University College London, 20/03/2018
“Time-periodic solutions of Einstein equations with a negative cosmological constant.”
I will show how to construct infinite dimensional families of time-periodic solutions of wave equations, including the Einstein equations with a negative cosmological constant and smooth conformal boundary at infinity. Based on arxiv:1711.1126.
“On foliations related to the center of mass in General Relativity.”
In many situations in classical physics, understanding the motion of the center of mass of a system is key to understanding the general “trend” of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in General Relativity.
However, while the definition of the center of mass via the mass density is straightforward in classical physics, there is a priori no definitive corresponding notion in General Relativity. We will pursue a geometric approach to defining the center of mass, using foliations by hypersurfaces with specific geometric properties. I will first illustrate this approach in the (easier) classical Newtonian setting and then review previous work in the relativistic situation, most prominently a fundamental result by Huisken and Yau from 1996. After introducing the foliation approach, I will discuss explicit counter-examples (joint work with Nerz) and discuss the analytic, geometric, and physical issues they illustrate. I will then present a new approach (joint work with Cortier and Sakovich) that remedies these issues.
“The Teukolsky equation on Kerr and te black hole stability problem.”
I will review first the formulation of the black hole stability problem in general relativity. As part of our previous proof of linear stability of Schwarzschild, we showed both boundedness and polynomial decay estimates for solutions of the spin ±2 Teukolsky equation by exploiting a physical space transformation to solutions of the Regge-Wheeler equation. We show how this procedure can be generalised to yield similar results for the Teukolsky equation on Kerr. This equation completely describes the gauge invariant part of gravitational perturbations and provides the key to a full understanding of the stability problem. This is joint work with G. Holzegel and I. Rodnianski.