Relativity and Singularities – a Short Introduction for Mathematicians

We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point set topology. Introduction Historically, much of the development of Riemannian geometry has been driven by General Relativity. This theory models spacetime as a Lorentzian manifold, which is analogous to a Riemannian manifold except that the positive definite metric is replaced by a metric with signature (−,+, . . . ,+). Not only is Lorentzian geometry similar to Riemannian geometry in many respects but also Riemannian manifolds arise naturally as submanifolds of Lorentzian manifolds. Physical considerations then give rise to conjectures in Riemannian geometry. Recent examples of results inspired by such conjectures include the mass positivity theorem (Schoen and Yau, [SY79], [SY81]) and the Penrose inequality (Bray, [Bra01], Huisken and Ilmanen, [HI01]). On the other hand, the effort involved in learning Lorentzian geometry is minimal once one has mastered Riemannian geometry. It therefore seems strange that many mathematicians (even geometers) choose not to do so. This may be in part due to the fact that most introductions to General Relativity start from first principles, developing the required differential geometry tools at length, and mostly focus on physical implications of the theory. A mathematician might prefer a shorter introduction to the subject from a more advanced starting point, focusing on interesting mathematical ideas. This paper aims to provide such an introduction, leading to a nontrivial result – the simplest of the Hawking-Penrose singularity theorems ([Pen65], [Haw67], [HP70]). These theorems basically state that physically reasonable Lorentzian manifolds (in a precise mathematical sense) must be singular (i.e. geodesically incomplete). Since the motions of free-falling particles are represented by geodesics, this has the physical interpretation that General Relativity cannot be a complete description of Nature. The paper is divided into three sections. The first section contains basic ideas of General Relativity and Lorentzian geometry: timelike, spacelike and null vectors and curves, matter models, the Einstein equation and its simplest solutions. Causality theory is developed in the second section, where we discuss time orientation, chronological and causal future and past sets, local causal structure, local maximizing properties of timelike geodesics (the Twin Paradox), the chronology This work was partially supported by FCT/POCTI/FEDER. 1