Geometrical Introduction

Abstract

It has become apparent in recent decades that in discussing fundamental properties of space and time the adequate language is Lorentzian geometry. Correspondingly, we start the book with a brief introduction to that discipline. In the first four sections of this chapter some basic notions—such as convexity, causal simplicity and global hyperbolicity—are defined and some basic facts about them are provided, such as Whitehead’s and Geroch’s theorems, the Gauss lemma, etc. Of course, this recapitulation cannot substitute a systematic presentation, so wherever possible the proofs of those facts are dropped, which is understood as the reference to classical monographs such as (O’Neill, Semi-Riemannian geometry. Academic Press, 1983) [141] or (Hawking and Ellis, The large-scale structure of spacetime. Cambridge University Press, Cambridge, 1973) [76]. Then, in Sect. 5, some new objects—perfectly simple sets—appear and the theorem is proved which states that they exist in arbitrary spacetime. We shall use this theorem in proving Theorem 2 in Chap. 5, but, as it seems, it is interesting also by itself (the perfectly simple sets are ‘as nice as possible’—they are both convex and globally hyperbolic, which makes them exceptionally useful in proving statements).