Lorentzian manifolds arise when one combines n-dimensional space and time to an (n+1)-dimensional manifold. An understanding of Lorentzian manifold is the key ingredient to understand the theoretical aspects of general relativity.

A Lorentzian metric is a symmetric (0,2)-tensor g on a manifold of dimension n+1, such that in every p∈M there is a basis (e₀,...,e_n) with g(e_ij)=0 for i ≠ j, g(e₀,e₀)=-1, and g(e_i,e_i)=1 for i>0. In other words, up to the minus sign, the definition coincides with the one of a Riemannian manifold.

Many aspects that you know from Riemannian geometry also hold for Lorentzian manifolds, we just have to add some signs at some places. These manifolds may be curved, and important notions of curvature are sectional curvature, Ricci curvature and scalar curvature. The famous Einstein equations are a statement about the Ricci curvature of the Lorentzian manifold describing our universe, e.g. vacuum spacetime is simply a Lorentzian manifold with vanishing Ricci curvature.

This allows to study important examples, as e.g. the Schwarzschild solution which is a (3+1)-dimensional manifold with vanishing Ricci-curvature, but non-zero sectional curvature.

Another example are so-called Robertson-Walker spacetimes which are used to model the evolution of the universe.

Two major goals of the lecture will be the singularity theorems by Penrose and Hawking.

More details are available on the lecture's web page.

Literature

• C. Bär, Vorlesungsskript "Lorentzgeometrie", SS 2004,
• Kriele, Marcus. Spacetime, Foundations of General Relativity and Differential Geometry. Springer 1999
• B. O'Neill, Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press
• Wald, Robert. General Relativity. University of Chicago Press
• Misner, C.W. and Thorne, K.S. and Wheeler, J.A.. Gravitation, Freeman New York, 2003
• Hawking, S.W. and Ellis, G.F.R., The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, 1973

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