A Propos de Quelques Métriques dans le Cadre de la Relativité Générale

Abstract

"The work of this Magister’s memory is in the framework of the General Theory of Relativity of Einstein. After a short introduction, followed by a concise but complete presentation of general relativity in order to dispense the reader to refer to an external document support, we tackled the rederivation of some interesting metrics of General Relativity , especially in the field of cosmology . We considered in turn: The FLRW metric, satisfying the cosmological principle , that is to say corresponding to a homogeneous isotropic universe, which is reflected by the invariance under rotations and translations (flat case ), or under rotations and quasi-translations (spherical and pseudo- spherical cases) . Shwarzshild metric, static and spherically symmetric , that is to say invariant under temporal translations and rotation, in absence of cosmological constant . The Kottler metric or Schwarzshild -de Sitter, also static and spherically symmetric, but with cosmological constant The work is done in two steps: In a first step and for each considered case , we impose the corresponding symmetries. The method used is that of the Killing vectors , which is an elegant and rigorous method to implement symmetries . Symmetries result in constraints on the form of the metric. During this first stage we do not appeal to Einstein's equations , that is to say we do not use dynamics . In a second step , having determined in the previous step the most general form of a metric satisfying some symmetries, we complete determnation of the metric by tacking into account the Einstein equations.To this end, we have conjectured some more or less justified forms for the energy-momentum tensor. In this way, we find the FLRW metric, the Shwarzshild metric inside and outside of the distribution of matter, and also Kottler metric inside and outside of the distribution of matter."