General Relativity and Cosmology

In this course, we study the theory of general relativity—that is, the modern description of universal gravitation. After a brief review of special relativity, we will familiarize ourselves with the basic concepts of general relativity by examining some accepted particular solutions to these equations in well-defined physical contexts: a weak field at the Earth’s surface, the 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 derive the field equations—that is, Einstein’s equations. The course will conclude with a discussion of black holes and gravitational waves.

The goal of this course is to provide the fundamentals of general relativity and cosmology, which will then be explored in greater depth in the advanced cosmology course offered in the^second year of the “Cosmos-Field-Particles” Master’s program.

Knowledge of Newtonian dynamics, electromagnetism, and special relativity.

Recommended prerequisites:

A strong penchant for abstraction.

Written exam (3 hours)

The overall course outline is as follows

• A Review of Special Relativity.
• The Equivalence Principle: Why Spacetime Cannot Be Minkowski Spacetime; Gravitational Spectral Shift in a Weak Field.
• 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- and light-like geodesics; radial energy equation; applications in the weak field: spectral shift, light deflection, and perihelion advance.
• Cosmology: the Friedmann–Lemaître–Robertson–Walker solution; expansion of the universe; redshift, distances.
• Dynamics: Einstein's equations.
• The Schwarzschild black hole: event horizon; maximum extension, white hole; Kruskal diagram.
• Gravitational waves: plane wave solutions; the effect of a gravitational wave on matter; sources (quadrupole formula).