ar X iv : m at h - ph / 0 30 90 40 v 1 1 7 Se p 20 03 dS 4 , 1 spacetime , projective embedded coordinates and associated isometry groups

dS 4,1 spacetime, projective embedded coordinates and associated isometry groups Abstract This work is intended to investigate the de Sitter metric and to give another description of the hyperbolical embedding standard formalism of the de Sitter space-time in a pseudoeu-clidean space-time E 4,1. Classical results in this mathematical formulation are reviewed in a more general setting together with the isometry group associated to the de Sitter space-time. It is known that, out of the Friedmann models that describe our universe, the Minkowski, de Sitter and anti-de Sitter space-times are the unique maximally isotropic ones, so they admit a maximal number of conservation laws and also a maximal number of Killing vectors. We introduce and discuss the well-known embedding of a 4-sphere and a 4-hyperboloid on E 4,1 , reviewing the usual formalism of spherical embedding and the way how it can retrieve the Robertson-Walker metric. With the choice of the de Sitter metric static frame, we write the so-called reduced model in suitable coordinates. We assume the existence of projective coordinates, since de Sitter space-time is orientable. From these coordinates, obtained by doing a stereographic projection of the (Wick-rotated) de Sitter 4-hemisphere, we consider the Beltrami geodesic representation, which gives a more general formulation of the semi-nal full model described by Schrödinger, concerning the geometry and topology of de Sitter space-time. Our formalism retrieves the classical one if we consider the linear metric terms over the de Sitter splitting on Minkowski space-time. From the covariant derivatives we find the acceleration of moving particles and review the Killing vectors and the isometry group generators associated to de the Sitter space-time.