Topics in General Relativity

Abstract

The course gives an introduction to some mathematical problems of general relativity. After a broad overview of the mathematical formulation of the Einstein equations and their role as a nonlinear hyperbolic system, the course will be shaped in part by the interests of the participants. The emphasis throughout will be on the rigorous analytical and geometric framework underlying modern research.

Objective

The main objective of the course is to familiarise students with the interplay between Lorentzian geometry and hyperbolic partial differential equations in the study of general relativity. Students will learn how geometric concepts and analytic techniques combine to yield precise formulations of fundamental questions concerning the Einstein equations. By the end of the course, successful students will be able to formulate mathematically rigorous problems in general relativity and understand the structure of modern research results in the field. As part of the course requirements, each student will present a portion of a proof from a research paper, thereby gaining hands-on experience with the technical tools, methods, and level of precision characteristic of modern work in mathematical general relativity.

Content

The course begins with an overview of the Cauchy problem for quasilinear wave equations, serving as a bridge between general PDE theory and the Einstein equations. We will then discuss the foundational results of Choquet-Bruhat on the local well-posedness of the Einstein equations and the geometric formulation of the initial value problem in general relativity. This introductory part of the course is intended to set the stage and provide a common language and framework. Consequently, we will not present full proofs of the foundational results, but rather focus on their statements, ideas, and significance. Subsequent parts of the course will delve into selected research-level topics based on the skills and interest of students, with greater emphasis on detailed arguments and techniques. Depending on these interests, we will study a selection of advanced topics in mathematical general relativity, which may include: (i) the formation of singularities in solutions to the Einstein equations; (ii) questions of extendability and uniqueness of spacetime developments; (iii) stability and instability properties of explicit solutions, such as Minkowski or black hole spacetimes.

Resources