Course of Special and General Relativity

This is a first course of relativity. We update in the context of special relativity the concepts of space, time, mass, momentum and energy. 

These concepts are illustrated with space-time diagrams, the famous twin paradox and various numerical exercises. We show how Lorentz's transformations make the laws of mechanics consistent with those of electromagnetism. We recall some mathematical tools of general relativity (vectors, differential forms, tensors, covariant derivatives). We then address the geodesics in order to determine 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. We demonstrate also the gravitational time dilation effect. We get briefly through different applications of relativity (advance of the perihelion of Mercury, gravitational deflection of light, gravitational waves, black holes).