Asymptotics of Linear Waves and Resonances with Applications to Black Holes

Abstract

Abstract. We apply the results of [Dy13] to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a| < M and M Λa 1. We prove that if the initial data is localized at frequencies ∼ λ 1, then the energy norm of the solution is bounded by O(λ 1/2 e −(νmin−ε)t/2 ) + O(λ −∞ ), for t ≤ C log λ, where ν min is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(λe −(νmin−ε)t ) + O(λ −∞ ) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.The subject of this paper are decay properties of solutions to the wave equation for the rotating Kerr (cosmological constant Λ = 0) and Kerr-de Sitter (Λ > 0) black holes, as well as for their stationary perturbations. In the recent decade, there has been a lot of progress in understanding the upper bounds on these solutions, producing a polynomial decay rate O(t −3 ) for Kerr and an exponential decay rate O(e −νt ) for Kerrde Sitter (the latter is modulo constant functions). The weaker decay for Λ = 0 is explained by the presence of an asymptotically Euclidean infinite end; however, this polynomial decay comes from low frequency contributions.We instead concentrate on the decay of solutions with initial data localized at high frequencies ∼ λ 1; it is related to the geometry of the trapped set K, consisting of lightlike geodesics that never escape to the spatial infinity or through the event horizons. The trapped set for both Kerr and Kerr-de Sitter metrics is r-normally hyperbolic, and this dynamical property is stable under stationary perturbations of the metric -see §3.6. The key quantities associated to such trapping are the minimal and maximal transversal expansion rates 0 < ν min ≤ ν max , see (2.9), (2.10). Using our recent work [Dy13], we show the exponential decay rate O(λ 1/2 e −(ν min −ε)t/2 ) + O(λ −∞ ), valid for t = O(log λ) (Theorem 1). This bound is new for the Kerr case, complementing Price's law.Our methods give a more precise microlocal description of long time propagation of high frequency solutions. In Theorem 2, we split a solution u(t) into two approximate solutions to the wave equation, u Π (t) and u R (t), with the rate of decay of u Π (t) between e −(νmax+ε)t/2 and e −(ν min −ε)t/2 and u R (t) bounded from above by λe −(ν min −ε)t , all 1 arXiv:1305.1723v1 [gr-qc]

Mathematical study of scattering resonances

“…From the mathematical point of view that makes infinity "larger" and provides exponential decay of waves which makes a rigorous formulation of expansions easier. When one adds frequency localization weaker expansions are still possible in the Kerr case-see [68,Theorem 2], [76, (13)]. References to the extensive mathematics literature in the case of = 0 can be found in [68].…”

“…The trapped set, K, consists of null geodesics that stay away from r = r + , r = r C for all times. We refer to Dyatlov [68,Prop.3.2] for the description of this trapped set and the constraints on the parameters a, M and . Here we remark that it has a particularly simple form when a = 0:…”

Quasilinear waves and trapping: Kerr-de Sitter space

“…There is more work on the linear equation on perturbations of de Sitter-Schwarzschild and Kerr-de Sitter spaces: a rather complete analysis of the asymptotic behavior of solutions of the linear wave equation was given in [46], upon which the linear analysis of [30], described here, is ultimately based. Previously in exact Kerr-de Sitter space and for small angular momentum, Dyatlov [20,19] has shown exponential decay to constants, even across the event horizon; see also the more recent work of Dyatlov [21]. Further, in de Sitter-Schwarzschild space (non-rotating black holes) Bachelot [3] set up the functional analytic scattering theory in the early 1990s, while later Sá Barreto and Zworski [4] and Bony and Häfner [7] studied resonances and decay away from the event horizon, Dafermos and Rodnianski in [12] showed polynomial decay to constants in this setting, and Melrose, Sá Barreto and Vasy [39] improved this result to exponential decay to constants.…”

On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 3. Mass Inflation and Extendibility of the Solutions

“…Strictly speaking, this problem does not address the strong cosmic censorship conjecture directly, because the data considered on the event horizon does not arise from the gravitational collapse of generic Cauchy initial data. However, since the Price law for Λ > 0 is widely expected to yield exponential decay of the scalar field along the event horizon (see for instance the linear analysis of [14,15]), we believe that our conclusions will provide valuable insights for this cosmological case.…”