General Relativity -- PHYS-GA 2060

Fall 2017

Professor: Yacine Ali-Haïmoud
Lectures: Tuesdays and Thursdays 9:30AM - 10:45AM, Physics building (726 Broadway) room 902
Office Hours: Mondays 9:30 - 10:30AM, Physics building office 939.
Grading: 30% homework, 30% midterm, 30% final exam, best of 3 gets 10% extra weight.

Course description: Manifolds, vectors and tensor fields, curvature and gravitation; applications to black holes, relativistic stars, cosmology and gravitational waves.

Recommended textbook: Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity.
Sean Carroll's lecture notes (an abridged, preliminary version of his book) are available here.

Other great textbooks:
Robert Wald's General Relativity
Steven Weinberg's Gravitation and Cosmology
Misner, Thorne and Wheeler's Gravitation (MTW)
Thorne and Blandford's Modern Classical Physics (chapters 1-2, 24-28).


--- Part I: Mathematical foundations and formulation of GR ---

lecture 1 (Sept 05): Linear algebra, vectors and tensors.
lecture 2 (Sept 07): Introduction to the tangent space, coordinate bases.
lecture 3 (Sept 12): Special relativity.
lecture 4 (Sept 14): Kinetic theory. The equivalence principle.
lecture 5 (Sept 19): Manifolds.
lecture 6 (Sept 21): Covariant derivatives.
lecture 7 (Sept 26): Curvature.
lecture 8 (Sept 28): The Riemann tensor.
lecture 9 (Oct 03): Einstein's field equation.
lecture 10 (Oct 05): Killing vectors, conservation laws, Lagrangian formulation of GR.

--- Part II: Linearized gravity; gravitational waves ---

lecture 11 (Oct 10): Gauge transformations, scalar-vector-tensor decomposition.
lecture 12 (Oct 12): Linearized Einstein field equations.
lecture 13 (Oct 17): Far-field metric of a quasi-Newtonian source.
lecture 14 (Oct 19): Lense-Thirring effect, gravitational redshift, deflection of light.
-- no lecture on Oct 24 (midterm exam).
lecture 15 (Oct 26): Gravitational lensing, Shapiro time delay, effect of gravitational waves.
lecture 16 (Oct 31): Generation of gravitational waves, application to a circular binary.
lecture 17 (Nov 02): Energy of gravitational waves. Quadrupole formula for the power radiated in GWs.
lecture 18 (Nov 07): Angular momentum radiated in gravitational waves. Introduction to the post-Newtonian expansion.

--- Part III: Symmetric spacetimes: relativistic stars and black holes; cosmology ---

lecture 19 (Nov 09): Symmetries. Spherically symmetric spacetimes.
lecture 20 (Nov 14): Birkhoff's theorem, the Schwarzschild metric, orbits of massive particles.
lecture 21 (Nov 16): Introduction to pulsar timing arrays (guest lecture by Dr. Chiara Mingarelli).
lecture 22 (Nov 21): Periastron precession. Tollmann-Oppenheimer-Volkov equations of stellar structure.
lecture 23 (Nov 28): Constant-density star. Embedding diagrams. Kruskal extension for Schwarzschild black holes.
lecture 24 (Nov 30): Conformal transformations and diagrams.
lecture 25 (Dec 05): Kerr metric. FLRW metric.
lecture 26 (Dec 07): Brief introduction to cosmology basics.


Homeworks will be posted on Tuesday and due the following Tuesday.
You can type them up or hand-write them, I only require that they are clearly legible.

homework 1 (due Sept 12). solution
homework 2 (due Sept 19). solution
homework 3 (due Sept 26). solution
homework 4 (due Oct 03). solution
homework 5 (due Oct 10). solution
homework 6 (due Oct 17). solution
-- no homework due Oct 24 (midterm exam)
homework 7 (due Oct 31). solution
homework 8 (due Nov 07). solution
homework 9 (due Nov 14). solution and mathematica notebook


Midterm (to be presented orally on Oct 24). solution

Instructions for the final project.

Last updated December 7th 2017