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A Positive-Definite Energy Functional for the Axisymmetric Perturbations

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A Positive-Definite Energy Functional for the Axisymmetric Perturbations A POSITIVE-DEFINITE ENERGY FUNCTIONAL FOR THE AXISYMMETRIC PERTURBATIONS OF KERR-NEWMAN BLACK HOLES VINCENT MONCRIEF AND NISHANTH GUDAPATI Abstract. We consider the axisymmetric, linear perturbations of Kerr-Newman black holes, allowing for arbitrarily large (but subex- tremal) angular momentum and electric charge. By exploiting the famous Carter-Robinson identities, developed previously for the proofs of (stationary) black hole uniqueness results, we construct a positive-definite energy functional for these perturbations and establish its conservation for a class of (coupled, gravitational and electromagnetic) solutions to the linearized field equations. Our analysis utilizes the familiar (Hamiltonian) reduction of the field equations (for axisymmetric geometries) to a system of wave map fields coupled to a 2+1-dimensional Lorentzian metric on the rel- evant quotient 3-manifold. The propagating ‘dynamical degrees of freedom’ of this system are entirely captured by the wave map fields, which take their values in a four dimensional, negatively curved (complex hyperbolic) Riemannian target space whereas the base-space Lorentzian metric is entirely determined, in our setup, by elliptic constraints and gauge conditions. The associated linearized equations are analyzed with insight derived from the so-called ‘linearization stability’ program for such (generally covariant) systems. In particular this program provides a natural connection between the (conserved, positive-definite) en- ergy defined for first order perturbations and the correction to the ADM mass induced therefrom at second order. A well-known technique allows one to generate, for sufficiently smooth perturba- tions, a sequence of higher order (conserved, positive-definite) en- arXiv:2105.12632v1 [gr-qc] 26 May 2021 ergies that, in turn, bound certain higher order (weighted) Sobolev norms of the linearized solutions. We anticipate that our results may prove useful in analyzing the dynamical stability of (arbi- trarily rapidly rotating) Kerr-Newman black holes with respect to axisymmetric perturbations. Establishing such stability at the lin- earized level is expected to be an essential first step in dealing, ultimately, with the nonlinear problem. Date: May 27, 2021. 1 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 pro- gram that began with the pioneering work of Regge and Wheeler [65], Vishveshwara [72] and Zerilli [75] for the case of Schwarzschild pertur- bations and with the discovery, by Teukolsky [70, 71], of a separable wave equation for Kerr perturbations. Subsequently the coupled gravi- tational and electromagnetic perturbations of (electrically charged but non-rotating) Reissner-Nordstr¨om black holes were analyzed by Zerilli through working in a special gauge [76] and by one of us who devel- oped 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-Nordstr¨om 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 lin- ear stability by Dafermos, Holzegel and Rodnianski [23] and, indepen- dently, 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 rotat- ing’ 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 anal- ysis 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 axisym- metric, 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 deriva- tion of a conserved, gauge-invariant, positive definite energy functional for the coupled, dynamical, gravitational and electromagnetic perturba- tions of Reissner-Nordstr¨om black holes. Using totally different (‘Hertz potential’) methods Wald and Prabhu have recently announced that the conserved, ‘canonical’ energy formula for purely gravitational per- turbations 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 pertur- bations [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 unper- turbed (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 combi- nations 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 intri- cate, 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 sta- bility 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 ini- tial 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 diver- gence term, positive definite. Upon discarding the boundary
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