Y → L Y (D.4) ω′ ω′ + (4) ω η′ η′ + (4) η → L Y → L Y (4) (4) µ ∂ where Y = Y ∂xµ is an arbitrary spacetime vector field that com- mutes with ψ. Recall that, in the absence of sources, both the electromagnetic 2- form field (4)F and its Hodge dual ⋆(4)F are closed, (D.5) d (4)F =0, d⋆ (4)F =0. Combined with its invariance under axial rotations, (D.6) ⋆ (4)F =0, Lψ the closure of ⋆(4)F implies the closure of the corresponding 1-form field (4) (4) µ µ (4) ν Ω= Ωµdx := ψ ⋆ Fµνdx (D.7) 1 1 µναβ (4) (4) γ = ψµ ǫ Fαβ gνγ dx 2 det (4)g ! − and thus, on any simplyp connected domain such as the domain of outer communications (DOC) of a black hole, the exactness of (4)Ω. In fact, by direct evaluation of the right hand side of the defining formula (D.7) in terms of our variables one arrives at (4) γ (D.8) Ω= dη = η,γ dx

5Note that λ is in fact also invariant with respect to electromagnetic gauge transformations since we only admit those transformations that preserve explicit axial symmetry. AXISYMMETRIC PERTURBATIONS 75 and thus concludes that our wavemaps ﬁeld η is indeed a spacetime scalar. Finally, consider the 1-form ﬁeld (4) (4) γ ∆= ∆γ dx :=

(D.9) 1 1 µναβ (4) γ ǫ ψµ(∂αψβ ∂βψα) gνγ dx 2 det (4)g − − µ constructed covariantlyp from the Killing 1-form ψµdx and its exterior derivative. Evaluating the right hand side of this expression in terms of our variables one arrives at (4)∆= dω + λdη (D.10) −{ } = ω dxµ + λη dxµ −{ ,µ ,µ } and thus concludes that the remaining wavemap ﬁeld, ω, does indeed transform as a spacetime scalar.

Appendix E. Electric Charge and Angular Momentum Conservation Laws The electric flux of a Maxwell field (4)F through a closed, connected and orientable 2-surface (2)Σ is defined by the integral of its dual 2-form, ⋆(4)F , over (2)Σ where, in coordinates, 1 (E.1) ⋆ (4)F = ⋆(4)F dxµ dxν 2 µν ∧ with 1 (E.2) ⋆(4)F = det (4)g ǫ (4)F αβ. µν 2 − µναβ If, for example, (2)Σ bounds a 3-ballp B lying in a spacelike hypersurface then the electric charge QB contained in that ball would be given, in our slightly non-standard conventions6, by

(4) (E.3) 8πQB = ⋆ F (2) Z Σ=∂B In the case of a black hole however the presence of non-vanishing flux through a 2-surface surrounding its event horizon may simply be a measure of ‘field lines trapped in the topology of space’ with no actual source current ⋆j for the Maxwell field necessarily existing in the spacetime (Wheeler’s ‘charge without charge’). This is indeed the case for the maximally analytically extended Kerr-Newman black hole

6Recall that we have absorbed a factor of 2 into (4)F and its ADM representatives (4)A, (3) , (3)B to ‘normalize’ the form of Hamilton’s equations. E 76 VINCENT MONCRIEF AND NISHANTH GUDAPATI spacetimes which are global solutions to the pure electrovacuum field equations. By the same (topological trapping) mechanism a stationary black hold solution can exhibit a non-vanishing magnetic flux (the integral of (4)F itself over a surface surrounding the event horizon) without the necessity of actual magnetic monopoles existing in the (topologically non-trivial) spacetime. But since one expects, on astrophysical grounds, that actual black holes in the Universe are created from the collapse of ordinary material sources preexisting in topologically trivial space (e.g., rotating stars), such objects could certainly be electrically, but presumably not magnetically, charged. For this reason we herein exclude the consideration of a non-vanishing magnetic flux, both for the background black hole spacetime and its perturbations. In view of the axial symmetry of the Kerr-Newman black holes we can exploit the formalism developed herein to evaluate the (electric) charge integral,

(E.4) 8πQ = ⋆(4)F, (2) Z Σ (over a surface (2)Σ surrounding the event horizon) in terms of the values of the wave map potential function η taken on the axes of symmetry. For simplicity let us evaluate this integral over the (topologically spherical) surface (2)Σ deﬁned in the Boyer-Lindquist type coordinates of Appendix A by R = R0 = constant > R+ and t = t0. Recalling that, in these coordinates, the axial Killing ﬁeld ψ = ψµ ∂ ∂ we ∂xµ −→ ∂ϕ get, by direct calculation

(4) 1 (4) (4) αβ ⋆ F = det g ǫθϕαβ F dθ dϕ (2)Σ (2)Σ 2 − ∧ Z Z p 1 1 (4) µναβ (4) = Fαβ ǫ gθµ ψν dθdϕ −2 (2) (4) Z Σ det g π − (E.5) 1 1 νµαβ (4) (4) =2π p ψν ǫ Fαβ gµθdθ 2 (4) Z0 det g π − =2π η,θpdθ Z0 =2π (η(R , π) η(R , 0)) 0 − 0 where we have, in the ﬁnal steps, appealed to Eqs. (D.7) and (D.8). This result reproduces the observation made incidentally in (C.47) while now justifying the identiﬁcation of the parameter Q occurring in the Kerr-Newman solution with electric charge. AXISYMMETRIC PERTURBATIONS 77

From the deﬁning formula (C.25) and the fact that the electric vec- a ∂ tor density ∂xa must, for reasons of regularity, have a vanishing θ- component alongE the axes of symmetry it follows that η must be constant along each of these axes so that both η(R, π) and η(R, 0) are independent of R. A straightforward linearization of the above argument leads to the corresponding perturbative formula

(E.6) 4Q′ = η′(t, θ = π) η′(t, θ = 0) − which, at first glance, would seem to allow for a time dependent perturbed charge. However, by combining the linearizations of Eqs. (C.28) and (C.29) with the axis regularity results of Ref. [68] one finds that bothr ˜′ andu ˜′ vanish to order O(sin θ) at the axes of symmetry and, ′ combined with a decay result for X η, that η,t′ actually vanishes to 2 L order O(sin θ) at these axes. It follows that η′(t, π) and η′(t, 0) are both independent of t. While one could thus allow the perturbation of η to incorporate a corresponding perturbation of the conserved electric charge there is little or no reason for doing so. One can simply insist that the given ‘unperturbed’ black hole have the full charge desired for the final, perturbed object and thus demand, without serious loss of generality, that η′ actually vanish on both symmetry axes. Only this choice is compatible with the natural perturbative boundary condition that η′ should vanish at infinity — an assumption that we shall impose herein. A similar argument can be given for the evaluation of the total angular momentum of a Kerr-Newman black hole and for that of its axisymmetric perturbations by appealing to Komar’s famous flux formula for such cases [73]. Komar’s formula states the total angular momentum J is given by the flux integral

(E.7) 16πJ = ⋆dψ, (2) Z Σ

µ where ψ = ψµdx , the covariant form of the axial Killing field, with the proviso that now, in order to include contributions from material sources such as the electromagnetic field, the integral should be evaluated in the limit that the ‘radius’ of the integration surface (2)Σ tends to µ infinity. Note that the Killing 1-form ψµdx plays here a role analogous µ to that of the ‘vector potential’ Aµdx in the case of electric charge. 78 VINCENT MONCRIEF AND NISHANTH GUDAPATI

A direct evaluation of this ﬂux integral over the (topologically spher- (2) ical) surface ΣR0 of Boyer-Lindquist ‘radius’ R0 gives (E.8)

1 1 νµαβ (4) ⋆dψ = (∂αψβ ∂βψα) ǫ ψν gθµ dθdϕ (2) (2) (4) ΣR 2 ΣR det g − Z 0 Z 0 − π 1 p 1 νµαβ (4) =2π ψν ǫ (∂αψβ ∂βψα) gµθ dθ 0 2 ( det (4)g − ) Z − R=R0 π p =2π (ω + λη ) dθ − ,θ ,θ Z0 R=R0 where we have, in the ﬁnal step, appealed to Eqs. (D.9) and (D.10).

In the limit that R0 the contribution proportional to λ (c.f. Eq. (C.40)) drops out leaving→ ∞

(E.9) 8J = lim ω(R0, 0) ω(R0, π) R0 →∞ { − } By an argument completely analogous to that given above for η though one ﬁnds that ω(R0, θ = π) and ω(R0, θ = 0) are both independent of R0 so that one recovers Eq. (C.48) together with the identiﬁcation that J = aM in terms of the Kerr-Newman parameter a. A straightforward linearization of the above argument leads to the corresponding perturbation formula

(E.10) 8J ′ = ω′(t, θ = 0) ω′(t, θ = π) { − } which would seem to allow for a time dependent perturbed angular momentum. But a straightforward linearization of Eq. (C.30), combined with the aforementioned results forr ˜′ andu ˜′ and an appeal to Ref. [68] 4 for the evaluation of ′ ω, shows that ω′ vanishes to order O(sin θ) LX ,t at the axes of symmetry. It follows that ω′(t, 0) and ω′(t, π) are both independent of t. While one could thus allow the perturbation of ω to reflect a corresponding perturbation in the conserved angular momentum there is, as was already noted for the case of electric charge, no reason for doing so. Again one can simply demand that the given ‘unperturbed’ Kerr-Newman black hole have the total angular momentum desired for the final, perturbed object and thus take ω′ to actually vanish on both symmetry axes. We thus assume herein, without any essential loss of generality, that the perturbations are taken to satisfy J ′ = 0 and Q′ = 0. The formulas, corresponding to Eqs. (E.4) and (E.5) above, for the (2) magnetic flux threading through a 2-surface Σ surrounding (at t = t0 AXISYMMETRIC PERTURBATIONS 79 and R = R0 > R+) the black hole’s event horizon are given (again in our slightly non-standard conventions) by

mag (4) (4) 8πQ = F = Fθϕ dθdϕ (2) (2) Z Σ Z Σ (E.11) π = (∂θλ) dθdϕ =2π ∂θλ dθ (2) Z Σ Z0 =2π (λ(R , π) λ(R , 0)) . 0 − 0 This expression of course vanishes for our (non magnetically charged) background solution since λ vanishes on the axes of symmetry . Linearizing the ADM formula for the magnetic ﬁeld, 1 (E.12) i = ǫijk(∂ A ∂ A ), B 2 j k − k j one arrives at the 2-dimensional vector density

a′ ab ab (E.13) = ǫ ∂ A′ = ǫ ∂ λ′ B b ϕ b which, for reasons of regularity, must have a vanishing θ-component along the axes of symmetry. It follows that λ′ must be independent of R along each of these axes and thus that the linearization for (E.11) yields

mag′ (E.14) 4Q = λ′(t, θ = π) λ′(t, θ = 0) − which, at ﬁrst glance, would seem to allow for a time dependent perturbation of the magnetic charge. However a straightforward linearization of Eq. (C.55), combined with the axis regularity results of Ref. [68], 2 shows that λ,t′ vanishes to order O(sin θ) at the axes of symmetry and hence that both λ′(t, π) and λ′(t, 0) are independent of t. As mentioned above we shall demand that these constants of motion both vanish so that even our perturbed black hole is not magnetically charged. Thus we demand that λ′ vanish on both the axes of symmetry.

Appendix F. Gauge Conditions for the Linearized Equations A fundamental result of Refs. [20] and [22] is that one can always i j express the induced metric, gij dx dx , on a Cauchy hypersurface for the DOC of an axisymmetric, non-degenerate,⊗ asymptotically ﬂat black hole in coordinates xi = xa,ϕ = ρ,z,ϕ such that (reexpressed in our notation) { } { } { }

i j 2γ a b 2γ a b (F.1) g dx dx = e− g˜ dx dx +e (dϕ+β dx ) (dϕ+β dx ) ij ⊗ ab ⊗ a ⊗ b 80 VINCENT MONCRIEF AND NISHANTH GUDAPATI where a b 2ν a b gãb dx dx = e hab dx dx (F.2) ⊗ ⊗ = e2ν (dρ dρ + dz dx), ⊗ ⊗ ψ = ∂ is the generator of (axial) rotations under which g dxi dxj is ∂ϕ ij ⊗ 2 2 invariant and where R := ρ + z takes a constant value, R R+ > 0, on the (topological) sphere corresponding to the black hole’s→ (non- degenerate) horizon (intersectedp with the chosen Cauchy surface). The coordinates introduced (via Eqs. (A.8)–(A.12)) for the Kerr-Newman ‘background’ solutions are clearly of this (Weyl-Papapetrou) type. The flexibility to arrange that the coordinate sphere R := ρ2 + z2 → R+ = constant > 0 coincide with a particular (topological) sphere of geometrical significance (e.g., the black hole’s horizon) resultsp from the fact that the (manifestly conformally flat) form (F.2) for the Rie- a b mannian 2-metricg ãb dx dx is preserved under arbitrary conformal transformations whereby the⊗ coordinates ρ and z can be replaced by arbitrary, conjugate harmonic functions thereof: ρ u(ρ, z), z v(ρ, z). To preserve this metric form under Einsteinian→ evolution,→ however, one would need to impose the condition

ab ab 1 ab ef 1 ab (F.3) (µ(2) g˜ ) = 2N˜ π˜ g˜ g˜ π˜ + (µ(2) g˜− ) =0 g˜ ,t − − 2 ef LX g˜ ˜ a ∂ as a restriction on the (2-dimensional) shift field X = N ∂xa . Reex- a b pressed in terms of the flat metric hab dx dx , Eq. (F.3) becomes (F.4) ⊗ ab (2) ab ab 1 ab ef (2) 1 √ h h = 2N˜ π˜ h hef π˜ + X √ h h− =0, ,t − − 2 L 1 ab ∂ ∂ (2) where h− = h ∂xa ∂xb and h := det (hab). Equation (F.4) ensures, of course, that the⊗ manifestly conformally flat form of this metric is preserved under the evolution but, even though we also demand that a b hab dx dx remain flat, it is not uniquely fixed by Eq. (F.4) since ⊗ λ (as was previously noted in Section 4.3) any metric of the form hab = 2λ e hab is also flat whenever the function λ is harmonic (with respect to a b hab dx dx or any metric conformal thereto). ⊗ a b In other words the requirement that hab dx dx be flat does not 2ν ⊗ uniquely fix the decomposition ofg ãb = e hab into a flat metric and a conformal factor but we can impose such uniqueness by fiat by absorbing the (harmonic logarithm) λ of any such deformation into the function ν, letting ν ν + λ and holding hab fixed. In this paper, of course,→ we shall not need to deal with this issue at the fully nonlinear level but the linearized form of Eq. (F.3), about a AXISYMMETRIC PERTURBATIONS 81

