Past lectures

Abstract

The course shall focus on the problem of the non-linear stability of the Minkowski spacetime of special relativity in the framework of general relativity. A number of basic results in differential geometry, partial differential equations and general relativity shall be covered along the way. At the end, the asymptotic behavior, at null infinity, of general spacetimes arising from asymptotically flat initial conditions shall be analyzed and the laws of gravitational radiation shall be deduced. A more detailed outline of the course follows.

Outline

The Einstein equations. The symbol. Characteristic (null) hypersurfaces.

The Cauchy problem. The constraint equations. The local existence theorem using wave coordinates.

Asymptotic flatness. The definition of energy, momentum and angular momentum. Positivity of the energy.

Maximal hypersurfaces. Local existence using maximal hypersurfaces.

Weyl fields and Bianchi equations.

Strategy of the proof of the stability theorem. The controlling quantity.

Comparison theorem. The geometry of null hypersurfaces.

Diversion: the Penrose singularity theorem.

The geometry of maximal hypersurfaces. The last slice.

Equations of motion of surfaces. Construction of an approximate conformal group.

The continuity argument. Error estimates. Completion of the proof of the stability theorem.

Asymptotic behavior. Asymptotic field equations at null infinity.

Gravitational waves. The non-linear memory effect.

Prerequisites

The topics covered in introductory courses in differential geometry and partial differential equations. Familiarity with special relativity.