The physical concepts of special and general relativity. The elements of differential calculus and tensorial calculus that are required by general relativity. The relativistic conceptions of time, space, mass, momentum and energy. The covariant formulation of electromagnetism. The geodesics. Einstein's equation. The metric of Schwarzschild. The "Newton's force" as outcome of general relativity. The slowing down of time by gravity. The gravitational lensing effect.
To acquire the physical concepts that lead to the theory of relativity. To be able to demonstrate the main results. To integrate the mathematical tools presented in this course. To be able to apply the concepts of this course to classical problems.
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.
The exam (written) is on the material presented in class (PowerPoint to be found on WebCampus). A list of questions will be given.
Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, "Gravitation" (W.H. Freeman and Company, New York, 1973).