Kerr-Newman background (for whichπ ˜ab = 0 and Xa = N˜ a = 0) is:

1 ab ab 1 ab ef (F.5) ′ (µ(2) g˜− ) =2N˜ π˜′ g˜ g˜ π˜′ LX g˜ − 2 ef or, equivalently,

(2) 1 ab ab 1 ab ef (F.6) ′ (√ h h− ) =2N˜ π˜′ h h π˜′ LX − 2 ef a a′ where X′ = N˜ . In this article, however, rather than attempt to solve Eq. (F.5) or (F.6) directly for the linearized shift X′ we shall, in Appendix H, construct the gauge transformation that carries one from an arbitrary gauge to the desired Weyl-Papapetrou gauge at the linearized level. (4) (4) µ ∂ From the vector field Y = Y ∂xµ that generates this gauge transformation (c.f., Eqs. (H.1)–(H.6)) one can then simply compute, among ˜ c′ ∂ other quantities, the transformed, linearized shift field, X′ = N ∂xc , via Eq. (H.24). Thus we may assume, without essential loss of generality, that the a b flat, ‘conformal’ metric, hab dx dx , preserves its (manifestly flat) form, ⊗ a b hab dx dx = dρ dρ + dz dz (F.7) ⊗ ⊗ ⊗ = dR dR + R2 dθ dθ ⊗ ⊗ under the perturbation and thus take hab′ = 0. Since, in principle, this (Weyl-Papaetrou) gauge condition can be imposed at the fully nonlinear level we may assume, a fortiori, that it holds to higher order at the perturbative level and thus, in particular, set hab′′ = 0.

Appendix G. Analysis of the Linearized Constraint Equations Upon introducing the ‘twist’ potentials η and ω we have solved the electromagnetic (Gauss law) constraint and the azimuthal projection of the (3+1 dimensional) momentum constraint leaving only (G.1) ˜ = 0 and ˜ =0 H Ha as constraints for the reduced field equations. A straightforward calculation using the reduced evolution equations (C.49)–(C.58) with arbi- ˜ ˜ a ∂ trary lapse N and shift X = N ∂xa shows that these quantities, if not already vanishing, satisfy the evolution equations ∂ (G.2) ˜ =(N˜ a ˜) + N˜ gãb ˜ +(N˜ gãb ˜ ) , ∂tH H ,a ,b Ha Ha ,b 82 VINCENT MONCRIEF AND NISHANTH GUDAPATI

∂ (G.3) ˜ =(N˜ b ˜ ) + N˜ b ˜ + N˜ ˜ ∂tHa Ha ,b ,aHb ,aH which are clearly at least consistent with the preservation of the constraints (G.1) in time. Linearizing Eqs. (G.2) and (G.3) about a background solution for which (as in the Kerr-Newman cases of interest here) X = 0 yields the corresponding propagation equations for the ﬁrst variations ( ˜′, ˜′ ) and ( ˜, ˜ ): H Ha H Ha ∂ ab ab (G.4) ˜′ = N˜ g˜ ˜′ +(N˜ g˜ ˜′ ) ∂tH ,b Ha Ha ,b ∂ (G.5) ˜′ = N˜ ˜′. ∂tHa ,aH These can also be derived by directly computing the time derivatives ˜ ˜ of ( ′, a′ ) by means of the linearized evolution equations. AsH aH subset of the linearized Einstein-Maxwell ﬁeld equations the linearized constraints

(G.6) ˜′ = 0 and ˜′ =0 H Ha are gauge invariant (provided always that the background, exact field equations are satisfied) and this is reflected in the fact that neither N˜ ′ ′ nor N˜ a appear in Eqs.(G.4) and (G.5). (c.f., the discussion in [54]). In a free evolution framework one would impose the linearized con- ˜ ˜ straints ′ = a′ = 0 on an initial Cauchy hypersurface and appeal to the propagationH H equations (G.4)–(G.5) to establish their preservation in time. Since these propagation equations however are apparently not of a standard type we prefer to adopt the strategy of constrained evolution whereby one enforces the linearized constraints on every time slice by solving them for certain ‘dependent’ variables in terms of the unconstrained, ‘dynamical’ variables, namely the first variations (γ′,ω′, η′,λ′) of the wave map functions and their conjugate momenta (˜p′, r˜′, u˜′, v˜′). In the class of gauges that we shall consider and recalling that the background, Kerr-Newman solutions of interest have vanishing canonical momenta, the linearized constraints reduce to:

(2) ab 2γ (G.7) ˜′ = √ h h 4γ γ′ e− γ′(η η + λ λ ) H ,a ,b − ,a ,b ,a ,b 2γ 4γ + e− (η η′ + λ λ′ ) 2e− γ′(ω + λη )(ω + λη ) ,a ,b ,a ,b − ,a ,a ,b ,b 4γ + e− (ω,a + λη,a)(ω,b′ + λη,b′ + λ′η,b) √(2) ab +2∂a h h ν,b′ =0,

(2) b 2ν (2) (G.8) ˜′ = 2 (h)r ˜′ e √ h τ ′ Ha − ∇b a − ,a +(˜p′γ,a +˜r′ω,a +˜v′λ,a +˜u′η,a)=0 AXISYMMETRIC PERTURBATIONS 83 where

b bd 1 bd ef r˜′ :=g ˜ π˜′ g˜ g˜ π˜′ a ad − 2 ef 2ν bd 1 bd ef (G.9) = e h π˜′ h h π˜′ ad − 2 ef 2ν bd 1 (2) bd = e h π˜′ √ h h τ ′ ad − 2 ab denotes the traceless part ofπ ˜′ and

gãb ab hab ab (G.10) τ ′ := π˜′ = π˜′ µ(2)g˜ √(2)h 2ν its (scalarized) trace. Here hab = e− gãb designates the flat metric (2) (2) on Mb introduced in Appendix F whereas a(h) and √ h denote covariant differentiation and ‘volume’ element∇ for this metric. Recall that in the Weyl-Papapetrou coordinates ρ,¯ z¯ first introduced in Ap- { } pendix A, Mb corresponds to the half plane (¯ρ, z¯) ρ¯ 0, z¯ R with 1 a { b | ≥ ∈ } the ‘cut’ (A.20) removed and habdx dx = dρ¯ dρ¯ + dz¯ dz¯. √(2)h ⊗ ⊗ ⊗ The Hamiltonian constraint (G.7) is an elementary (flat space) Pois- a b son equation on Mb, hab dx dx for the first variation, ν′, of the logarithm of the conformal{ factor⊗ e2ν}. As discussed in Section 4.4, however, regularity at the axes of symmetry requires that we impose the Dirictlet boundary condition (c.f., Eq. (4.84)):

(G.11) ν′ θ=0,π = 2γ′ θ=0,π . |R R+ |R R+ ≥ ≥ Additional considerations, such as those discussed in Section 4.3, can lead to the imposition of a Neumann boundary condition such as the (minimal surface preserving) condition

(G.12) ν′ =0 ,R R=R+ at the event horizon. Thus one can be naturally led to a mixed, elliptic boundary value problem for ν′ with Dirichlet data required along the axes of symmetry and complementary Neumann data needed along the horizon boundary. Though such problems can be notoriously diﬃcult to solve in general we shall be able to exploit the special features of our particular problem to solve it by elementary means. In this way we simultaneously remove the ambiguity in the construction of ν′ (which would otherwise be undetermined up to the addition of a harmonic function) and cancel the ﬂux contributions that could otherwise lead to a violation of the conservation of energy. 84 VINCENT MONCRIEF AND NISHANTH GUDAPATI

A standard (Green’s theorem) argument shows that if indeed a solution vanishing at inﬁnity exists for this (mixed, elliptic) problem then it will necessarily be unique. Our strategy for constructing this hypo- thetical solution will be to seek to express it as

(G.13) ν′ = νD′ + νN′ where νD′ is the solution to an associated, inhomogeneous Dirichlet problem chosen to solve Eq. (G.7) with the boundary condition (G.11) imposed, whereas νN′ will be the harmonic solution to a complementary, homogeneous Neumann problem chosen to impose the boundary condition (G.12) and constructed in such a way as to leave the Dirichlet condition on the axes of symmetry undisturbed. The special (2-dimensional, conformally covariant) nature of our problem is what allows this last step to be carried out.

We begin by imposing suitable Dirichlet conditions for νD′ on the boundary of the closure M¯ b of Mb (i.e., on the fullz ¯-axis of the half- plane (¯ρ, z¯) ρ¯ 0, z¯ R ) and with suitable ‘regularity’ assumed for { | ≥ ∈ } the free data γ′,ω′, η′,λ′ appearing in Eq. (G.7). More precisely we { } choose Dirichlet data for νD′ along the upper and lower axis components to cancel the unwanted ﬂux contributions identiﬁed previously (i.e., so as to impose (G.11)) and, as an intermediate step, interpolate along the ‘strut’ separating these disjoint axes with smooth but arbitrarily chosen, complementary Dirichlet data. One could, for example, choose

νD′ = 2γ′ along this strut. Using the explicitly known fundamental solution (Green’s function) for this problem (see, for example [35]), we solve the corresponding Dirichlet problem (i.e., solve Eq. (G.7) for νD′ in place of ν′ with the boundary data so chosen). The solution for νD′ will of course fail in general to satisfy the Neu- mann condition (G.12) along the horizon but if, as in the asymptotically pure gauge problem discussed in Section 4.1, γ′ has the property that

(G.14) γ,R′ R=R+ =0 θ=0,π then, from the chosen condition,

(G.15) νD′ =2γ′ along thez ¯-axis we shall automatically have

(G.16) νD,R′ R=R+ =0. θ=0,π

We now revert to the ‘half-plane with half disk removed’ picture for Mb discussed in Appendix A and extend this to a ‘full plane with full AXISYMMETRIC PERTURBATIONS 85 disk removed’ by reﬂection across thez ¯-axis. We now choose Neumann data for νN′ on the circle at R = R+ by setting

(G.17) νN,R′ R=R+ = νD,R R=R+ θ [0,π] − |θ [0,π] ∈ ∈ and then anti-reflecting this data across thez ¯-axis to complete the specification on the full circle (i.e., choosing the value of νN,R′ to be the negative of that at its mirror image on the circle). While it is not strictly needed for our construction Eq. (G.16) will ensure that this extension of the Neumann data will be continuous at those points where the horizon meets the axes of symmetry. The fundamental solution for the Neumann problem on the plane with a disk removed is explicitly known (c.f., [29]). Using it together with the chosen Neumann data on the circle R = R+ we construct the harmonic function νN′ . From the uniqueness of this solution and the reflection (anti-) symmetry of its boundary data we see that νN′ will in fact vanish on the axes of symmetry. Expressing νD′ and νN′ in a common coordinate system, adding them and restricting the result ¯ to Mb we see that ν′ := νD′ + νN′ satisfies (G.7) together with the mixed boundary conditions (G.11) and (G.12). By construction it is the unique function vanishing at infinity that has these properties. We thus conclude that7

Theorem 3. Equation (G.7) has, for each choice of regular data γ′,ω′, η′,λ′ , { } a unique solution ν′ that vanishes at inﬁnity and satisﬁes the mixed (Dirichlet/Newmann) boundary conditions (G.11) and (G.12).

