General Relativity -- PHYS-GA 2060
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
bare grade = 50% homework + 25% midterm + 25% final.
Attendance below 50% is penalized, attendance above 50% is rewarded:
renormalized grade = (bare grade)1/(attendance + 0.5)
Examples: If you have a perfect bare grade, attendance will not affect it.
If you have a bare grade of 50% but have attended all
lectures, your renormalized grade will be 63%.
If you have a bare
grade of 50% but only attended 1/10 of the lectures, your renormalized
grade will be 31%.
Course description: The first ~5 weeks will be dedicated to
the mathematical foundations of GR, leading to Einstein's field equations (review of special relativity, manifolds, vectors and tensor fields,
differential geometry). The next ~5 weeks will focus on immediate
applications: linearized gravity, gravitational waves, spherically symmetric
stars and black holes, and will be followed by the midterm
exam. In the last ~5 weeks will explore
spinning stars and black holes, frame dragging, conformal diagrams,
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).
--- Lectures ---
--- Part I: Mathematical foundations and formulation of GR ---
lecture 1 (Sept 3): warm up
with spatial distances, notion of spatial metric, coordinate transformations.
lecture 2 (Sept 5):
introduction to inertial coordinate systems and spacetime
lecture 3 (Sept 10):
inverse metric, measurements in a particle's rest frame, geodesic
lecture 4 (Sept 12):
the equivalence principle, gravitational redshift, preliminaries on
lecture 5 (Sept 17):
manifolds and tangent vectors.
lecture 6 (Sept 19): dual
vectors and tensors.
lecture 7 (Sept 24): basic
operations on tensors, tensor fields.
lecture 8 (Sept 26): covariant
lecture 9 (Sept 30): parallel
transport, geodesics again, generally covariant equations.
lecture 10 (Oct 1): charge
conservation, electromagnetism, stress-energy tensor.
lecture 11 (Oct 8): the
Riemann tensor: definition and basic properties.
lecture 12 (Oct 10): Riemann
as a measure of curvature; Fermi normal coordinates.
lecture 13 (Oct 11): geodesic
deviation; Einstein field equations, Lagrangian formulation.
--- Part II-a: Linearized gravity, gravitational waves ---
lecture 14 (Oct 15): warm-up
with electromagnetism; gauge transformations and scalar-vector-tensor decomposition.
lecture 15 (Oct 17):
linearized Einstein field equations.
lecture 16 (Oct 25):
far-field metric of a quasi-Newtonian source.
lecture 17 (Oct 29):
gravitational redshift, deflection of light and Shapiro time
lecture 18 (Nov 1):
power radiated by gravitational waves; merger of a circular binary.
--- Part II-b: Spherically-symmetric solutions ---
lecture 19 (Nov 5):
symmetries; spherically symmetric spacetimes; Schwarzschild
lecture 20 (Nov 7):
timelike geodesics of Schwarzschild.
lecture 21 (Nov 12):
spherically symmetric and stationary stars.
lecture 22 (Nov 14):
Schwarzschild black holes: extended Kruskal coordinates, Penrose diagrams.
--- Part III: Advanced topics ---
lecture 23 (Nov 21):
polarizations of gravitational waves and generation by a circular
lecture 24 (Nov 25): Kerr
(i.e. spinning) black holes.
lecture 25 (Dec 3):
introduction to gravitational lensing and the post-Newtonian expansion.
lecture 26 (Dec 5): introduction to cosmology (guest
lecture by Masha Baryakhtar)
lecture 27 (Dec 10): introduction to numerical relativity (guest
lecture by Masha Okounkova)
lecture 28 (Dec 12): student presentations
--- Homeworks ---
Posted on Tuesdays, due the following Tuesday in class.
Hand-written homeworks are acceptable only if they are clearly and cleanly written.
homework 1 (due Sept 10). solution.
homework 2 (due Sept
homework 3 (due Sept 24). solution.
homework 4 (due Oct 1). solution.
homework 5 (due Oct 8). solution.
homework 6 (due Oct 15). solution.
homework 7 (due Oct 25). solution.
homework 8 (due Nov 5). solution.
homework 9 (due Nov 12). solution.
--- Exams ---
The midterm exam will take place on November 19 (mark
The final will consist of a project, with a written
report and an oral presentation on December 12 (mark
• You may pick a topic of your choice, and could for instance explore
one of the topics covered in class in greater depth.
You must choose your topic by November 19 so you get three full weeks
to work on it.
Below is a non-exhaustive list of possible topics. For more ideas,
and for references, a great resource is Living reviews in
-Post-Newtonian perturbation theory and evolution of a binary
-Black hole perturbations.
-The gravitational self-force and point-particles in relativity.
-Laser-interferometer gravitational-wave detectors.
-Pulsar timing arrays.
-Relativistic accretion disks.
-Experimental tests of general relativity.
-Non-Schwarzschild black holes.
-Modified theories of gravity.
-Cosmological perturbation theory.
-Explain the EHT observations.
• The written report must be typed up in latex, and be of comparable
length and depth as one of the lecture notes, i.e. around 4-5
It must include an exhaustive list of references. Here is an example latex
Please email me the report by Tuesday, 12/10, 9:30 am.
• The oral presentation on Thursday 12/12 will be 20 minutes + 5 minutes of
You will not have time to derive everything that is
in your project, but are expected to highlight the major steps and
I strongly encourage you to rehearse before presentation day .