Vorträge in der Woche 14.10.2024 bis 20.10.2024

Donnerstag, 17.10.2024: Rigidity Theorems in General Relativity: Birkhoff's Theorem via Existence and Uniqueness of Quasilinear Hyperbolic PDEs

Vindula Kumaranayake (Universität Münster)

In this talk, I begin by assuming the existence of a spherically symmetric solution to the Einstein vacuum equations and proceed to derive a system of hyperbolic partial differential equations (PDEs) that the metric components satisfy in null coordinates. Then I establish the local existence and uniqueness of solution for this system of hyperbolic PDEs using the Banach Fixed Point Theorem, demonstrating that the system is well-posed in an appropriate function space. Furthermore, by prescribing suitable data in the Kruskal spacetime, we obtain a proof of Birkhoff's theorem, which states that any spherically symmetric vacuum solution must be locally isometric to the Schwarzschild solution. Time permitting, I will also discuss the Lichnerowicz theorem and Israel’s theorem regarding the uniqueness of static, asymptotically flat spacetime solutions.