This course is aimed at students with a basic knowledge of:
1) Differential geometry, as in, for example, the following lecture notes by G. Heckman and I. Marcut
Previous knowledge of Riemannian geometry is not required, but would be helpful, see e.g. the following course in by G. Heckman:
However, theoretical physics students with a working knowledge of differential geometry and previous exposure to GR are also welcome to try this course. For mathematicians no previous knowledge of general relativity or even physics is required.
2) Partial differential equations, at the level of e.g. the following notes by G. Sweers:
These notes in fact contain more than we need; in particular, practical calculus-like methods of solution will be irrelevant. In any case almost everything from PDE’s we need will be repeated and explained, though usually without proof.
Aim of the course
Einstein’s theory of General Relativity (which he completed in 1915) describes gravity and the cosmos through a system of ten coupled nonlinear partial differential equations (called the Einstein equations), which may be split into four elliptic constraint equations and six hyperbolic evolution equations.
This theory describes the expansion of the universe, black holes, and the recently discovered gravitational waves (“ripples in space-time"). GR is (according to many) not only the most beautiful physical theory we have, it also has great mathematical beauty and interest, lying at the interface between (pseudo-) Riemannian geometry and PDE theory. The aim of the course is to introduce this theory to mathematicians, and also to deepen the mathematical understanding of this theory for theoretical physicists who have had some previous exposure to GR (which will not hurt mathematicians either but is not required). The emphasis will be on the PDE aspects of the theory (notably the initial value problem), and hopefully we will also reach the singularity theorems of Hawking and Penrose. We will not cover aspects of GR typically covered by physics courses (classical tests, elementary solutions, etc.).
This course is new and some improvising will be necessary, partly in order to cater well for students with diverse backgrounds. Please bear with us!
Homework and exam
Weekly homework contributing to bonus points. Written exam and/or take-home essay(s).
Lecture Notes / Literature
Alan Rendall, Partial Differential Equations in General Relativity (OUP, 2008)
Hans Ringström, The Cauchy Problem in General Relativity (EMS, 2009)
K. Landsman (RU)
Jeremie Joudioux (RU), exercise class