To solve the momentum constraint (G.8) we exploit the fact that, under suitable boundary and asymptotic conditions (discussed in detail in Appendix I), symmetric transverse traceless tensors on M¯ b vanish identically and thus that (the mixed form of) a symmetric traceless b ∂ a density,r ˜′ b dx , can be expressed as a ∂x ⊗

b (2) (2) b (2) b c b(2) c (G.18)r ˜′ = √ h (h)Y ′ + (h)(h Y ′ ) δ (h)Y ′ a ∇a ∇ ac − a ∇c h i

7By construction our solution satisﬁes the necessary condition, G.11, for regularity at the axes. That it is, moreover, fully regular at the axes follows indirectly from its uniqueness as a solution to the relevant Hamiltonian constraint and an independent, purely 3+1-dimensional treatment of the corresponding Lichnerowicz equation with axisymmetric boundary data. The well-known existence of a unique, globally smooth solution to the latter ensures that our solution, with which it must agree, has the required regularity. 86 VINCENT MONCRIEF AND NISHANTH GUDAPATI

c ∂ a for a suitably chosen vector field Y ′ = Y ′ a . Recalling that h dx ∂x ab ⊗ dxb is flat one finds easily that Eq. (G.8) reduces to (G.19) (2) (2) (2) b c 2ν (2) 2√ h (h) (h)(h Y ′ ) = e √ hτ ′ +(˜p′γ +˜r′ω +˜v′λ +˜u′η ) ∇b ∇ ac − ,a ,a ,a ,a ,a which, in Weyl-Papapetrou coordinates, takes the form of elementary, decoupled Poisson equations for the components of Y ′. With suitable boundary and regularity conditions imposed upon the relevant data these can be solved explicitly for Y ′. Again the relevant elliptic theory is presented in detail in Appendix H below. By exploiting the background field equations (C.64)–(C.71), satisfied by an arbitrary, Kerr-Newman black hole, it is straightforward to show that (G.20)

∂ (2) ab N˜ ˜′ = N˜√ h h 4γ γ′ +2ν′ H ∂xb ,a ,a 2γn 4γ (2) ab + e− (η η′ + λ λ′)+ e− (ω + λη )(ω′ + λη′) 2√ h h N˜ ν′ ,a ,a ,a ,a − ,a o for arbitrary (γ′,ω′, η′,λ′, ν′). That this expression is a spatial divergence reflects the fact, discussed briefly in Section 3.2, that (C,Z) = (N,˜ 0) is an element of the kernel of the adjoint of the linearized con- ∂ straint operator, corresponding to the occurrence of ζ = ∂t as a Killing field on the quotient manifold V/U(1) = R Mb (c.f., Appendix C). To fully appreciate its implications for the perturbative× analysis it is essential to consider the 2nd variation of the Hamiltonian constraint. Let us abbreviate by q := γ,ω,η,λ the wave map variables and by p := p,˜ r,˜ u,˜ v˜ {their} canonically{ conjugate} momenta. These are the{ unconstrained,} { dynamical} ‘degrees of freedom’ for the reduced, axisymmetric Einstein-Maxwell system. The flat, conformal metric a b habdx dx and (2+1-dimensional) mean curvature function, τ, are restricted,⊗ in our reduced Hamiltonian framework, through the imposi- a b tion of suitable gauge conditions by setting, for example, habdx dx = dρ¯ dρ¯ + dz¯ dz¯ and (in the simplest case) taking τ = 0⊗ (2+1- dimensional⊗ maximal⊗ slicing). The canonically conjugate partners of these gauge variables, namely the tracefree component of the gravitational momentum, ∂ (G.21)r ã dxb := ˜r, b ∂xa ⊗ and conformal factor, e2ν, are to be determined (subject to suitable boundary conditions) through the solution of the elliptic momentum and Hamiltonian constraints on each time slice. Preservation of the AXISYMMETRIC PERTURBATIONS 87 gauge conditions throughout the evolution necessitates a corresponding fixation of the lapse and shift fields (N,X) via the solution of an auxiliary set of (linear) elliptic equations (c.f., Appendices F, H and K for details). Treating black hole stability problems at this fully nonlinear level is currently out of reach but closely related methods have been suc- cessfully used to prove the fully nonlinear stability (in the direction of cosmological expansions) of a family of U(1)-symmetric, spatially compact, vacuum cosmological models [19]. To derive the linearized and higher order perturbation equations (for axisymmetric perturbations of Kerr-Newman backgrounds, in particular) one can imagine having a smooth one-parameter family of exact solutions, containing the desired background, and differentiating the field equations one or more times with respect to this curve parameter, e, and then fixing it to the background value, say e = 0. Thus we now write q′, p′ for (γ′,ω′, η′,λ′), (˜p′, r˜′, u˜′, v˜′) where { } { } ∂ ∂ (G.22) q′, p′ := q(e, ), p(e, ) { } ∂e · ∂e · e=0 and denote by q′′, p′′ the corresponding 2nd variations { } ∂2 ∂2 (G.23) q′′, p′′ := q(e, ), p(e, ) { } ∂e2 · ∂e2 · e=0 etc. (2) (2) The gauge choice made for the flat metric h implies that h′ = (2) h′′ = 0, etc. (c.f., Appendix F), whereas that for τ (in the simplest, 2+1-dimensional, maximal case) yields τ ′ = τ ′′ = 0 as well. To allow however for more general time gauge conditions (3+1-dimensional maximal slicing, for example) we shall retain τ ′ and τ ′′ in the formu- a′ ˜ las to follow. The corresponding perturbations ν′, r˜b , N ′,X′ and a ˜ { } ν′′, r˜b ′′, N ′′,X′′ of the remaining, dependent variables are determined {(with suitable boundary} conditions) by solving the corresponding perturbed elliptic equations and are thus, in effect, known functionals of q′, p′, q′′, p′′ . { Let us now} denote the 1st variations (G.7) and (G.8), of the constraints more explicitly as first order linear operators acting on the relevant linearized variables, via (2) (G.24) D ˜(q, h) (q′, ν′) := ˜′ H · H and (2) (G.25) D ˜ (q, h, ν) (p′, ˜r′, τ ′) := ˜′ Ha · Ha 88 VINCENT MONCRIEF AND NISHANTH GUDAPATI

(2) so that ND˜ ˜(q, h) (q′, ν′) is the total divergence given explicitly by Eq. (G.20). H · A straightforward calculation, utilizing Eqs. (C.33,C.34,G.20–G.25) now yields (G.26) (2) 2 (2) N˜ ˜′′ = ND˜ ˜(q, h) (q′′, ν′′)+ND˜ ˜(q, h, ν) ((q′, p′, ˜r′, τ ′), (q′, p′, ˜r′, τ ′)) , H H · H · where (G.27) 2 (2) D ˜(q, h, ν) ((q′, p′, ˜r′, τ ′), (q′, p′, ˜r′, τ ′)) H · 2ν 2e− b a 1 2 1 4γ 2 := r˜′ r˜′ + (˜p′) + e (˜r′) √(2) a b 8 2 h 1 2γ 2 2 2ν (2) 2 + e (˜v′) +(˜u′ λr˜′) e √ h(τ ′) 2 − − √(2) ab 2γ 2 + h h 4γ,a′ γ,b′ +2e− (γ′) (η,aη,b + λ,aλ,b) 2γ 2γ 4e− γ′(η η′ + λ λ′ )+ e− (η′ η′ + λ′ λ′ ) − ,a ,b ,a ,b ,a ,b ,a ,b 4γ 2 +8e− (γ′) (ω,a + λη,a)(ω,b + λη,b) 4γ 8e− γ′(ω + λη )(ω′ + λη′ + λ′η ) − ,a ,a ,b ,b ,b 4γ + e− (ω,a′ + λη,a′ + λ′η,a)(ω,b′ + λη,b′ + λ′η,b)

4γ + e− (ω,a + λη,a)(2λ′η,b′ ) , and ˜ ˜ (2) a′′ = D a(q, h, ν) (p′′, ˜r′′, τ ′′) (G.28) H H · 2ν (2) 2e √ h ν′τ ′ +(˜p′γ′ +˜r′ω′ +˜v′λ′ +˜u′η′ ). − ,a ,a ,a ,a ,a Combining (G.20), (G.26), and (G.27) we see that the constraint equations of second order, namely

(G.29) ˜′′ = 0 and ˜′′ =0, H Ha imply that the density Alt deﬁned by E ˜ Alt N 2 (2) (G.30) := D ˜(q, h, ν) ((q′, p′, ˜r′, τ ′), (q′, p′, ˜r′, τ ′)) E 2 H · is equal to a spatial divergence and thus that the integral

(G.31) EAlt := d2x Alt {E } ZMb AXISYMMETRIC PERTURBATIONS 89 is equal to a boundary integral when the field equations are satisfied. In the limiting case of purely electromagnetic perturbations of a Kerr back-ground EAlt reduces to the functional HAlt defined via Eqs. (2.30) and (2.31). It is clear from the divergence form for N˜ ˜′ given by Eq. (G.20) that, H when the linearized Hamiltonian constraint, ˜′ = 0, is imposed, the H integral of N˜ ˜′ over Mb will imply the vanishing of a sum of potential ‘boundary flux’H terms arising at the boundary components corresponding to R , R R and θ 0, π. By exploiting the asymptotic ր ∞ ց + → behaviors of the perturbations η′,λ′,ω′,γ′, ν′ given via Eqs. (4.9), (4.10), (4.63), (4.36), (4.38), (4.40){ and (4.42)} it is straightforward to show that the flux integrand vanishes pointwise as R yielding a separately vanishing contribution to the net boundary flux.ր∞ By exploiting the regularity of the various perturbations at the axes of symmetry, including especially the condition (4.84) on ν′ 2γ′ and the fact dis- − 2 cussed in Section 4.4 that each of ω′,λ′, η′ vanishes to order O(sin θ) as θ 0, π, it follows that the boundary{ } flux integrand also vanishes pointwise→ along the (artificial) boundary components corresponding to θ 0, π. → Finally, by exploiting the regularity of the perturbations γ′,λ′, η′,ω′ at the horizon discussed in Section 4.1 together with the (minimal{ sur-} face preserving) condition (G.12) and the pointwise vanishing of the 2γ 2γ 4γ factors N,γ˜ ,R, e− η,R, e− λ,R, e− (ω,R + λη,R) as R R+, it is straightforward{ to show that the only potential} flux contributionց at this ‘inner’ boundary component must come from the only remaining term in the flux integrand:

(2) Ra ∂ (2) θ (G.32) 2√ hh N˜,aν′ = 4R+ sin θ Y . − R+ − ∂θ R+

While not pointwise vanishing as the other terms were this clearly has vanishing integral with respect to θ when integrated over the interval θ [0, π] corresponding to the horizon component at R+. ∈Proceeding now to the second variation of the Hamiltonian constraint, ˜′′ = 0, it is now clear from Eqs. (G.20), (G.24)–(G.27) that the sumH of boundary ﬂux contributions resulting from the integral of 90 VINCENT MONCRIEF AND NISHANTH GUDAPATI the divergence expression

(G.33)

(2) ∂ (2) ab ND˜ ˜(q, h)(q′′, ν′′)= N˜√ h h 4γ γ′′ +2ν′′ H ∂xb ,a ,a 2γ n 4γ +e− (η,aη′′ + λ,aλ′′)+ e− (ω,a + λη,a)(ω′′ + λη′′)

(2) ab 2√ h h N˜ ν′′ − ,a o over Mb must equate to the volume integral over this same domain of

Alt 2 (2) (G.34) 2 = ND˜ ˜(q, h, ν) ((q′, p′, ˜r′, τ ′), (q′, p′, ˜r′, τ ′)) − E − H · which, by Eqs. (G.30) and (G.31) is equal to 2EAlt. It is straightforward to verify that mere boundedness− (or even mild blowup) of the perturbations λ′′, η′′,ω′′ as R is sufficient to ensure their pointwise vanishing{ flux contributions} ր∞ at the ‘outer’ boundary. Furthermore their regularity as smooth scalar fields at the axes of symmetry and at the horizon guarantees their (pointwise) vanishing contributions to the flux integrands at these boundary components as well. This leaves only possible contributions of γ′′ and ν′′ to be considered. On the other hand the demand for regularity at the axes of symmetry leads, upon appealing again to the Rinne/Stewart results [68], to the restriction

(G.35) ν′′ =2γ′′ |θ=0,π |θ=0,π upon the second order perturbations and this suffices to ensure their (pointwise) vanishing contributions to the flux integrands along these axes. Following up on the seminar work of D. Brill [13], Sergio Dain derived an elegant integral expression for the ADM mass of an asymptotically flat, axisymmetric Einstein spacetime [26]. Its first variation (about a Kerr-Newman background) vanishes, for the class of perturbations considered herein, in view of the flux integral results described above but its second variation, expressed in our notation, yields the formula

π 1 2 1 2 (G.36) MADM′′ = lim R sin θ ν,R′′ ν′′ + γ′′ dθ −4 R − R R ր∞ Z0 AXISYMMETRIC PERTURBATIONS 91

Alt and thus allows us to express the ‘volume’ integral of over Mb (c.f., Eqs. (G.30)–(G.31)) as follows: E (G.37) π 1 Alt 1 M ′′ = E + dθ(R sin θ ν′′) ADM 4 2 + Z0 R R+ ց 1 2 (2) = ND˜ ˜(q, h, ν) ((q′ , p′, ˜r′, τ ′), (q′, p′, ˜r′, τ ′)) dRdθ 8 H · ZMb 1 π + dθ(R sin θ ν′′) . 2 + Z0 R R+ ց Recalling the discussion at the beginning of Section 4 we see that, at least for the class of (asymptotically-pure-gauge) perturbations considered herein, EAlt can be replaced by the manifestly positive definite expression EReg. Whether this perturbative contribution to the ADM mass is further ‘shifted’ by the boundary integral over the horizon hinges, of course, upon the boundary condition chosen for ν in ′′ R R+ | ց the second variation of the Hamiltonian constraint. If, for example, the perturbations considered are chosen to be symmetric under the map- 2 2 R+ R+ ping (‘inversion in the sphere’) R R′ (for which r = R + M + R 2 → → R+ r′ = R′ + M + R′ ) which maps one ‘end’ of the Kerr-Newman solution isometrically to the other, then the resulting boundary integral over the horizon could not distinguish one end from the other and would have to vanish.

