Math 237. Topics in PDE: General Relativity
- Winter 09 - Hans Lindblad

First we will introduce Einstein's equations and try to motivate them from a physical and geometric point of view. The formulation of the equations requires geometric concepts, like tensors and curvature, which we will review. Einstein's equations describe how space curves under the influence of gravity. We will motivate the equations by studying the Newtonian approximation, special relativity, matter models and special solutions like cosmological space times and the Schwarzschild solution.

Then we will study how solutions of Einstein's equations behave. We will show that the initial value problem for Einstein's equations has a local unique solution (in harmonic coordinates). We will also study how singularities (black holes and big bang) may develop, if the curvature or the concentration of mass is large. 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.

Jacob Sterbenz will continue with the course in the Spring quarter.

We will use material from Wald "General Relativity". The first part of the course corresponds to ch. 1-6 and the second to ch. 8-12 and my recent papers. An alternative is Hawking and Ellis "The large scale structure of space-time", or the last part of O'Neill "Semi-Reimannian Geometry with applications to Relativity". It is also useful to first read a physics undergraduate/graduate text book like Carroll "Spacetime and Geometry".

The lectures are MWF 1-2 in APM 7218. The lecture notes can be downloaded below from the schedule. There will be no exams but you are expected to do some problems. Problem session F 3-4 in APM 7241.

wk  date  Monday  Wednesday  Friday
  1  1/5 1.1-4 Introduction-Overview  2.1-2 Manifolds-Tangent Vectors  2.3-4 Cotangent Space-Tensors
  2  1/12  3.1 Covariant Derivative  3.2 Curvature  3.3 Geodesics
  3  1/19  Holiday  3.4 Curvature calculation  4.1-2 Pre and Special Relativity
  4  1/26 4.2 Energy-momentum tensor SR 4.3 Energy-momentum tensor GR 10.2 Harmonic Coordinates  4.4 Harmonic coordinates and Linearized Gravity.
  5  2/2 5.1-2 Homogeneous Isotropic Cosmology 5.3 Particle Horizon 6.1 Schwarzschild space-time 6.2 Interior 6.4 Kruskal Extension 6.3 Geodesics in Schwarzschild
  6  2/9  8.1  8.2  8.3
  7  2/16  Holiday  9.2  9.3
  8  2/23  9.3  9.4-5  10.1-2
  9  3/2  10.2  11.1-2  12.1
10  3/9 12.2   12.3   12.4  

Homework Ch2: 1,3a,5,6,7,8a, Ch3: 2,3,4a,5,6,7,8