### Math 712. Topics in Mathematical Physics: General Relativity

- Spring 18 -
Hans Lindblad

We will show that the initial value problem for Einstein's
equations has a local unique solution (in harmonic coordinates).
We will also show that Einstein's equations have global
solutions (in harmonic coordinates) if initial conditions are close to flat space,
as in my recent work with Rodnianski. Einstein's equation in harmonic coordinates
become a system of nonlinear wave equations. We will therefore develop the tools
needed to show existence and estimates for nonlinear wave equations.
Long time existence for nonlinear wave equations will therefore be
the starting point of theses seminars. A good place to read about this is
Chapter 6 in Hormander's book 'Lectures on Nonlinear Hyperbolic Differential Euqations'.
A more geometric text is
Alinhac's book
'Geometric Analysis of Hyperbolic Equations an introduction'.
A place to learn some of the techniques for simple nonlinear wave equations is my
paper Lindblad.
My paper(s) about global existence for Einstein's equations can be found at
arXiv
See in particular the original papers with Rodnianski and a recent
simplifications
with Taylor and Tohaneanu. The course is assuming that you know local existence for nonlinear wave
equations as in Evans PDE book Chapter 7 that is taught in the second part of the
graduate PDE sequence Math 632 PDE.

It is also useful to first read a physics undergraduate/graduate text book like
Carroll "Spacetime and Geometry".

The lectures are MW 3-4 in Krieger 406 or the chair's office.