Appendix H. Transforming Compactly Supported Perturbations to Weyl-Papapetrou Gauge As discussed previously (c.f., the discussion near the end of Appen- dix B) one can evolve a large class of compactly supported solutions to the linearized constraint equations in a hyperbolic gauge and appeal to finite propagation speed to show that such perturbations remain bounded away from the horizon and spatial infinity for all finite (Boyer-Lindquist) time t. On the other hand our energy flux derivation has assumed that the perturbations be expressed in a Weyl-Papapetrou gauge in order to make them amenable to an application of Robinson’s identity in its traditional form. Thus we need to consider the transformation of perturbations expressed in say a hyperbolic gauge of Lorenz type to a Weyl-Papapetrou gauge of the ‘elliptic type’ needed for our analysis. (4) (4) µ ν (4) (4) µ Let g = gµνdx dx , A = Aµdx be a Kerr-Newman black hole{ solution expressed⊗ in coordinates x0 =} t, x1, x2, x3 = ϕ of { } 92 VINCENT MONCRIEF AND NISHANTH GUDAPATI

∂ the (Boyer-Lindquist) type introduced in Appendix A (wherein ζ = ∂t ∂ and ψ = ∂ϕ are the Killing fields corresponding to the given black hole’s stationarity and axial symmetry). Relative to this background (4) (4) (4) µ ν (4) (4) (4) µ let g′ := k = kµν dx dx , A′ := ℓ = ℓµdx designate{ an axisymmetric, spatially⊗ compactly supported solution} to the (4) (4) µ ∂ corresponding linearized equations. If Y = Y ∂xµ is a (sufficiently smooth) vector field invariant with respect to the rotations generated by ψ, i.e., such that µ (H.1) (4)Y = (4)Y µ =0, Lψ ,ϕ then the gauge transformed perturbations (H.2) (4)k˜ = (4)k˜ dxµ dxµ, (4)ℓ˜ = (4)ℓ˜ dxµ { µν ⊗ µ } defined by (4) (4) (4) (H.3) k˜ = k + (4) g µν µν L Y µν (4) ˜ (4) (4) (H.4) ℓµ = ℓµ + (4) A L Y µ will also satisfy the linearized field equations and preserve explicit axisymmetry, i.e., obey

(4) (4) (H.5) ψ k˜ = k˜µν,ϕ =0, L µν and (4) (4) (H.6) ψ ℓ˜ = ℓ˜µ,ϕ =0. L µ Recalling that, in our notation, (4) 2γ 2γ gab = e− g˜ab + e βaβb (H.7) 2γ+2ν 2γ = e− hab + e βaβb

a b where habdx dx is a ﬂat 2-metric which, in Weyl-Papapetrou spatial coordinates ⊗xa = ρ, z , satisﬁes the (conformally invariant) condition { } { }

1 a b (H.8) habdx dx = dρ dρ + dz dz √(2)h ⊗ ⊗ ⊗ and recalling as well that βa = 0 on a Kerr-Newman background, we (4)˜ (4) see that a gauge transformed perturbation kµν of gµν will preserve this Weyl-Papapetrou form to linearized order if and only if it satisﬁes the gauge conditions (H.9) (4)k˜ (4)k˜ =0, (4)k˜ =0. ρρ − zz ρz AXISYMMETRIC PERTURBATIONS 93

Appealing to Eq. (H.3) one can reexpress these conditions in the form (H.10) (4) 1 cd (4) (4) 1 cd (4) (4) g h h (4) g = k h h k L Y ab − 2 ab L Y cd − ab − 2 ab cd But utilizing the fact that the background metric also satisﬁes

(4) (4) (4) (4) (H.11) gta = gϕa =0, gµν,ϕ = gµν,t =0

(4) µ and that Y ,ϕ = 0 by assumption one can rewrite Eq. (H.10) in the 2-dimensionally covariant form (H.12) (2) 1 cd (2) 2γ 2ν (4) 1 cd (4) (2) h h h (2) h = e − k h h k L Y ab−2 ab L Y cd − ab − 2 ab cd (2) (2) a ∂ (4) a ∂ where Y := Y ∂xa = Y ∂xa . Note that in the complement of the support of (4)k (i.e., in the ‘asymptotic regions’ near the horizon and near spatial inﬁnity) Eq. (H.12) reduces to the conformal Killing equation for the ﬂat 2-metric (2)h. As we shall show below this equation (with its inhomogeniety included) can be solved explicitly for (2)Y thus determining the (2-dimensional) ‘spatial components’ of (4)Y . These 2-dimensional ‘spatial’ components of (4)Y will play a distinc- tive role in that the induced gauge transformations of the linearized (4) wave map scalars γ′,ω′,λ′, η′ generated by Y , namely { }

(H.13) γ˜′ := γ′ + (4) γ, ω˜′ := ω′ + (4) ω, L Y L Y (H.14) λ˜′ := λ′ + (4) λ, η˜′ := η′ + (4) η, L Y L Y simplify to

(2) a (2) a (H.15) γ˜′ = γ′ + Y γ,a, ω˜′ = ω′ + Y ω,a, (2) a (2) a (H.16) λ˜′ = λ′ + Y λ,a, η˜′ = η′ + Y η,a in view of the invariance of the background ﬁelds γ,ω,λ,η with respect to t and ϕ translations. In particular, in{ the complement} of the support of the (compactly supported) perturbations γ′,ω′,λ′, η′ { } their gauge transformed counterparts γ˜′, ω˜ ′, λ˜′, η˜′ , though no longer in general having compact support, will{ nevertheless} simplify to their pure gauge forms

(2) a (2) a (H.17) γ˜′ Y γ , ω˜′ Y ω −→ ,a −→ ,a (2) a (2) a (H.18) λ˜′ Y λ , η˜′ Y η . −→ ,a −→ ,a 94 VINCENT MONCRIEF AND NISHANTH GUDAPATI

On the other hand, to compute the gauge transformations of the linearized canonical momenta p˜′, r˜′, v˜′, u˜′ we shall need the time component (4)Y 0 of (4)Y . To see{ how this} is determined recall that the m ∂ (3+1-dimensional) lapse function N and shift field N ∂xm are defined by 1 N m (H.19) = (4)g00 and = (4)g0m. − N 2 N 2 (4) Thus a first variation δN of N induced by k˜αβ is given by

2 (4) 0α (4) 0β (4)˜ 3 δN = g g kαβ (H.20) N − (4) 0α (4) 0β (4) (4) = g g k + (4) g . − αβ L Y αβ Evaluating the Lie derivative and recalling that, in our notation, N = γ e− N˜ so that γ γ (H.21) δN = e− δN˜ Ne˜ − δγ − (4) 2γ whereas, since gϕϕ = e ,

1 2γ (4) δγ = e− k˜ϕϕ (H.22) 2 1 2γ (4) (2) a = e− k + Y γ 2 ϕϕ ,a we arrive at (H.23) ˜ ˜ N ′ 1 2γ (4) 2 (4) 0α (4) 0β (4) (4) 0 (2) a N,a = e− kϕϕ N˜ g g kαβ + Y + Y N˜ 2 − ,0 N˜ where we now write N˜ ′ for δN˜ in accordance with our previously established notation. Thus given a choice for the linearized lapse function (4) 0 N˜ ′ in the desired (elliptic) gauge, Eq. (H.23) determines Y by direct time integration. In a completely analogous way one ﬁnds that the components of the linearized shift are given by c′ c′ ac 2γ (4) (4) N˜ = N =g ˜ e k0a β0 kaϕ (H.24) − + (4)Y c N˜ 2 g˜ac (4)Y 0 ,0 − ,a ϕ′ a N =(β N˜ β )′ 0 − a 2γ (4) (4) = e− k0ϕ β0 kϕϕ (H.25) − (4) 0 (4) ϕ (2) c + Y ,0β0 + Y ,0 + Y β0,c a β′ (since N˜ = β = 0 in the background) −→ 0 a AXISYMMETRIC PERTURBATIONS 95

Note that this last equation provides a means of computing the ‘last’ ′ component, (4)Y ϕ, of (4)Y provided that a gauge condition for N ϕ is specified. However a different way of computing (4)Y ϕ (that would then ′ fix the corresponding choice for N ϕ ) arises from noting that 2γ 2γ (4) (e β )′ e β′ = k˜ a −→ a aϕ (4) (4) (H.26) = k + (4) g aϕ L Y aϕ (4) 2γ (4) 0 (4) ϕ = kaϕ + e Y ,aβ0 + Y ,a so that (4) ϕ (4) 0 2γ (4) (H.27) βa′ = Y ,a + β0 Y ,a + e− kaϕ. Thus a choice for (4)Y ϕ allows one to control the ‘longitudinal part’ of βa′ whereas its ‘transversal part’ is governed independently by the linearized wave map momentum variabler ˜′ (c.f., Eq. (C.28)) To actually solve Eq. (H.12) let us first reexpress it in the more convenient form (H.28) cd (2) (2) (2) ac bd 2γ (4) 1 ef (4) (2) √ h h = g˜ g˜ g˜ e k g˜ g˜ k L Y ab − 2 ab ef :=p cd M Evaluating this (traceless, symmetric) equation in the R, θ coordinates of Appendix A, for which { } ∂ ∂ ∂ ∂ 1 ∂ ∂ (H.29) √(2)h hcd = R + , ∂xc ⊗ ∂xd ∂R ⊗ ∂R R ∂θ ⊗ ∂θ one gets the two independent components (2)Y R 1 (H.30) (2)Y θ = R + RR ,θ R RM ,R 1 (2)Y R 1 (H.31) (2)Y θ = Rθ ,R −R R − RM ,θ The integrability condition for this first order system for (2)Y θ is the (2)Y R Poisson-type equation for R given by (H.32) 1 ∂ (2)Y R 1 ∂2 (2)Y R 1 1 1 R + = Rθ RR R ∂R R R2 ∂θ2 R −R2 M ,θ−R RM ,R ,R (2)Y R Note that the operator acting on R in this equation is identical to the scalar Laplacian for the flat metric (2)f (conformal to (2)h) given 96 VINCENT MONCRIEF AND NISHANTH GUDAPATI by (H.33) (2)f = dR dR + R2dθ dθ ⊗ ⊗ For reasons of regularity the radial component, (2)Y R, of (2)Y must admit a Fourier expansion of the form

∞ (2) R (H.34) Y = a0(R, t)+ an(R, t)cos(nθ) n=1 X so that, in particular, its θ-derivative vanishes on the axes of symmetry corresponding to θ = 0 and θ = π. For the same reasons (2)Y θ must itself vanish at these axes and thus admit an expansion of the form

∞ (2) θ (H.35) Y = bn(R, t) sin (nθ) n=1 X Similar considerations for the vector density resulting from pairing the ab ∂ ∂ one-form dR with the tensor density ∂xa ∂xb lead to Fourier explansions of the latter’s components givenM by⊗ 1 ∞ (H.36) RR = c (R, t)+ c (R, t)cos(nθ) −RM 0 n n=1 X ∞ (H.37) R Rθ = d (R, t) sin (nθ) − M n n=1 X Substituting these expansions into Eqs. (H.30–H.31) leads to the following system for the Fourier coeﬃcients 1 (H.38) a a = c , 0,R − R 0 0 1 (H.39) a a nb = c , n,R − R n − n n (H.40) R2b na = d n,R − n n for n =1, 2,.... While we shall show below how to solve this system explicitly using the method of ‘variation of parameters’ this will not, by itself, deal with the convergence issues presented by the resultant (formal) Fourier series. To prove that global, bounded solutions to Eq. (H.28) for (2)Y do indeed exist we shall instead ﬁrst solve the Poisson equation for (2)Y R R which, together with the solution of Eq. (H.38), will serve to ,θ determine (2)Y R and, at the same time, provide the needed integrability condition for the complementary component (2)Y θ. AXISYMMETRIC PERTURBATIONS 97

For convenience extend the domains of definitions (at fixed t which, for simplicity, we suppress in the following) of the source components ab to the full plane R2 with the open disk of radius R = 1 M 2 (a2 + Q2) {M } + 2 − removed. This corresponds to ‘reflecting’ RR and ‘anti-reflecting’ p Rθ through the z-axis or, equivalently, throughM taking the range of θMin expansions (H.36–H.37) to now be [0, 2π). Note accordingly that the source term on the right-hand side of Eq. (H.32) will automatically be reflection symmetric whereas its θ-derivative, which provides the (2)Y R source in the Poisson equation for R , namely ,θ (H.41) 1 ∂ ∂ (2)Y R 1 ∂2 (2)Y R 1 1 1 R + = Rθ RR , R ∂R ∂R R R2 ∂θ2 R −R2 M ,θθ−R RM ,θ ,θ!! ,θ! ,R will be reflection anti-symetric and thus have a vanishing net ‘charge’ as well as having compact support. The fundamental solution (Green’s function) for this Dirichlet problem is explicitly known and, for arbitrary sufficiently smooth, reflection anti-symmetric Dirichlet data specified on the circle R = R+, provides a (2)Y R unique, globally bounded, reflection anti-symmetric solution, R , ,θ that decays asymptotically like 1 [29, 66]. Note that terms of the ∼ R form α + β ln (R/R+) that might otherwise be expected to occur are excluded by the reflection anti-symmetry of the source and boundary conditions. To complete the determination of (2)Y R we must solve Eq. (H.38) for the Fourier component a0 which, in view of Eq. (H.34), is defined by 1 2π (H.42) a (R)= dθ (2)Y R(R, θ). 0 2π Z0 From Eq. (H.36) we see that the source, c0(R), for this quantity is in turn given by 1 2π 1 (H.43) c (R)= dθ RR(R, θ) . 0 2π −RM Z0 This solution to Eq. (H.38) is simply R a0(R+) 1 (H.44) a (R)= R + dR′ c (R′) 0 R R 0 + ZR+ ′ but only the unique choice

∞ 1 (H.45) a (R )= R dR′ c (R′) 0 + − + R 0 ZR+ ′ 98 VINCENT MONCRIEF AND NISHANTH GUDAPATI yields a globally bounded solution for a0(R) which in fact vanishes outside the support of c0. (2) θ After fixing (for reasons of regularity at the axes) Y (R+, 0) = 0 one could now integrate the first order system (H.30)–(H.31) to determine (2)Y θ(R, θ). A more elegant approach however is to combine this regularity condition with the integral of Eq. (H.30) with respect to θ at R = R+ to determine (reflection anti-symmetric) Dirichlet data, (2) θ Y (R+, θ) for the solution to the Poisson equation 1 1 1 1 (H.46) R (2)Y θ + (2)Y θ = Rθ + RR . R ,R ,R R2 ,θθ −RM ,R R3 M ,θ which, in turn, results from Eqs. (H.30)–(H.31). Since the source term in Eq. (H.46) and its associated Dirichlet data are both reflection anti- symmetric this Poisson equation has a unique, globally bounded, reflection anti-symmetric solution. From the explicit form of Green’s function combined with the source’s compact support it further follows (2) θ 1 that Y (R, θ) decays asymptotically as R . The above argument has shown that a∼ unique, globally bounded, regular solution for (2)Y is determined from specifying Dirichlet data for (2)Y R at the horizon. On the other hand it is still of interest to see more explicitly how the Fourier coefficients a0, an, bn for this solution behave, especially in the asymptotic regions.{ We have} already solved Eq. (H.38) and found that 1 2π dθ (2)Y R(R, θ)= a (R) 2π 0 (H.47) Z0 ∞ 1 = R dR′ c (R′) − R 0 ZR+ ′ near R = R+ and that

