Course of Special and General Relativity

This is a first course of Special and General Relativity. We update in the context of Special Relativity the concepts of space, time, mass, momentum and energy. We show how Lorentz's transformations make the laws of mechanics consistent with those of electromagnetism. We develop next the mathematical tools of General Relativity (vectors, differential forms, tensors, covariant derivatives, etc). We then address the geodesics in order to determine the trajectories in a space-time that is curved by gravitation. We introduce the Riemann tensor, the Ricci tensor and the Einstein tensor in order to describe the curvature of space-time. The energy-momentum tensor is introduced in order to describe the densities and fluxes of momentum and energy. These different concepts are finally related by the equation of Einstein. We then determine Schwarzschild's metric in order to describe space-time around a central mass. We can then establish "Newton's force" as an outcome of General Relativity. The theory explains the advance of the perihelion of Mercury and the deflection of light by a gravitational field. The course ends with a lesson on black holes. We get briefly through relativistic aspects of the GPS, gravitational lensing and gravitational waves.