Frequency space analysis in General Relativity

Abstract

This thesis is divided into two completely independent parts, each addressing a distinct problem for the vacuum Einstein equations in General Relativity. The common theme is the use of frequency analysis as a tool to exploit fine structures within the partial differential equations. However, the two parts are structured so as to be read independently. The first part concerns the classical stability problem for black holes in the Kerr family which, conjecturally, exhausts the class of stationary asymmetric asymptotically flat vacuum black hole solutions. We study the Teukolsky master equation, a covariant wave-type PDE which, for different values of integer and half-integer s, describes the dynamics of perturbations of Kerr black holes in the linearized setting. We begin by showing that the Teukolsky equation is modally stable, i.e. that it admits no exponentially growing or bounded but non-decaying separable solutions. This result is novel for so-called extremal Kerr black holes, which rotate at the maximum allowed speed and for which stability is expected to be more delicate. For the subextremal ones, which rotate below this threshold, the result goes back to Whiting in 1989, but we provide here a new proof relying on previously unknown symmetries of the spectrum of the separated ODEs. Then, we focus on the most relevant cases s=±1,±2, where the Teukolsky equation governs the dynamics of, respectively, electromagnetic and curvature components that are gauge-independent at the level of the linearized, respectively, Maxwell and Einstein equations around the Kerr family. By a precise frequency space analysis, we obtain estimates for separable solutions which are uniform in the separation parameters. We also show that our methods are sharp, in the sense that they cannot be extended to |s|>2: we prove that the nature of the conserved energy associated to the Teukolsky equation changes drastically, as it can exhibit new forms of non-coercivity for |s|>2. This result dispels a prevailing myth in the literature. As a corollary of our frequency space estimates, we then show that solutions of the Teukolsky equation for s=±1,±2 arising from regular initial data on any subextremal black hole in the Kerr family remain bounded and decay in time. We also establish some first basic results for the scattering properties of this equation. Our results are a key step in establishing forward-in-time and scattering stability of Kerr under electromagnetic and gravitational perturbations. This first part of the thesis contains joint work with Marc Casals (Centro Brasileiro de Pesquisas Físicas and University College Dublin) and Yakov Shlapentokh-Rothman (Princeton University). The second part of the thesis concerns a homogenization problem for the vacuum Einstein equations. We fix a manifold and consider a weakly converging sequence of Lorentzian metrics on the manifold which solve the vacuum Einstein equations. Following the physics literature, we study a setting where the weak convergence allows oscillations, but no concentrations. We show that, under symmetry and gauge assumptions, the effective model for the limit metric is an Einstein--massless Vlasov system. We thus prove a conjecture of Burnett from 1989, and improve recent work of Huneau--Luk which required stricter assumptions. This second part of the thesis is joint work with André Guerra (University of Oxford).