(H.48) a0(R)=0 for all R outside the support of c0. We shall prove below in Appendix J that, for the perturbations of interest herein, the ‘integral invariant’ a0(R+) defined by Eq. (H.45) actually vanishes. It follows then from Eqs. (H.44) and (H.48) that a0(R) will vanish both inside and outside the support of c0 (i.e., throughout both asymptotic regions). To solve Eqs. (H.39) and (H.40) first note that they imply 1 n2b nc 1 d (H.49) (R b ) n = n + n R n,R ,R − R2 R2 R R ,R AXISYMMETRIC PERTURBATIONS 99 and that independent solutions to the corresponding homogenous equa- n n tions (for n =1, 2, ) are given by R and R− . It is therefore straightforward to apply the··· method of variation of parameters to show that, for each n, there is a unique, globally bounded solution bn(R) determined by boundary data bn(R+) specified at the horizon. These are of course nothing but the Fourier coefficients for the corresponding solutions (2)Y θ to Eq. (H.46) found previously. For large R, outside the source’s support, these solutions take the form (at fixed time t)

( ) n (H.50) bn(R)= βn− R− ( ) for suitable constants βn− whereas for R suﬃciently near R+ (inside the source’s support) they{ have} the form

(+) n ( ) n (H.51) bn(R)= αn R + αn− R− (+) ( ) (+) ( ) ( ) for suitable constants αn ,αn− . The constants αn ,αn− , βn− are all determined explicitly{ in terms} of the chosen Dirichlet{ data specified} at R+ and by the source functions cn(R),dn(R) . By now simply setting n 1 { } ∀ ≥ R2b d (R) (H.52) a (R)= n,R n n n − n one readily verifies that all of Eqs. (H.39) and (H.40) are satisfied and that the a (R) take the asymptotic forms { n } ( ) n+1 (H.53) a (R)= β − R− n − n for R sufficiently large and

(+) n+1 ( ) n+1 (H.54) a (R)= α R α − R− n n − n ( ) for R sufficiently near R . Note in particular that a (R) β − + 1 −→ − 1 for large R whereas the higher order coefficients an(R); n =2, 3,... decay as increasingly negative powers of R. { } While it may not be specifically needed for our analysis to go through we shall focus henceforth on those particular gauge transformations generated by vector fields (2)Y satisfying the ‘homogeneous’ Dirichlet condition (2) R (H.55) Y ,θ(R+, θ)=0. From Eqs. (H.34) and (H.54) this boundary condition clearly corresponds to setting a (R )=0 n 1 or, equivalently n + ∀ ≥ ( ) 2n (+) (H.56) αn− = R+ αn . 100 VINCENT MONCRIEF AND NISHANTH GUDAPATI

Designating the ‘source’ term for Eq. (H.49) by σn(R) so that nc (R) 1 ∂ d (R) (H.57) σ (R) := n + n n R2 R ∂R R one readily ﬁnds the unique, globally bounded solution to this equation to be

n ∞ σn(R′) 1 n b = R (R′) − dR′ n − 2n ZR n 2n ∞ σn(R′) 1 n (H.58) + R− R (R′) − dR′ − + 2n ZR+ R σn(R′) n+1 (R′) dR′ . − 2n ZR+ Specializing this formula to the asymptotic regions corresponding to (+) ( ) ( ) R R+ and R one easily discovers that the coeﬃcients αn ,αn− , βn− areց given by ր∞ { }

(+) ∞ σn(R′) 1 n α = (R′) − dR′ n − 2n ZR+ ( ) αn− (H.59) = 2n R+ and

( ) 2n ∞ σn(R′) 1 n β − = R (R′) − dR′ n − + 2n ZR+ ∞ σn(R′) n+1 (H.60) (R′) dR′ − 2n ZR+ wherein, as above, we have suppressed their time dependence to simplify the notation. Since we have already argued (c.f., the proof given in Appendix J) that a0(R+) = 0 for the perturbations of interest herein, it follows from Eqs. (H.55) and (H.56) that (2)Y R satisﬁes the Dirichlet condition (2) R (H.61) Y (R+, θ)=0 at the horizon boundary. At several points in our discussion we have encountered occasions wherein the leading order term in an expansion of the form

∞ ( ) 1 (H.62) Ψ (R, θ) := β − sin (kθ) 1 k Rk Xk=1 AXISYMMETRIC PERTURBATIONS 101 cancels out in the expression of interest leaving what appears to have a faster rate of decay as R . While this higher rate of decay would be self-evident for a finiteր∞series it is not obvious in the case of an infinite series that the ‘remainder’ does indeed decay faster than the leading order term. For the functions considered herein however we shall see that this is indeed the case. In the asymptotic region near the functions of interest in this context are harmonic, hence analytic∞ and have convergent expansions of the type indicated above. If for some reason the first N 1 terms (for N 2) cancelled from a quantity being computed we’d− be left with a (convergent)≥ expansion of the form

∞ ( ) 1 (H.63) Ψ = β − sin (kθ) . N k Rk kX=N We wish to consider this in the asymptotic region R > R0 > R+. For this purpose define, for convenience, the coordinate x by R 1 1 (H.64) R = 0 , x , 1 R x ∈ −R R − 0 0 0 so that x 0 R R and x 1 R . 0 R0 Writingց ⇔ ց ր ⇔ ր∞ k ∞ ( ) 1 R0x (H.65) ΨN = βk− sin (kθ) − R0 kX=N and recalling that the analyticity of ΨN implies the absolute convergence of its series expansion we get k ∞ ( ) 1 R0x ΨN = βk− sin (kθ) − | | R0 kX=N k ∞ ( ) 1 R0x βk− sin (kθ) − ≤ R0 k=N X N k N (H.66) ∞ 1 R 0x ( ) 1 R0x − = − βk− sin (kθ) − R0 R0 k=N N X ℓ ∞ 1 R0x ( ) 1 R0x = − βℓ+−N sin ((ℓ + N)θ) − R0 R0 ℓ=0 X < . ∞ Every term in the final summation has positive sign and either remains constant or decays monotonically in R as R (i.e., x 1 ) for → ∞ ր R0 102 VINCENT MONCRIEF AND NISHANTH GUDAPATI any fixed θ. It follows that the resultant expression for ΨN decays at N | | 1 R0x 1 least of order O − = O N for N 2 as R . R0 R ≥ ր∞ Recalling that ∂r R2 (H.67) = 1 + ∂R − R2 and that the partial derivatives γ ,γ ,ω ,ω ,λ ,λ , η , η are all { ,r ,θ ,r ,θ ,r ,θ ,r ,θ} bounded at R = R+ we see that the corresponding gauge transformed perturbations γ˜′, ω˜′, λ˜′, η˜′ are all regular at the horizon. (c.f., Eqs. (H.15)– (H.18)). { } To evaluate the relevant ‘flux’ integrals resulting from, for example, the integrated form of Eq. (4.2) we shall need the asymptotic forms of the linearized canonical momenta. These are given by the linearized field equations (c.f., Eqs. (C.49), (C.51), (C.53), (C.55) and (C.57)) ˜ Np˜′ ˜ a′ (H.68) = 4(γ,t′ N γ,a), (2)g˜ − ˜ 2γ Nep v˜′ ˜ a′ (H.69) = λ,t′ N λ,a, (2)g˜ − ˜ 2γ Ne (˜u′ pλr˜′) ˜ a′ (H.70) − = η,t′ N η,a, (2)g˜ − ˜ 4γ ˜ 2γ p Ne r˜′ ˜ a′ Ne = ω,t′ N ω,a + λ(˜u′ λr˜′), (H.71) (2)g˜ − (2)g˜ − a′ a′ = ω′ N˜ ω + λ(η′ N˜ η ), p ,t − ,a p ,t − ,a 2N˜ ′ cd ′ (2) (H.72) (˜gacg˜bd gãbg˜cd)˜π =g ãb,t′ N˜ c ∂ g˜ (2)g˜ − − L c ab Using Eqs.p (H.17), (H.18) and (H.24) to evaluate these in the asymp- (4) totic regions (where kαβ = 0) we obtain, thanks to a fortuitous can- (2) a cellation of the terms involving Y ,t, ˜ Np˜′ ˜ 2 ac (4) 0 (H.73) 4N g˜ γ,a Y ,c, (2)g˜ −→ ˜ 2γ Nep v˜′ ˜ 2 ac (4) 0 (H.74) N g˜ λ,a Y ,c, (2)g˜ −→ ˜ 2γ Ne p ˜ 2 ac (4) 0 (H.75) (˜u′ λr˜′) N g˜ η,a Y ,c, (2)g˜ − −→ p AXISYMMETRIC PERTURBATIONS 103

˜ 4γ Ne r˜′ ˜ 2 ac(4) 0 (H.76) N g˜ Y ,c(ω,a + λη,a), (2)g˜ −→ and p ˜ 2N ′cd (H.77) (˜gacg˜bd gãbg˜cd)˜π ( (2) g˜)ab (2)g˜ − −→ L D where p ∂ (H.78) (2) := N˜ 2g˜cd(4)Y 0 . D ,d ∂xc Note that each of the above expresses the desired (linearized) momentum asymptotically in terms of Lie derivatives with respect to (2) . In deriving the above we have made use of the formula D 2γ (4) 2γ (4) (H.79)g ˜′ = g˜ e− k + e k +( (2) g˜) ab ab ϕϕ ab L Y ab which results from linearizing the defining equation 2γ 4γ gãb = e gab e βaβb (H.80) − = (4)g (4)g (4)g (4)g ϕϕ ab − aϕ bϕ about the chosen background (c.f. Eqs. (2.11–2.14).

Appendix I. Compactly Supported Solutions of the Linearized Constraint Equations As we have already discussed near the end of Appendix B, the use of hyperbolic gauge conditions for the linearized field equations allows one to exploit the corresponding, causal propagation of the perturbations to conclude that compactly supported initial data on a Cauchy hypersurface of constant Boyer-Lindquist time, t, evolves so as to preserve this property for all finite t. Thus data initially bounded away from the horizon and from spacelike infinity evolves to remain so throughout the evolution — a feature which reflects the fact that Boyer-Lindquist time slices for Kerr-Newman spacetimes are ‘locked down’ at i0 (spacelike infinity) and at the bifurcation 2-sphere lying in the horizon. While this property of compactly supported evolution will ultimately be lost upon transformation to an elliptic gauge of the type adopted herein, it will be noteworthy to recognize that the transformed perturbations, though no longer in general having compact support, will necessarily be of ‘pure gauge type’ near the horizon and near infinity. The utilization of hyperbolic gauge conditions to secure causal evolution for the perturbations does not, however, preclude the need to solve the linearized constraint equations, at least on the initial Cauchy 104 VINCENT MONCRIEF AND NISHANTH GUDAPATI hypersurface. Since these latter are normally treated as an elliptic system for certain dependent or constrained variables, it is not immediately clear how to ensure the desired compact support of their resulting solutions. While one could presumably guarantee this outcome by imposing suitable restrictions upon the otherwise ‘free data’ occurring in these equations we shall herein adopt a different strategy whereby one solves the constraints algebraically for a subset of this normally regarded free data, reversing somewhat the usual roles of free and constrained variables. This will allow us to ensure the compact support of the solutions so obtained without otherwise unduly restricting their generality. Consider first the reduced momentum constraints, Eqs. (G.8), and assume for definiteness that the background has charge Q = 0. Assume also that a = 0 since otherwise the background spacetime6 would be a Reissner-Nordstrom6 solution which is treatable by much more elementary methods [54, 55, 56]. Under these assumptions the functions λ and η (c.f. Eqs. (C.40) and (C.45)) are both non-vanishing and one can reexpress the momentum constraints as an algebraic system of the form

η λ u˜′ (I.1) ,R ,R = SR η λ v˜′ ,θ ,θ Sθ where ′ (2) b 2ν √(2) (I.2) R := 2 b(h)r ˜ R + e h τ,R′ p˜′γ,R r˜′ω,R S ∇ ′ − − (2) b 2ν (2) (I.3) := 2 (h)r ˜ + e √ h τ ′ p˜′γ r˜′ω . Sθ ∇b θ ,θ − ,θ − ,θ ′b The idea is choose the data p˜′, r˜′, τ ′, r˜ to have compact support { a} on Mb and to solve equations (I.1) for the electromagnetic momenta u˜′, v˜′ . Clearly the feasibility of this approach hinges upon the invert- {ibility} of the matrix function η λ (I.4) := ,R ,R . D η,θ λ,θ By a straightforward computation one ﬁnds that its determinant det = η λ λ η D ,R ,θ − ,R ,θ 2 (I.5) R+ 2 2 2 3 1 R2 (r + a )4Q a sin (θ) = − , (r2 + a2 cos2 θ)2

2 which is thus non-vanishing except on the horizon (where 1 R+ = − R2 0) and on the axes (having sin (θ) = 0). The formal solution to Eq. (I.1) AXISYMMETRIC PERTURBATIONS 105 is given explicitly by

u˜′ 1 R (I.6) = − S v˜′ D θ S where

4ra cos θ 2a(r2 a2 cos2 θ) − R2 (r2+a2) sin θ 1 + sin2 θ − R2 1 1   (I.7) − = D 4Qa 2(r2 a2 cos2 θ) 4ra2 cos θ  − −  R2 (r2+a2) sin θ  1 + sin2 θ   R2   −    By choosing the free data occurring in on to have not only SR Sθ compact support on Mb but also to vanish at the axes as suitable powers of sin θ one ensures both the compact support of the resulting solution and its regularity at the axes. Note by contrast that one normally thinks of Eqs. (I.1)–(I.3) as an elliptic system to be solved for ′b r˜ a instead of an algebraic one for u˜′, v˜′ . Now, however, suppose that Q ={ 0 (but} a = 0 since otherwise the background would simply be Schwarzchild).6 The functions γ and ω (given by Eqs. (C.43) and (C.46) in the limiting case Q 0) are still non-vanishing and one can now express the momentum constraints→ in the alternative form:

p˜′ ˜ (I.8) ˜ = SR D r˜′ ˜ Sθ where γ ω (I.9) ˜ := ,R ,R D γ,θ ω,θ Q=0

and where ′ ˜ (2) b 2ν √(2) (I.10) R := 2 b(h)r ˜ R + e h τ,R′ S ∇ Q=0 n o and ′ ˜ (2) b 2ν √(2) (I.11) θ := 2 b(h)r ˜ θ + e h τ,θ′ S ∇ Q=0 n o

