The de Sitter and anti-de Sitter Sightseeing Tour

Abstract

While celebrating the 100 anniversary of the discovery of special relativity [1], it may not be inappropriate to open a window on the de Sitter universes, as their importance in contemporary physics is gradually increasing. Just to mention two examples, the astronomical evidence for an accelerated expansion of the universe gives a central place to the de Sitter geometry in cosmology [2] while the so-called AdS/CFT correspondence [3] supports a major role for the anti-de Sitter geometry in theoretical physics. From the geometrical viewpoint, among the cousins of Minkowski spacetime (the class of Lorentzian manifolds) de Sitter and anti-de Sitter spacetimes are its closest relatives. Indeed, like the Minkowski spacetime, they are maximally symmetric, i.e. they admit kinematical symmetry groups having ten generators. Maximal symmetry also implies that the curvature is constant (zero in the Minkowski case). Owing to their symmetry, it is possible to give a description of the de Sitter universes without using the machinery of general relativity at all. However, it is worth saying right away that, even if they share important features with Minkowski spacetime, their physical interpretation is quite different and the technical problems to be solved in order to merge de Sitter spacetimes with quantum physics are considerably harder. The aim of this note is to give a simple and short geometrical introduction to the de Sitter and anti-de Sitter universes and to briefly comment on their physical meaning.