Decay of Solutions of the Wave Equation in the Kerr Geometry

Abstract

As was recently pointed out to us by Thierry Daudé, there is an error on the last page of our paper [2]. Namely, the inequality (8.3) cannot be applied to the function (t) because it does not satisfy the correct boundary conditions. This invalidates the last two inequalities of the paper, and thus the proof of decay is incomplete. We here fill the gap using a different method. At the same time, we will clarify in which sense the sum over the angular momentum modes converges in [1, Theorem 1.1] and [2, Theorem 7.1], an issue which in these papers was not treated in sufficient detail. The arguments in these papers certainly yield weak convergence in L 2 loc ; here we will prove strong convergence. Our method here is to split the wave function into the high and low energy components. For the high energy component, we show that the L 2 -norm of the wave function can be bounded by the energy integral, even though the energy density need not be everywhere positive (Sect. 1). For the low energy component we refine our ODE techniques (Sect. 2). Combining these arguments with a Sobolev estimate and the Riemann-Lebesgue lemma completes the proof (Sect. 3).We begin by considering the integral representation of [2, Theorem 7.1], for fixed k and a finite number n 0 of angular momentum modes, The online version of the original article can be found under

A Rigorous Treatment of Energy Extraction from a Rotating Black Hole

“…In this paper we shall consider the Cauchy problem analytically, giving a mathematically rigorous treatment of superradiance for scalar waves. Our analysis is based on the integral representation for the wave propagator obtained in [8,9].…”

“…The corresponding eigenfunctions Θ ω,k n are referred to as spheroidal wave functions. In order to bring the radial equation into a convenient form, we introduce a new radial function φ(r) by 9) and define the Regge-Wheeler variable u ∈ R by 10) mapping the event horizon to u = −∞. The radial equation then takes the form of the Schrödinger equation…”

“…Using the notation in [9], we can explain his ideas and results as follows. We fix the separation constants k > 0, ω and λ n .…”

“…Thus there are fundamental solutionsφ andφ of (1.11) which behave asymptotically like plane waves, 15) (see [9] for details). The solution φ =φ has, near the event horizon u = −∞, the asymptotics φ(u) = e −iΩu .…”

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

“…We add that there have been many papers on decay of linear waves for Schwarzschild and Kerr black holes-see [2,9,18,19,22,23,28,29,46,47,49] and references given there. In that case the cosmological constant is 0 (unlike in the de Sitter case, where it is positive), and the methods of scattering theory are harder to apply because of an asymptotically Euclidean infinity.…”