Algebraic solvability now hinges on the invertibility of the matrix function ˜. A straightforward computation of the determinant, D (I.12) det ˜ := γ ω ω γ ) , D { ,R ,θ − ,R ,θ |Q=0 106 VINCENT MONCRIEF AND NISHANTH GUDAPATI of ˜ yields D (I.13) 2 3 R+ Ma sin θ 1 R2 det ˜ = − − D (r2 + a2 cos2 θ)3 (r2 + a2)2 a2∆ sin2 θ  − (r3 Ma2) 6r6 +2 a2 r4 a4 cos6 (θ)  × − − +6a6r(Mr a2) cos6 (θ) − + 10r3(Mr a2)+2M(r4 a4)+2r2(r3 Ma2) a4 cos4 (θ) − − − + 10a2r4(r3 a2M)+4r2Ma2(r4 a4)+2r5a2(rM a2) cos2 (θ) − − − This is easily seen to be non-vanishing except on the axes (where it vanishes as sin3 (θ)) and at the horizon where, in the subextremal cases, 2 it vanishes like 1 R+ as R R . Curiously, in the extremal cases − R2 ց + ( a = M), every term in the brackets also vanishes at the horizon | | { } r r+ = M = a . →Thus one can| now| solve the momentum constraints for the gravita- ′b tional momenta, p˜′, r˜′ , taking the ‘free data’ r˜a , τ˜′ to have compact support and to vanish{ at} the axes as suitable powers{ of} sin (θ) to ensure regularity of the solution. Turning now to the (linearized) Hamiltonian constraint, ˜′ = 0, one sees from Eq. (G.20) that this can be expressed in divergenceH form as

∂ (2) ab N˜ ˜′ = N˜√ h h 4γ γ′ +2ν′ H ∂xb ,a ,a 2γ n 4γ (I.14) + e− (η,aη′ + λ,aλ′)+ e− (ω,a + λη,a)(ω′ + λη′)

(2) ab 2√ h h N˜ ν′ =0. − ,a o Since Mb is simply connected the vector density appearing in the b bc { } brackets must take the form = ǫ σ′ for some function σ′. Thus { } ,c any solution to (I.14) must satisfy

4γ 2γ 4γ,aγ′ + e− (ω,a + λη,a)(ω′ + λη′)+ e− (η,aη′ + λ,aλ′) bc (I.15) habǫ 2N˜,a = σ,c′ + ν′ 2ν,a′ N˜√(2)h N˜ −

Now if Q = 0 (and, as always a = 0) we deﬁne Ω′ := ω′ + λη′ and 6 6 regard (I.15) as an algebraic system for η′,λ′ , taking the ‘free data’ { } σ′, ν′,γ′, Ω′ in this case to have compact support and to vanish suf- {ﬁciently rapidly} at the axes. AXISYMMETRIC PERTURBATIONS 107

Since the matrix of coefficients for this algebraic problem is nothing other than the defined previously one solves for η′,λ′ and then D { } sets ω′ = Ω′ λη′ to complete the solution. If on the other− hand Q = 0 then (I.15) reduces to the form (I.16) bc ˜ 4γ habǫ 2N,a ′ − ′ ′ ′ ′ γ,a(4γ )+ ω,a(e ω ) Q=0 = σ,c + ν 2ν,a (N˜√(2)h N˜ − ) Q=0 ˜ and one can exploit the fact that is invertible (assuming as always D 4γ that a = 0) to solve this system algebraically for 4γ′, e− ω′ . Thus all of the6 reduced constraints can be solved algebraically{ for compa} ctly supported data that is regular at the axes of symmetry for the background black hole. There is however a remaining subtlety that must be dealt with. We need to ‘lift’ the Cauchy data defined on the quotient manifold Mb back up to the actual, 3-dimensional Cauchy surface for the black hole’s DOC and ensure that it all has compact support there as well. The potential obstructions to this are the first variations, ab ab (I.17)r ˜′ = ǫ βa,b′ , u˜′ = ǫ Ca,b′ a a of the defining equations (C.28) for the one-forms βadx and Cadx . a Even ifr ˜′ andu ˜′ have compact support the one-forms βa′ dx and a Ca′ dx need not inherent this property without further restrictions upon the ‘sources’r ˜′ andu ˜′. By contrast note that the first variations of Eqs. (C.24) and (C.25),

a′ ab a′ ab (I.18) f˜ = ǫ ω′ , = ǫ η′ ,b E ,b ′ ′ automatically yield lifted vector densities f˜a ∂ and a ∂ of compact ∂xa E ∂xa support provided only that the base space potentials ω′ and η′ have this property. Since both equations (I.17) are identical in form it suﬃces to show what further restrictions uponr ˜′ are needed to solve for a compactly a a supported βa′ dx since the argument for the pair u˜′,Ca′ dx will follow the same pattern. { } Guided by the Hodge decomposition of one-forms on simply con- ab nected 2-manifolds we seek a solution tor ˜′ = ǫ βa,b′ of the form

hab bc (I.19) βa′ = ζ,a + ǫ ψ,c √(2)h for some undetermined functions ζ, ψ . The equation to be solved now takes the u form of Poisson’s{ equation} for the unknown function 108 VINCENT MONCRIEF AND NISHANTH GUDAPATI

ψ, 1 1 b √(2) bc bc (I.20) b(h) (h)ψ := ∂b( hh ψ,c)= ǫ βb,c′ . ∇ ∇ √(2)h √(2)h

In terms of the coordinates R and θ introduced for Mb in Appendix A, and for which the ﬂat metric h dxa dxb takes the form ab ⊗ (I.21) h dxa dxb = dR dR + R2dθ dθ, ab ⊗ ⊗ ⊗ any smooth source function

1 bc (I.22) s := ǫ βb,c′ √(2)h that is regular at the axes of Mb and that has compact support on this space will admit a Fourier expansion of the form

∞ (I.23) s(R, θ)= σm(R)cos(mθ) m=0 X where each of the Fourier coeﬃcient functions σm will vanish for all R such that { } (I.24) R R R > R ≥ 2 ≥ 1 + and that (I.25) R < R R R + ≤ 1 ≤ 2 for suitably chosen R1 and R2. Any smooth solution ψ to (I.20) must admit a corresponding Fourier expansion,

∞ (I.26) ψ(R, θ)= ψm(R)cos(mθ) m=0 X with coefficients satisfying the associated ordinary differential system d2ψ 1 dψ m2 (I.27) m + m ψ = σ ; m =0, 1, 2,... dR2 R dR − R2 m m Each of these equations can be readily solved using the method of variation of parameters. It is straightforward to show that the resulting solution ψ will take constant values in the two asymptotic regions (I.28) R < R R and R R , + ≤ 1 ≥ 2 and thus have compactly supported gradient on Mb if and only if the source functions σm satisfy the following (definite) integral conditions: { } R2 1 m (I.29) R − σm(R)dR = 0 (no sum on m) ZR1 AXISYMMETRIC PERTURBATIONS 109 and

R2 1+m (I.30) R σm(R)dR =0. (no sum on m) ZR1

The fact that these two conditions coincide for m = 0 corresponds to the ﬂexibility of allowing ψ to have two distinct, constant values in the two asymptotic regions. The remaining function ζ arising in the decomposition (I.19) is un- restricted by Eq. (I.17) and thus can be chosen arbitrarily to have compactly supported gradient. The freedom to add an arbitrary gradi- a ent to the one-form βa′ dx corresponds to that of making a coordinate transformation of the form

(I.31) x3 = ϕ ϕ + ζ → in the U(1) bundle over Mb. While we chose above to solve the reduced (Hamiltonian and momentum) constraints algebraically it is straightforward to see from the preceding example that we could, alternatively, have treated them as ′b Poisson type equations for the ‘usual’ unknowns ν′, r˜ a and still en- sured compact support for the solutions by imposing{ suitable} integral constraints (as well as compactness of support) upon the ‘free data’ (γ′, p˜′), (ω′, r˜′), (η′, u˜′), (λ′, v˜′), τ ′ . This follows from the fact that, when{ expressed in terms of the ‘Cartesian’} coordinates ρ,¯ z¯ for the flat metric (2)h (wherein (2)h = dp¯ dp¯ + dz¯ dz¯), the{ linearized} constraints reduce to decoupled equations⊗ of precisely⊗ the (flat space) ′b Poisson type that we have just dealt with for the unknowns ν′, r˜ a . At various stages in our analysis (e.g., solving the momentum{ con-} straint in Appendix G, preserving Weyl-Papapetrou gauge conditions with a suitably chosen perturbed shift in Appendix F and in the proof of vanishing of the ‘integral invariant’ a0(R+) presented in the Appen- dix below) we have (implicitly or explicitly) exploited the claim that transverse-traceless symmetric 2-tensors, subject to suitable asymptotic and boundary conditions on Mb, vanish identically. To establish this claim let us first work in ‘isothermal’ coordinates ρ, z for which the flat metric (2)h = h dxa dxb takes the form { } ab ⊗

(I.32) (2)h = h dxa dxb = dρ dρ + dz dz. ab ⊗ ⊗ ⊗ 110 VINCENT MONCRIEF AND NISHANTH GUDAPATI

(2) tr tr a b An arbitrary traceless symmetric 2-tensor, k = kabdx dx can be expressed in these coordinates as ⊗ (2)ktr = ktr dxa dxb ab ⊗ (I.33) = u(dρ dρ dz dz) ⊗ − ⊗ v(dρ dz + dz dρ) − ⊗ ⊗ and is obviously traceless with respect to any metric conformal to (2)h as well. Imposing the independent (and equally conformally invariant) condition that the covariant divergence of (2)ktr vanish is well-known (and straightforwardly seen) to be equivalent to requiring that the component functions u, v satisfy the Cauchy-Riemann equations { } u,p = v,z (I.34) u = v ,z − ,p which in turn of course imply that each of u, v is harmonic with respect to the metric (2)h (or to any metric conformal thereto):

(I.35) ∆(2)hu = ∆(2)hv =0. (2) Reverting to polar coordinates R, θ on Mb for which h takes the form { } (I.36) (2)h = dR dR + R2dθ dθ ⊗ ⊗ with R > R+ and θ [0, 2π) one easily ﬁnds that globally harmonic functions that vanish∈ on the horizon as R R and are bounded on ց + Mb must in fact vanish identically. Hence we have that Theorem 4. Globally deﬁned transverse traceless symmetric 2-tensors, (2) TT k , which are bounded on Mb and which vanish at the horizon corresponding to R R > 0 vanish identically. ց +

Appendix J. The Vanishing of a0(R+) As discussed in Appendix H, the successful implementation of our chosen (Weyl-Papapetrou) gauge condition hinges upon proving that a certain ‘integral invariant’, a0(R+), actually vanishes for the class of perturbations considered. In the course of carrying out such a proof we shall see that this quantity is in fact gauge-invariant (with respect to the relevant class of such transformations) and thus justify its char- acterization as such. In terms of the (spatially compactly supported) 4-metric perturbation, (4)k = (4)k dxµ dxν of (4)g introduced in Appendix H (and µν ⊗ AXISYMMETRIC PERTURBATIONS 111 assumed therein to be expressed in a ‘hyperbolic’ gauge), a0(R+) was deﬁned by the integral formula (c.f., Eq. (H.45))

∞ 1 (J.1) a (R )= R dR′ c (R′) 0 + − + R 0 ZR+ ′ wherein c0(R) was in turn given by (c.f., Eq. (H.43))

1 2π 1 (J.2) c (R)= dθ RR(R, θ) 0 2π −R M Z0 with cd defined via (c.f., Eq. (H.28)) M 1 (J.3) cd = (2)g˜ gãcg˜bd e2γ (4)k g˜ g˜ef (4)k . M ab − 2 ab ef p Evaluating RR on a Kerr-Newman background and exploiting Eq. (2.12) to express theM relevant components of (4)k in terms of (first variations of) our 2+1 dimensional quantities γ, ν, g˜ , h we arrive at { ab ab}

RR R 2γ 2ν (4) 1 (4) e − k k M → 2 RR − R2 θθ (J.4) R 1 = (δh ) (δh ) 2 RR − R2 θθ where δhab designates the first variation (also signified by a ) of the flat ‘conformal metric’ introduced in Appendix F. In view of′ the axis regularity requirements discussed in Appendix H this perturbation has an expansion (with its t-dependence suppressed, as before, to simplify the notation) of the form, setting ℓab := δhab

∞ (J.5) ℓRR = γ0(R)+ γn(R)cos(nθ), n=1 X ∞ (J.6) ℓRθ = ℓθR = δn(R) sin (nθ), n=1 X ∞ (J.7) ℓθθ = σ0(R)+ σn(R)cos(nθ). n=1 X 112 VINCENT MONCRIEF AND NISHANTH GUDAPATI

