General relativity and cosmology

In this course, we study the theory of general relativity, which is the modern description of universal gravitation. After reviewing some concepts from special relativity, we will familiarize ourselves with the basic concepts of general relativity using a few specific solutions to these equations in well-defined physical contexts: weak fields at the Earth's surface, geometry around an isolated spherical star, and the universe on large scales. This will allow us to generalize our understanding and construct 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 teaching unit is to provide the foundations of general relativity and cosmology, which will then be explored in greater depth in the advanced cosmology course in the^second year of the Master's program "Cosmos-Field-Particles."

Knowledge of Newtonian dynamics, electromagnetism, and special relativity.

Recommended prerequisites:

A strong taste for abstraction.

Written exam (3 hours)

The overall plan for the course is as follows

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