401-3830-21L  Wave Equations on Black Hole Spacetimes

SemesterFrühjahrssemester 2021
DozierendeC. Kehle
Periodizitäteinmalige Veranstaltung
KommentarNumber of participants limited to 12.

KurzbeschreibungIntroduction to Lorentzian geometry, to the notion of a black hole, and to the study of linear wave equations on such spacetimes.
LernzielWe will learn about the basics of Lorentzian geometry, the geometric framework which incorporates space and time as one geometric entity---spacetime. Then, we will briefly introduce the Einstein equations of General Relativity and study the Schwarzschild and Reissner--Nordström black holes solutions. We will further discuss tools to study linear wave equations on black holes and other spacetimes.
InhaltBlack holes are among the central theoretical predictions of general relativity which is governed by the celebrated Einstein's equations. The notion of a black hole has a clean mathematical definition, and the concept is already exhibited by the simplest non-trivial solution of the Einstein vacuum equation: the Schwarzschild solution. These “black hole spacetimes” give rise to many natural mathematical problems in the analysis of (hyperbolic) PDE which in turn describe physical phenomena related to black holes. More specifically we will cover the following topics: Basic Lorentzian geometry, the Schwarzschild and Reissner-Nordström black hole, the wave equation on general Lorentzian manifolds, the wave equation on black hole backgrounds. We will also adapt the content to the prior knowledge of the students.
LiteraturMain reference: Lecture Notes of Mihalis Dafermos: Link

Further references (going beyond the scope of the seminar):
- Dafermos, Mihalis, and Igor Rodnianski. "Lectures on black holes and linear waves." Clay Math. Proc 17 (2013): 97-205. (see also arXiv:0811.0354)
- Aretakis, Stefanos. "General Relativity". Link
- Christodoulou, Demetrios. Mathematical problems of general relativity I. Vol. 1. European Mathematical Society, 2008.
Voraussetzungen / BesonderesIdeally, participants have some familiarity with the basics of differential manifolds (definition of smooth manifolds, tangent space, vector fields, as well as the formal apparatus of Riemannian geometry: connections, curvature, geodesics) and basic functional analysis (Sobolev spaces, etc.).