A Short Intro to General Relativity

Pre-requisites: Knowledge of multi-dimensional Differential Calculus and Linear Algebra is necessary. Desirable is some background in linear PDE’s (Wave and Poisson equations), Differential Geometry (which will be reviewed) and most basic Classical Mechanics.

Summary: This mini course is aimed at introducing General Relativity to students and researchers coming from an Analysis background. I motivate and introduce first Special and then General Relativity. I present some basic properties of the Einstein equations, such as special solutions important for physics, and discuss the Cauchy problem for the Einstein equations. In more detail, the content of the class is as follows:

  • Newtonian Physics and its shortcomings: Newtonian equation of motion; Galilean transformation; Maxwell’s equations and speed of light; Inconsistency with measurements of speed of light. [5, 1]
  • Special Relativity: Lorentz transformations; time dilatation and length contraction; axioms of Special Relativity and their physical motivation; relativistic equations of motion and E = mc2; Minkowski spacetime; energy-momentum tensor; shortcoming of Special Relativity and necessity of a curved spacetime. [5, 2, 1]
  • A short review of Differential Geometry: Manifolds; tangent vectors; Riemann and Lorentz metrics; covariant derivatives; geodesic equation; Riemann curvature; Jacobi equations. [3, 5]
  • General Relativity and Einstein equations: Accelerated coordinates and gravity; axioms of General Relativity and their motivation; light cone; Einstein equations (including their derivation); linearized Einstein equations and gravitational waves. [5, 2, 1]
  • Solutions of the Einstein equations: Friedmann-Robertson-Walker metric (of cosmology); Schwarzschild and Kerr metrics (black hole spacetimes); gravitational collapse to black holes. [3, 5, 6]
  • Cauchy Problem for the Einstein equations: Cauchy problem for Maxwell’s equations; Einstein constraint equations; wave gauge and reduced Einstein equations; local existence of solutions of vacuum Einstein equations. [5, 1, 4]

Lecture notes covering the content of this mini course will be provided and the following literature is suggested for further reading:

[1] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, 2009.
[2] A. Einstein, Relativity: The Special and General Theory, Methuen & Co. Ltd., 1920.
[3] S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Spacetime, Cambridge University Press, 1973.
[4] H. Ringstrom, The Cauchy Problem in General Relativity, European Mathematical Society, 2009.
[5] R. M. Wald, General Relativity, The University of Chicago Press, 1984.
[6] S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, New York, 1972.