Wintersemester 2012/13: Student seminar on the Initial Value Problem in General Relativity

The class will take place on Thursdays, 12 noon - 2pm in Seminarraum A, Währinger Strasse 17. The first class will be on October 4.

We will go through material in geometry and partial differential equations required to formulate and solve the initial value problem for the vacuum Einstein equations. The course will be split into individual presentations by students on the following topics:

  1. wave equation (Minkowski) in dimension 1, 2 and 3
    • spherical means, uniqueness, domains of influence & dependence
    • Fourier methods
  2. General Cauchy problems (Sections 1-5 of Ehlers et al)
  3. Geometry of hypersurfaces in Lorentzian manifolds
  4. Cauchy problem for Einstein equations
  5. Solution of vector constraint equation
  6. Solution of scalar constraint equation
  7. Strong Cosmic Censorship (time permitting)

The wave equation on Minkowski space

Notes from the introductory lectures on wave equations may be found here.

General references on partial differential equations are the the books of Evans and Taylor. More specifically, the notes above are derived from material found in the following references:

  • Spherical means approach, etc: pp. 65-85 of Evans. (See, also, Chapter 1 of Sogge.)
  • Fourier methods for the Minkowski wave equation: Selected topics from Taylor sections 3.3, 3.4 and 3.5.

General Cauchy problems

Notes of Ehlers-Friedrich-Rendall-Schmidt on the Cauchy problem, Garching preprint "Das Cauchy Problem", 1989: p1-p19.pdf p20-p32.pdf p27-p42.pdf p43-p56.pdf



Cauchy problem in GR



Piotr Chruściel's lecture notes on the Cauchy problem.

Notes from the introductory lectures on the conformal method may be found here.
Notes from the introductory lectures on the initial value problem for the vacuum Einstein equations may be found here.

(Note that the notes on the conformal method and the initial value problem are really just summaries of the relevant parts of the lecture notes of Piotr. Chruściel mentioned above.)

Further reading:

  • Robert Bartnik and Jim Isenberg, The constraint equations. Available here.
  • Yvonne Choquet-Bruhat, General relativity and the Einstein equations (Oxford University Press, Oxford, 2009).
  • Lawrence C. Evans, Partial differential equations (American Mathematical Society, Providence, RI, 2010).
  • H. Friedrich and A. D. Rendall, The Cauchy problem for the Einstein equations, Lecture Notes in Physics 540 (2000) 127-224. Available here.
  • S. Klainerman, Lecture notes in analysis. Available here.
  • Hans Ringström, The Cauchy problem in general relativity (European Mathematical Society, Zürich, 2009).
  • Christopher D. Sogge, Lectures on nonlinear wave equations (International Press, 1995).
  • Michael E. Taylor, Partial differential equations I. Basic theory (Springer, New York, 2011).

This page was modified on December 13, 2012 by James Grant.