General relativity and cosmology

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

The aim of this teaching unit is to give the basics of general relativity and cosmology which will then be deepened in the teaching of advanced cosmology in the^2nd year Master "Cosmos-Field-Particles".

Knowledge of Newtonian dynamics, electromagnetism and special relativity.

Recommended Prerequisites:

A pronounced taste for abstraction.

Written exam (3 h)

The overall course outline is as follows

• Reminder of special relativity.
• The equivalence principle: why the space-time cannot be Minkowski's; gravitational spectral shift in weak field.
• Kinematics: curvilinear coordinates; metrics; affine connection; parallel transport; geodesic deflection equation, curvature and tidal forces.
• Metric around an isolated spherical body: the Schwarzschild solution; geodesics of time and light; 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.
• The 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).