(2) To preserve its flatness this perturbation of the hab metric must satisfy the (necessary and sufficient) condition D √(2)h (2)R((2)h) ℓ = √(2)h (2) a((2)h)(2) b((2)h)ℓ · ∇ ∇ ab (2) ((2)h)(2) a((2)h)(hcdℓ ) − ∇a ∇ cd 1 2 (J.8) = ℓ + ℓ + ℓ −R RR,θθ RR,R R Rθ,θR 2 2 1 ℓ + ℓ ℓ − R3 θθ R2 θθ,R − R θθ,RR =0. This condition is, of course, automatically satisfied by the pure gauge perturbations (2) vab := (2)Y h (J.9) L ab (2) c (2) c (2) c = Y hab,c + Y ,ahcb + Y ,bhac where (J.10) h dxa dxb = dR dR + R2 dθ dθ ab ⊗ ⊗ ⊗ and wherein the vector field (2)Y admits an expansion of the form given by Eqs. (H.34) and (H.35), namely

∞ (2) R (J.11) Y = a0(R)+ an(R)cos(nθ), n=1 X ∞ (2) θ (J.12) Y = bn(R) sin (nθ). n=1 X In view of the simple formula for RR (c.f., Eq. (J.4)) which now gives M RR 1 1 (J.13) M = ℓ ℓ R 2 RR − R2 θθ and the angular integral in Eq. (J.2) for c0(R) we see that only the (rotationally-invariant) n = 0 terms in the expansions (J.5)–(J.7) contribute to c0(R) and hence to a0(R+). For these quantities it is convenient to deﬁne a new set of variables I σ0 (J.14) k0 := γ0 − R ,R II (J.15) k0 := σ0 for which the inverse transformation is clearly II (J.16) σ0 = k0 , AXISYMMETRIC PERTURBATIONS 113

kII (J.17) γ = kI + 0 . 0 0 R ,R

I II It is easily veriﬁed that k0 is gauge invariant whereas k0 is, in effect, pure gauge. Furthermore, the rotationally invariant component of Eq. (J.8) (i.e., its integral with respect to θ over the circle) yields the (gauge invariant) constraint

I (J.18) k0,R =0

I so that k0,R is, at most, a (possibly t-dependent) constant. Substituting the above results into the formula for a0(R+) we now arrive at:

I II R ∞ k k (J.19) a (R )= + dR 0 + 0 . 0 + 2 R R2 ZR+ " ,R#

I But the term in k0 can only give a ﬁnite contribution if this constant II vanishes whereas the (boundary) contributions of k0 will vanish for any compactly supported perturbation. We conclude that

Theorem 5. The integral invariant a0(R+) vanishes when evaluated upon compactly supported perturbations (that vanish on the asymptotic regions near the horizon on inﬁnity).

A simpler, more explicit proof of the above result can be given in the non-rotating (a = 0) case by exploiting the utility of expanding the perturbations of the (spherically symmetric) background Reissner- Nordström solution in (Regge-Wheeler) tensor harmonics. It is clear from the structure of c0(R) (c.f., Eq. (H.43)) and a0(R) (c.f., Eqs. (H.44)– (H.45)) that only the spherically symmetric ‘mode’ of the perturbations contributes in this case and, as is well-known, this non-dynamical mode decouples from all of the ‘higher harmonic’ modes. Because of the dynamical triviality of this (spherically symmetric) perturbative mode, as guaranteed by the (generalized) Birkhoff theorem, it was not treated in detail in the earlier, Hamiltonian stability analyses of the Reissner- Nordström spacetime (c.f. Refs. [54, 55, 56]). We therefore provide those ‘missing’ details in the following. 114 VINCENT MONCRIEF AND NISHANTH GUDAPATI

In the t,R,θ,ϕ coordinates of Appendix A the Reissner-Nordström line element{ takes} the form 2 2 R+ 2 1 R2 dt ds2 = − − 2 2 M R+ (J.20) 1+ R + R2 M R2 2 + 1+ + + dR2 + R2dθ2 + R2 sin2 θ dϕ2 R R2 where 1 (J.21) R = M 2 Q2 + 2 − with the remainingp ADM variable (c.f., Ref. [54]) given by8 (J.22) R =2Q sin θ, θ = ϕ =0, Bi =0, πij =0. E E E The axisymmetric perturbations of such a background may be conveniently expanded in the usual way (c.f., [54, 55, 56]) in terms of Regge-Wheeler tensor harmonics, which in turn, are constructed explicitly in terms of the standard (scalar) spherical harmonics YL0 . Since we shall only here be concerned with the (spherically symmetric{ }) 1 case corresponding to L = 0 and since Y00 = 4π we shall absorb this ubiquitous constant multiplicative factor intoq the perburbative func- R tions that it multiplies (i.e., into the quantities H2,K,PH,PK,Y , etc. defined below) to simplify the notation. Defining M R2 2 (J.23) e2λ = 1+ + + R R2 we expand the ADM spatial metric perturbation (hij) := (δgij) as 2λ e H2(R, t)0 0 2λ 2 (J.24) (hij)= 0 e R K(R, t) 0  0 0 e2λR2 sin2 θK(R, t)    The gauge transformations of (hij) in this case are generated entirely i ∂ by spatial vector fields, Y = Y ∂xi , of the form (J.25) (Y i)= Y R(R, t), 0, 0

8 We assume throughout that the magnetic fieldBi is derivable from a vector potential and thus vanishes identically (together with its first variation) in the spherically symmetric case of interest here. Recall also the slightly non-standard conventions for the designation of the electromagnetic field introduced in Appen- dix C. AXISYMMETRIC PERTURBATIONS 115 and induce the (pure gauge) first variations (c.f., [54])

R R (J.26) δH2 =2λ,RY +2 Y ,R 2 R+ 2 1 2 − R R (J.27) δK = 2 Y . R+ R + M + R It is therefore natural to introduce the new variables k1,k2 deﬁned by { }

2 2 M 2R+ R+ R + R2 K R + M + R K (J.28) k := H + 1 2 R2 R2 + −  +  1 R2 1 R2 − − ,R 2   R+ 1 K R + M + R (J.29) k := 2 R2 2 1 + − R2 for which the inverse transformation is easily found to be

2 M 2R+ 2k2 R + R2 (J.30) H = k +2k 2 1 2,R R2 − + R + M + R R2 2k 1 + 2 − R2 (J.31) K = 2 R+ R + M + R and for which the pure gauge variations take the form

R (J.32) δk1 =0, δk2 = Y showing that k1 is gauge invariant. i In view of the Gauss law constraint, ,i = 0, and its ‘linearization’ about the chosen background, the only allowedE spherically syummetric perturbation of i ∂ must take the form E ∂xi R R′ (J.33) δ := =2Q′ sin θ, E E ′ (J.34) δ θ := θ =0, E E ′ (J.35) δ ϕ := ϕ =0. E E Since we shall eventually require that the perturbations of interest have compact support this will necessitate taking the charge perturbation Q′ = 0 but we shall retain this for now. 116 VINCENT MONCRIEF AND NISHANTH GUDAPATI

In terms of these new variables the linearized Hamiltonian constraint, ′ = 0, now takes the form H

∂ 2k1 2 2 ′ = sin θ − R R H ∂R R − + sin θ 4QQ k 8R2 2MR2 (J.36) + ′ 1 2M + + + + R2 2 2 M + R − R R R 1+ R + R2 =0 which, of course, is gauge invariant (c.f., the discussion in Section IV of [54]). Given a choice for Q′, this constraint is clearly a ﬁrst order linear equation for the invariant perturbation k1 whose general solution is given by

′ ′ 2M ′ (MM QQ ) − ∂ R + R2 (J.37) k = R 1 R2 − ∂R  1 +  − R2   where M ′ is the corresponding ‘constant’ of integration which, at this point, could conceivably be a function of time (as could Q′). As we shall see however, the linearized evolution equations can be exploited to show that both M ′ and Q′ are both necessarily true constants which, not surprisingly, designate 1st order variations to the mass and charge parameters of the (Reissner-Nordström) ‘background’ solution. Indeed the most straightforward way of solving Eq. (J.36) is simply to evaluate k1 for this ‘trivial’ perturbation which, by the generalized Birkhoff theorem, is the most general, spherically symmetric perturbation that could induce a variation of this gauge invariant quantity. It is now clear however that the only such compactly supported per- i′ ∂ turbations must have Q′ = M ′ = 0 with k1 = 0 and ∂xi = 0 and i′ ∂ E ∂xi = 0 accordingly. Note furthermore that these quantities must vanishB t since their otherwise non-compact support at any finite value of t would∀ contradict the causal propagation of perturbations in ‘hyperbolic’ gauge (c.f., the discussion in Appendix B). Below however we shall give an independent proof of the ‘conservation’ of M ′ and Q′. AXISYMMETRIC PERTURBATIONS 117

Evaluating RR (c.f., Eq. (H.28) on these spherically symmetric perturbationsM (prior to imposing their compact support) one arrives at RR 1 k −M = k R 2 R −2 1 − R ,R ′ ′ (J.38) M ′ (MM QQ ) − ∂ R + 2R2 k = R 2 R2 ∂R  1 + − R   − R2  so that (c.f., Eqs. (H.43)–(H.45))  2π 1 1 RR c0(R)= dθ 2π 0 −RM Z ′ ′ M ′ (MM QQ ) (J.39) − ∂ R + 2R2 k = R 2 R2 ∂R  1 + − R   − R2  and, consequently,   ′ ′ M ′ (MM QQ ) ∞ − R + 2R2 k (J.40) a (R )= R 2 0 + + R2 −  + − R  1 R2  −  R+ which clearly thus vanishes for any allowed perturbation of compact   support. R′ The linearized Maxwell equations for ∂t gives immediately the d E expected result that dt Q′ = 0 (i.e., conservation of charge). To derive directly the corresponding result for M ′ we introduce the linearized, spherically symmetric, gravitational momenta (δπij) := (pij) with (J.41) λ 2 e R sin θPH(R, t)0 0 ij λ (p )= 0 e sin θPK (R, t) 0  eλ  0 0 sin θ PK(R, t) and deﬁne the ‘new variables’  2 3λ (J.42) p1 := R e PH, R2 M 2R2 p := 4Re2λP 1 + 2Re2λP + + 2 K − R2 − H R R2 (J.43) ∂ 2R2e3λP − ∂R H so that p ,p are (after absorbing the normalization factor of Y = { 1 2} 00 1 ) precisely the canonical momenta conjugate (respectively) to k ,k . 4π { 1 2} q 118 VINCENT MONCRIEF AND NISHANTH GUDAPATI

In terms of these quantities the linearized momentum constraint becomes

(J.44) ( ′)=(p sin θ, 0, 0) Hi 2 which, as expected (c.f. section IV of [54]) reveals this constraint as the generator of the gauge transformations (J.32). The linearized evolution equations for k ,k yield { 1 2} p (J.45) k = − 2 0, 1,t 2Re3λ ≈ p ′ (J.46) k = − 1 + XR 2,t 2Re3λ ′ ′ where (Xi )=(XR , 0, 0) is the linearized shift field. Note that the first dM ′ of these gives the independent proof that dt = 0. Appendix K. Maximal Slicing Gauge Conditions For the ‘background’ Kerr-Newman metric, expressed in Boyer-Lindquist coordinates via Eq. (A.1), both the 2+1-dimensional mean curvature of the constant time hypersurfaces, g˜ πãb (K.1) τ := ab , µ(2) g˜ and its 3+1-dimensional analogue, 1 ij (3) 2 gijπ γ p˜ (K.2) tr(3)g K := = e τ + µ(3) 4µ(2) g g˜ vanish so that these slices are ‘maximal’ in both senses of the term. To impose, on the other hand, a (linearized) maximal slicing gauge condition on the perturbations one must choose between setting τ ′ =0 p˜′ (maximal slicing in the 2+1-dimensional sense) or τ ′ + 4µ = 0 (its (2)g˜ 3+1-dimensional analogue) since, in general, these are inequivalent. The linearized field equations yield 1 cd ˜ (K.3) τ,t′ = ∂c(µ(2)g˜g˜ N,d′ ) −µ(2)g˜ so that, to enforce 2+1-dimensional maximal slicing, one needs to require that the linearized lapse function N ′ satisfy the ‘harmonic’ condition √(2) cd ˜ (K.4) ∂c hh N,d′ =0. Taken together with the simplest (homogeneous) boundary conditions this equation has the unique, trivial solution N˜ ′ = 0. This is the AXISYMMETRIC PERTURBATIONS 119 gauge condition we have exploited above in our discussion of energy conservation (c.f., Section 4) since it automatically ‘kills off’ several of the terms in the energy flux formula (c.f., Eq. (4.2)) that would, otherwise, need to be evaluated and dealt with. Consider however the alternative condition needed to preserve 3+1- p˜′ dimensional maximal slicing, namely τ,t′ + 4µ = 0. In this case (2)g˜ ,t the linearized field equations yield the more intricate elliptic equation for N˜ ′ given by:

(K.5) 2γ (2) ab γ γ (2) ab e− ∂ √ hh e (e− N˜ ′) + N˜ ′√ hh (η η + λ λ ) − a ,b 4 ,a ,b ,a ,b h i e 4γ + − (ω + λη )(ω + λη ) 2 ,a ,a ,b ,b e 2γ ′ + N˜√(2)hhab − (η η + λ λ ) 4 ,a ,b ,a ,b e 4γ ′ + N˜√(2)hhab − (ω + λη )(ω + λη ) 2 ,a ,a ,b ,b ˜√(2) ab + N hh γ,b′ ,a =0.

