We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equation: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the linearization of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equation have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications.
In the algebraically more complicated but analytically less degenerate context of cosmological black holes, we recall that S 'a Barreto--Zworski @cite_113 studied the distribution of resonances of SdS black holes; exponential decay of linear scalar waves to constants was proved by Bony--H "afner @cite_36 and Melrose--S a Barreto--Vasy @cite_112 on SdS and by Dyatlov @cite_70 @cite_31 on KdS spacetimes, and substantially refined by Dyatlov @cite_73 to a full resonance expansion. (See @cite_56 for a physical space approach giving superpolynomial energy decay.) Tensor-valued and nonlinear equations on KdS spacetimes were studied in a series of works by Hintz--Vasy @cite_63 @cite_24 @cite_102 @cite_116 @cite_28 . For a physical space approach to resonances, see Warnick @cite_110 , and for the Maxwell equation on SdS spacetimes, see Keller @cite_66 .
@cite_36 @cite_70 @cite_28 @cite_112 @cite_102 @cite_113 @cite_56 @cite_24 @cite_116 @cite_63 @cite_110 @cite_31 @cite_73 @cite_66
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We describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an ( ) derivative loss in the angular directions. We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole rotates slowly enough, we show that these poles form a discrete subset of ( C ) . As an application we prove that the local energy of linear waves in that background decays exponentially once orthogonality to the zero resonance is imposed. Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space. We study asymptotics for solutions of Maxwell’s equations, in fact of the Hodge-de Rham equation (d+ δ)u = 0 without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which in particular include Schwarzschild-de Sitter spaces of all spacetime dimensions n ≥ 4. We prove that solutions decay exponentially to 0 or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and in particular prove analogous results on Schwarzschildde Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell’s equations, the Hodge-de Rham equation and the wave equation on differential forms on Kerr-de Sitter spacetimes with small angular momentum. We consider solutions to the linear wave equation @math on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may contain the sphere of bifurcation of the black white hole horizons and the cosmological horizons.) We prove that in the region bounded by a set of black white hole horizons and cosmological horizons, solutions @math converge pointwise to a constant faster than any given polynomial rate, where the decay is measured with respect to natural future-directed advanced and retarded time coordinates. We also give such uniform decay bounds for the energy associated to the Killing field as well as for the energy measured by local observers crossing the event horizon. The results in particular include decay rates along the horizons themselves. Finally, we discuss the relation of these results to previous heuristic analysis of Price and We consider quasilinear wave equations on manifolds for which infinity has a structure generalizing that of Kerr-de Sitter space; in particular the trapped geodesics form a normally hyperbolic invariant manifold. We prove the global existence and decay, to constants for the actual wave equation, of solutions. The key new ingredient compared to earlier work by the authors in the semilinear case [33] and by the first author in the non-trapping quasilinear case [30] is the use of the Nash-Moser iteration in our framework. We establish a Bohr–Sommerfeld type condition for quasi-normal modes of a slowly rotating Kerr–de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a = 0 (Schwarzschild–de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves. In this work, we consider solutions of the Maxwell equations on the Schwarzschild-de Sitter family of black hole spacetimes. We prove that, in the static region bounded by black hole and cosmological horizons, solutions of the Maxwell equations decay to stationary Coulomb solutions at a super-polynomial rate, with decay measured according to ingoing and outgoing null coordinates. Our method employs a differential transformation of Maxwell tensor components to obtain higher-order quantities satisfying a Fackerell-Ipser equation, in the style of Chandrasekhar and the more recent work of Pasqualotto. The analysis of the Fackerell-Ipser equation is accomplished by means of the vector field method, with decay estimates for the higher-order quantities leading to decay estimates for components of the Maxwell tensor.