Teaching

This page hosts the syllabus, homework assignments, and other relevant information for AS.171.646 (General Relativity).

Syllabus

Homework will be announced here in due time.

Homework #1 (due 9/25): Solve the following problems from Poisson-Will: Problems 1.5, 1.6; Problem 3.1. Read carefully and reproduce the calculations in this article (Gravitational waves on the back of an envelope by Bernard Schutz). Show your work - that’s the whole point! You learned the basic techniques from the first few classes. In particular, reproduce Eqs. (16), (19), (22), (24), and - perhaps the most important of all - the unnumbered equation before Eq. (39). Extra points if you recover the order-of-magnitude estimates for supernovae and binaries in Sections III.E and III.F.

Homework #2 (due 10/14): Solve the following problems from Poisson-Will: Problems 4.1, 4.2, 4.4, 4.8, 4.9, 4.10
Homework #3 (due 10/30): Poisson problem 1.5 (check that Lx and Ly satisfy Killing’s equation); PW problems 5.5, 5.7, 5.9, 5.10, 5.12, 5.15.
Homework #4 (due 11/20): Problem 1: Show that the four scalar fields defined below Eq. (5.174) satisfy the wave equation in Schwarzschild spacetime. Solve also PW problem (5.16), i.e., show that isotropic coordinates and spherical polar harmonic coordinates do not satisfy the harmonic coordinate condition. Problem 2: Use a computer algebra program (e.g. Mathematica) to find the vacuum Einstein equations in spherical symmetry (step by step!) and the Schwarzschild metric. Show that you can reproduce the intermediate results given (e.g.) in Eqs. (5.39), (5.40), (5.41) of Carroll. Compute the Kretschmann invariant and verify that it is given by (5.50) of Carroll’s book. Problem 3: Complete the derivation of the equation of state for degenerate fermions [cf. Eq. (2.3.6) and (2.3.8) of Shapiro-Teukolsky]. Write a numerical code to integrate the Lane-Emden equation. Plot the mass-radius curve and verify the Chandrasekhar limit. Try also to recover the Oppenheimer-Volkoff maximum mass of an “ideal” neutron star.