We begin by introducing tensors in non-relativistic physics. We then give an overview of Special Relativity, and discuss the special nature of gravity as an “inertial force”. With this motivation, we develop the differential geometry necessary to describe curved spacetime and the geodesic motion of free-falling particles. We then proceed to Einstein’s field equations, which we analyze in the Newtonian limit and in the linearized limit (gravitational waves). Finally, we study two iconic solutions to the field equations: the Schwarzschild black hole and Friedman-Robertson-Walker cosmology. We will use Sean Carroll’s textbook as the main reference, but we will not follow it strictly.

This is an alternating years course.

1. Tensors in 3d: moment of inertia and magnetic field

2. Special Relativity in 3d language

3. Special Relativity in 4d language: Minkowski spacetime

4. Gravity as an inertial force: the equivalence principle

5. Curved spacetime: metric and Christoffel symbols

6. Geodesic motion: Newtonian limit, redshift, deflection of light

7. Curved spacetime: The Riemann tensor and its components

8. The Einstein field equations and their Newtonian limit

9. Linearized limit and gravitational waves

10. The Schwarzschild black hole

11. More on the Schwarzschild metric: precession of planets, black hole thermodynamics

12. Friedmann-Robertson-Walker cosmology

Prerequisites: Maxwell’s equations in differential form. Solving Maxwell’s equations to obtain electromagnetic waves. Linear algebra of vectors and matrices.