We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein-Maxwell equations: If one perturbs the initial data of a slowly rotating Kerr-Newman-de Sitter (KNdS) black hole, then in a neighborhood of the exterior region of the black hole, the metric and the electromagnetic field decay exponentially fast to their values for a possibly different member of the KNdS family. This is a continuation of recent work of the author with Vasy on the stability of the Kerr-de Sitter family for the Einstein vacuum equations. Our non-linear iteration scheme automatically finds the final black hole parameters as well as the gauge in which the global solution exists; we work in a generalized wave coordinate Lorenz gauge, with gauge source functions lying in a suitable finite-dimensional space. We include a self-contained proof of the linear mode stability of Reissner-Nordstrom-de Sitter black holes, building on work by Kodama-Ishibashi. In the course of our non-linear stability argument, we also obtain the first proof of the linear (mode) stability of slowly rotating KNdS black holes using robust perturbative techniques.
There is a large amount of literature on asymptotically flat black hole spacetimes; we only give a brief account here. The seminal papers by Wald @cite_60 and Kay--Wald @cite_78 proved the boundedness of linear scalar waves on the Schwarzschild spacetime; Dafermos and Rodnianski @cite_94 gave a more robust proof, exploiting the red-shift effect, and established polynomial decay. Polynomial decay for linear scalar waves on Kerr was proved by Andersson--Blue @cite_14 and in the aforementioned @cite_13 ; see also @cite_82 . The precise decay rates (Price's law @cite_16 ) were obtained by Tataru @cite_105 , see also @cite_72 , as well as @cite_107 in a spherically symmetric but non-linear context. Strichartz estimates were proved by Tataru and Tohaneanu @cite_28 , also in collaboration with Marzuola and Metcalfe @cite_80 . For non-linear results, we refer to Luk's work on Kerr @cite_48 , Ionescu--Klainerman @cite_22 and Stogin @cite_5 for a wave map equation, and Dafermos--Holzegel--Rodnianski @cite_39 for a scattering construction of dynamical Kerr black holes. Many of these works rely on the very influential vector field method' developed by Klainerman @cite_17 ; see also @cite_20 for a discussion of recent developments.
@cite_14 @cite_107 @cite_78 @cite_60 @cite_28 @cite_48 @cite_22 @cite_20 @cite_17 @cite_39 @cite_5 @cite_72 @cite_80 @cite_16 @cite_13 @cite_105 @cite_94 @cite_82
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These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spacetimes themselves as solutions to the Einstein equations, one of the central open problems of general relativity. After an introductory discussion of the Schwarzschild geometry and the black hole concept, the classical theorem of Kay and Wald on the boundedness of scalar waves on the exterior region of Schwarzschild is reviewed. The original proof is presented, followed by a new more robust proof of a stronger boundedness statement. The problem of decay of scalar waves on Schwarzschild is then addressed, and a theorem proving quantitative decay is stated and its proof sketched. This decay statement is carefully contrasted with the type of statements derived heuristically in the physics literature for the asymptotic tails of individual spherical harmonics. Following this, our recent proof of the boundedness of solutions to the wave equation on axisymmetric stationary backgrounds (including slowly-rotating Kerr and Kerr-Newman) is reviewed and a new decay result for slowly-rotating Kerr spacetimes is stated and proved. This last result was announced at the summer school and appears in print here for the first time. A discussion of the analogue of these problems for spacetimes with a positive cosmological constant follows. Finally, a general framework is given for capturing the red-shift effect for non-extremal black holes. This unifies and extends some of the analysis of the previous sections. The notes end with a collection of open problems. This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| M, general or (II2) |a| < M, axisymmetric. In either case, we prove a version of 'integrated local energy decay', specifically, that the 4-integral of an energy-type density (degenerating in a neighborhood of the Schwarzschild photon sphere and at infinity), integrated over the domain of dependence of a spacelike hypersurface connecting the future event horizon with spacelike infinity or a sphere on null infinity, is bounded by a natural (non-degenerate) energy flux of through . (The case (II1) has in fact been treated previously in our Clay Lecture notes: Lectures on black holes and linear waves, arXiv:0811.0354.) In our forthcoming Part III, the restriction to axisymmetry for the general |a| < M case is removed. The complete proof is surveyed in our companion paper The black hole stability problem for linear scalar perturbations, which includes the essential details of our forthcoming Part III. Together with previous work (see our: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, in XVIth International Congress on Mathematical Physics, Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arxiv:0910.4957), this result leads, under suitable assumptions on initial data of , to polynomial decay bounds for the energy flux of through the foliation of the black hole exterior defined by the time translates of a spacelike hypersurface terminating on null infinity, as well as to pointwise decay estimates, of a definitive form useful for nonlinear applications. This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal |a| < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. (ed.), World Scientific, Singapore, 2011, pp. 132-189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincare], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notations so that it can be read independently of previous work. We review our recent work on linear stability for scalar perturba- tions of Kerr spacetimes, that is to say, boundedness and decay properties for solutions of the scalar wave equation 2g = 0 on Kerr exterior backgrounds (M,ga,M). We begin with the very slowly rotating caseSaS ≪M, where first boundedness and then decay has been shown in rapid developments over the last two years, following earlier progress in the Schwarzschild case a= 0. We then turn to the general subextremal range SaS <M, where we give here for the first time the essential elements of a proof of definitive decay bounds for solutions . These developments give hope that the problem of the non-linear stability of the Kerr family of black holes might soon be addressed. This paper accompanies a talk by one of the authors (I.R.) at the 12th Marcel Grossmann Meeting, Paris, June 2009.