One would want to solve this equation, if possible, with boundary conditions chosen so that no non-vanishing energy flux contributions result from the terms involving N˜ ′ in Eq. (4.2). Equation (K.5) will be more recognizable and tractible to analyze if we first ‘lift’ it back to 3-dimensions and reexpress it as an equation for the first variation, N ′, of the 3+1-dimensional lapse function N = γ e− N˜, namely

γ γ γ (K.6) N ′ = e− N˜ ′ γ′e− N˜ = e− N˜ ′ γ′N. − −

At this point of course γ′ and N will be known quantities that can be ‘shifted’ into the ‘source terms’ for the single unknown N ′. The lifted equation, expressed in terms of the ADM spatial metric (3)g = 120 VINCENT MONCRIEF AND NISHANTH GUDAPATI g dxi dxj (c.f., Appendix C) takes the form ij ⊗ 2γ (3) ij (3) ij e− ∂ gg N ′ + N ′ gg (η η + λ λ ) − i ,j 4 ,i ,j ,i ,j p ep4γ + − (ω + λη )(ω + λη ) 2 ,i ,i ,j ,j γ γ (3) ij e− (K.7) = Ne− gg (η η + λ λ ) − 4 ,i ,j ,i ,j p 1 3γ ′ + e− (ω + λη )(ω + λη ) 2 ,i ,i ,j ,j (3) ij ∂i gg γ′N,j wherein, for the sake of uniform notation,p we have included terms that actually vanish by virtue of axisymmetry e.g. η,3 = η,ϕ, η,ϕ′ , etc. . By the same token we are only interested in{ axisymmetric solutions} for (3) ij (3) ab which of course ∂ gg N ′ ∂ gg N ′ . Equation (K.7) i ,j → a ,b is of course nothing butp the linearized versionp of the usual 3+1-dimensional lapse equation for maximal slicing reexpressed in terms of our variables and restricted to a Kerr-Newman background solution. We anticipate that well-known arguments (c.f., [12]) can be modified to establish the existence and uniqueness of smooth, axisymmetric solutions to this equation that vanish at infinity with homogeneous Dirichlet data specified on the horizon boundary (i.e., N ′ R+ = 0). In fact a standard uniqueness argument would suffice to guarantee| that any such (i.e., smooth, bounded with vanishing Dirichlet data) solution would automatically be axisymmetric and hence project naturally to the original quotient space whereon Eq. (K.5) was formulated. But would such a solution contribute unwanted flux terms to Eq. (4.2) and disrupt the argument for conservation of energy? The terms in Eq. (4.2) involving N˜ ′ can be expressed as the divergence of the vector density b ˜ ˜ ˜ ˜ ab Ξ := NN,a′ N ′N,a 2˜π′ (K.8) − γ 2 2γ ab = e NN˜ ′ + N˜ γ′ N ′e N 2˜π′ ,a ,a − ,a where

ab bc a 1 a (K.9)π ˜′ =g ˜ (˜r )′ + δ τ ′µ(2) c 2 c g˜ and N ′ is given by (K.6). AXISYMMETRIC PERTURBATIONS 121

From Eqs. (H.77–H.78) one sees that, in the asymptotic regions near R R and R , one has ց + ր∞ ac bd cd g˜ g˜ ef (K.10)π ˜′ = µ(2)g˜ ( (2) g˜)ab gãbg˜ ( (2) g˜)ef 2N˜ L D − L D where ∂ (K.11) (2) := N˜ 2g˜cd (4)Y 0 . D ,d ∂xc In terms of the ‘conformal data’ ν and hab (for which, as before,g ãb = 2ν e hab) this becomes (K.12) ac bd cd h h (2) e ef π˜′ = µ(2)h 2 ν,e hab +( (2) h)ab habh ( (2) h)ef . 2N˜ − D L D − L D Utilizing the asymptotic properties of (2) derived in Section 4.1 and D imposing the (homogeneous) Dirichlet boundary condition N ′ R+ = 0 upon the desired solution of Eq. (K.7) one can show that, if a| regular such solution exists, then one has (K.13) ΞR =0 |R+ i.e., pointwise vanishing of the energy flux integrand at the horizon boundary. Furthermore the corresponding flux integrand vanishes as R for any solution N ′ that grows sufficiently slowly. In particular anyր∞ solution that is bounded with bounded first derivatives would yield a (pointwise) vanishing energy flux integrand as R . Finally, by exploiting the regularity results for axi-symmetricր∞ fields and their perturbations derived in [68], it is straightforward to verify that potential flux contributions at the (artificial) boundaries provided by the axes of symmetry at θ = 0, π vanish (pointwise) as O(sin2 θ). Thus, modulo the aforementioned need for an existence proof for Eq. K.7, it follows that conservation of our energy functional holds as well in the 3+1-dimensional maximal slicing gauge.

Appendix L. The Weyl Tensor for Vacuum Axisymmetric Spacetimes In section 2 we analyzed the (axisymmetric) purely electromagnetic perturbations of a Kerr black hole spacetime by introducing a complete set of (electromagnetic) gauge and infinitesimal diffeomorphism- invariant canonical variables for the (linearized) Maxwell field and deriving a conserved, positive definite energy functional expressible in terms of these quantities. An advantage of the use of such variables is 122 VINCENT MONCRIEF AND NISHANTH GUDAPATI their insensitivity to the non-local features of any elliptic gauge condition that one might choose to employ. By contrast the variables we introduced later for the full, linearized Kerr-Newman problem were gauge dependent—a feature directly reflected in the dependence of their evolution equations on the elliptically determined (hence non-local) lin- a′ earized lapse and shift fields N˜ ′, N˜ . It is therefore natural to ask{ whether,} at least for the purely gravitational perturbations of a Kerr background, a corresponding set of fully gauge-invariant canonical variables might also be available for the (linearized) metric component of the problem. Since the (complex) field satisfying Teukolsky’s equation is gauge-invariant one might well expect that it provides (upon specialization to the axisymmetric setting considered here) a natural answer to this question. If one could affect a canonical transformation to a new set of variables that includes Teukolsky’s field and its conjugate momentum as a gauge invariant subset than one would expect that our energy functional, which is itself gauge-invariant, could be reexpressed purely in terms of this (invariant) subset. Since Teukolsky’s field is defined in terms of the linearization of the Weyl tensor about a Kerr background we present here the actual Weyl tensor for vacuum, axisymmetric metrics expressed in terms of our symmetry reduced canonical variables from Appendix C. Since, in the case of a vacuum background, the linearized gravitational and electromagnetic perturbations decouple from one another and since we have already dealt with the Maxwell component in Section 2, we focus exclusively here on the pure metric component and specialize the formulas of Appendix C accordingly. As is well-known [5, 61] the Weyl tensor for a vacuum spacetime can be expressed in terms of ADM Cauchy data (3)g = g dxi dxj , (3)π = { ij ⊗ πij ∂ ∂ on a 3-manifold M as a pair of (traceless, symmetric) ∂xi ⊗ ∂xj } tensor densities, an ‘electric’ field (3) = ij ∂ ∂ given by E E ∂xi ⊗ ∂xj ij (3) ij (3) 1 i mj 1 ij m (L.1) := µ(3)g R ( g) π mπ π π m E − µ(3) − 2 g and a corresponding ‘magnetic’ field (3) = ij ∂ ∂ defined by B B ∂xi ⊗ ∂xj mℓj ij ǫ m 1 i k (L.2) := πi ℓ δm(π k) ℓ . B µ(3) | − 2 | g Note that (3) , though manifestly symmetric, is traceless only by virtue of the (vacuum)E Hamiltonian constraint (L.3) m = 0 (in vacuum) E m −H → AXISYMMETRIC PERTURBATIONS 123 whereas (3) , though identically traceless, is symmetric only by virture of the (vacuum)B momentum constraint

ij 1 k (L.4) ǫijk = H 0 (in vacuum). B −2 µ(3)g →

One can now evaluate (3) and (3) in terms of the canonical pairs, ab a E B (˜gab, π˜ ), (βa, e˜ ), (γ, p˜) , for the (axial-) symmetry reduced system deﬁned in Eqs. (C.13–C.17). A ﬁnal transformation to wave map variables would then result from the substitution

(L.5) e˜a ǫabω , → ,b (L.6) ǫabβ r˜ a,b →

a On the other hand the one-form ﬁeld βadx , which appears in the spatial metric (3)g, is non-local in terms of the wave map ﬁeldr ˜ and, moreover, incorporates a (longitudinal) component that varies as

(L.7) βa βa + λ a → | under a coordinate transformation of the form

(L.8) x3 = ϕ ϕ + λ → whereasr ˜ is invariant with respect to such a transformation. For this reason we prefer to express the results in terms of the intermediate canonical pairs listed above. Only a certain set of ‘mixed’ components of (3) and (3) are invariant E B a with respect to the aforementioned ‘gauge’ transformation of βadx , namely,

(L.9) ab, , a, ab, , a . E E33 E3 B B33 B3 Using the spatial metric (3)g to raise or lower indices one can easily express all of the contravariant or covariant components of these ﬁelds in terms of the speciﬁed ‘mixed’ components but only at the expense of foregoing the aforementioned invariance. Without further ado we present here the relevant, ‘mixed’ components of the Weyl tensor expressed in terms of the symmetry reduced 124 VINCENT MONCRIEF AND NISHANTH GUDAPATI canonical variables:

ab 3γ (2) ab (2) b(2) a ab (2) (2) c = e µ(2) R˜ ˜ ˜ γ +˜g ˜ ˜ γ E g˜ − ∇ ∇ ∇c ∇ 3 (2) ˜ aγ (2) ˜ bγ +˜gab (2) ˜ γ (2) ˜ cγ − ∇ ∇ ∇c ∇ 1 e4γ g˜acg˜df g˜be (β β )(β β ) (L.10) − 2 d,e − e,d f,c − c,f γ e 1 2γ ab 1 cd e π˜ p˜ +2˜gcdπ˜ − µ(2) −2 2 g˜ 2γ ad bc 1 2γ a b + e g˜ π˜ π˜ + e− e˜ e˜ , cd 4

γ 2γ e 1 2γ a b e 1 ab 33 = e− g˜abe˜ e˜ + p˜ p˜ +˜gabπ˜ E −µ(2) 4 4 2 g˜ 3γ ab ab (L.11) e ∂ µ(2) g˜ γ + µ(2) g˜ γ γ − a g˜ ,b g˜ ,a ,b 1 4γ ac bd + e µ(2) g˜ g˜ (β β )(β β ) , 4 g˜ c,b − b,c d,a − a,d

a 5γ 1 (2) bd ac = e µ(2) ˜ g˜ g˜ (β β ) E3 g˜ 2 ∇b d,c − c,d 5 (L.12) (2) ˜ bγ g˜ac (β β ) − 2 ∇ b,c − c,b γ e 1 c ab 1 a e˜ g˜bcπ˜ + p˜ e˜ , − µ(2) 2 8 g˜

ab bc3 1 5γ g˜ce de af mn = ǫ e π˜ g˜ ǫfd (ǫ βm,n) B 2 µ(2) g˜ d γ d e˜ γ af 1 e e˜ a e g˜ γ,f g˜dc + γ,dδc − µ(2)g˜ 2 µ(2)g˜ (L.13) 5γ 1 e 1 mn af rs p˜ +˜gmnπ˜ g˜ ǫfc (ǫ βr,s) − 2 µ(2) 2 g˜ a 1 γ (2) e˜ e ˜ c , − 2 ∇ µ(2) g˜ AXISYMMETRIC PERTURBATIONS 125

γ γ ac3 2γ 1 e− 1 e eã 33 = ǫ e eã γ,c B 2 µ(2) − 2 µ(2) ( g˜ ,c g˜ 5γ e 1 bd (L.14) + (βa,c βc,a) p˜ +˜gbdπ˜ 2µ(2) − 2 g˜ 5γ 1 e f πã (βf,c βc,f ) , − 2 µ(2) − g˜ 3γ a 3ca e b b mn 3 = ǫ γ,b π˜c δcg˜mnπ˜ B −µ(2) − g˜ (L.15) 3γ 3γ 1 e b 1 e p˜ e˜ (βb,c βc,b)+ . − 4 µ(2) − 4 µ(2) g˜ g˜ ,c) In these formulas indices a, b, c, . . . are raised and lowered with the (2) a b (2) Riemannian 2-metric g˜ =g ãbdx dx , ˜ a designates covariant ⊗ ∇ (2) ãb differentiations with respect to this metric whereas µ(2)g˜ and R are its ‘volume’ element and Ricci tensor. Whereas the explicit symmetry of (3) implies, for example, that E (L.16) a = a and ǫ ab =0 E3 E 3 abE the corresponding equations for (3) only hold ‘weakly’ (i.e., modulo the momentum constraints). MoreB precisely one finds that 3γ a a 1 e ab3 ˜ (L.17) 3 3 = ǫ b, B − B −2 µ(2)g˜ H γ ab 1 e c (L.18) ǫab = e˜ ,c B 2 µ(2)g˜ and ac3 γ 3a a3 1 ǫ e ˜ d (L.19) = c βce˜ ,d . B − B −2 µ(2)g˜ H − where 3a 2γ a ba (L.20) = e− β B B3 − bB To see that all of the components of (3) are indeed determined by the mixed set we have presented above oneE computes that a3 3a 2γ a ab (L.21) = = e− β E E E 3 − bE and that 33 4γ 2γ a ab (L.22) = e− 2e− β + β β . E E33 − aE3 a bE 126 VINCENT MONCRIEF AND NISHANTH GUDAPATI

As we have already mentioned, the trace of (3) only vanishes weakly since, in fact E (L.23) g ij = = eγ ˜. ijE −H − H Similar formulas hold for the (contravariant) components of (3) , allowing for the fact that it, unlike (3) , is not explicitly symmetric.B A straightforward further calculationE gives ij 4γ 2 a b ij = e− ( 33) +2˜gab (L.24) E E E E 3E 3 +˜g g˜ ab cd ac bdE E which, being independent of βa, is invariant under the ‘gauge’ transformation βa βa + λ a. A similar formula can of course be derived for ij → | (3) ij, again allowing for the lack of explicit symmetry of . Taken BtogetherB these quantities constitute the ‘Bel Robinson energy deB nsity’. AXISYMMETRIC PERTURBATIONS 127

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Department of Physics and Department of Mathematics, Yale Uni- versity, P.O. Box 208120, New Haven, CT 06520, USA. Email address: [email protected] Center of Mathematical Sciences and Applications, Harvard Uni- versity, 20 Garden Street, Cambridge, MA-02138, USA Email address: [email protected]