Math 257. Topics in Differential Geometry: General Relativity
- Winter 06 - 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 equation's decsribe how space curves under the influence of gravity. We will motivate the equations by studying the Newtonian approximation, special relativity and special solutions like cosmological spacetimes and the Schwarzschild solution.

In the second part of the course will be to 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 course continues in the Spring we will go on to study global existence for Einstein's equations in harmonic coordinates with initial conditions close to flat space, as in my recent work with Rodnianski.

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. An alternative is Hawking and Ellis "The large scale structure of spacpe-time". For the first part we will also pick material from a couple of short nice books Dirac "General Theory of Relativity" and Frankel "Gravitational Curvature" as well as physics texts like Carroll "Spacetime and Geometry" and the not so short classic Misner, Thorne and Wheeler "Gravitation"

There will be no exams. The lectures are MWF 10.00-10.50 in APM 5829. The lecture notes can be downloaded below from the schedule.

 wk  date  Monday  Wednesday  Friday
  1  1/9  1.2-4  2.1-2  2.2-2.3
  2  1/16  Holiday  2.4, 3.1  3.1-3.2
  3  1/23  3.4a,3.2-3.3  3.3  4.1-4.2
  4  1/30  4.2  4.3  4.4-4.4a
  5  2/6  5.1-5.2,5.3b  6.1,6.4  6.3
  6  2/13  7.1,12.3  8.1  8.2
  7  2/20  Holiday  8.3  9.2
  8  2/27  9.3  9.4  9.5
  9  3/6  10.1  10.2  10.2
 10  3/13  11.1-2  12.1  12.2