General relativity and cosmology

In this course, we study the theory of general relativity, i.e. the modern description of universal gravitation. After a few reminders of special relativity, we'll familiarize ourselves with the basic concepts of general relativity, based on a few particular accepted solutions of these equations in well-identified physical contexts: weak field at the Earth's surface, geometry around an isolated spherical star, large-scale universe. This will enable us to generalize our understanding and build the theory, then deduce the field equations, i.e. Einstein's equations. The course will conclude with a discussion of black holes and gravitational waves.

The aim of this course is to provide a basic grounding in general relativity and cosmology, which will be further developed in the advanced cosmology course of the^2nd year "Cosmos-Champ-Particles" Master's degree.

Knowledge of Newtonian dynamics, electromagnetism and special relativity.

Recommended prerequisites :

A pronounced taste for abstraction.

Written exam (3 h)

The course outline is as follows

• A reminder of special relativity.
• The equivalence principle: why space-time can't be Minkowski space-time; gravitational spectral shift in weak fields.
• Kinematics: curvilinear coordinates; metrics; affine connection; parallel transport; geodesic deviation equation, curvature and tidal forces.
• Metric around an isolated spherical body: Schwarzschild's solution; time and light geodesics; radial energy equation; weak-field applications: spectral shift, light deviation, perihelion advance.
• Cosmology: the Friedmann-Lemaître-Robertson-Walker solution; expansion of the universe; spectral shift, distances.
• Dynamics: Einstein's equations.
• Schwarzschild black hole: event horizon; maximum extension, white hole; Kruskal diagram.
• Gravitational waves: plane wave solutions; effect of a gravitational wave on matter; sources (quadrupole formula).