Browsing by Subject "null geodesic incompleteness"

In this thesis we present and prove Roger Penrose’s singularity theorem, which is a fundamental result in mathematical general relativity. In 1965 Penrose showed that in Einstein’s theory of general relativity, under certain general assumptions on the topology, curvature, and causal structure of a Lorentzian spacetime manifold, the spacetime manifold is null geodesically incomplete. At the time, Penrose’s theorem was highly topical in a longstanding debate on the question whether singularities are formed in the process of gravitational collapse. In the proof of the theorem, novel mathematical techniques were introduced in the study of Einstein’s theory of gravity, leading to further important developments in the mathematics of general relativity. Penrose’s theorem is built on the methods of semi-Riemannian geometry, in particular Lorentzian geometry. To lay the basis for later constructions, we therefore review the basic concepts and results of semi-Riemannian geometry needed in order to understand Penrose’s theorem. The discussion includes semi-Riemannian metrics, connection, curvature, geodesics, and semi-Riemannian submanifolds. Second, calculus of variations on semi-Riemannian manifolds is introduced and a set of results pertinent to Penrose’s theorem is given. The notion of focal point of a spacelike submanifold is defined and a proposition stating sufficient conditions for the existence of focal points is presented. Furthermore, we give a series of results that establish a relation between focal points of spacelike submanifolds and causality on a Lorentzian manifold. In the last chapter, we define a family of concepts that can be used to analyze the causal structure of Lorentzian manifolds. In particular, we define the notions of global hyperbolicity, Cauchy hypersurface, and trapped surface, which are central to Penrose’s theorem, and show some important properties thereof. Finally, Penrose’s theorem is stated and proved in detail.