### Emanuele Berti

Professor, Johns Hopkins University

# Teaching

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

## Preliminary Lecture Plan

• Homework #1 (due 9/21): Solve the following problems from Poisson-Will: Problems 1.5, 1.6; Problems 3.1, 3.2, 3.3 (Note: in problems 3.2 and 3.3 you can ignore the part that asks about computing the perturbing force f acting on a planetary orbit and computing the secular changes in the planet’s orbital elements, since we will not cover the theory of perturbations of Keplerian orbits in class. Extra points if you try to work that out, too!)

• Homework #2 (due 10/5): Solve the following problems from Poisson-Will: Problems 4.1, 4.2, 4.4, 4.8, 4.9, 4.10

• Homework #3 (due 10/24): 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 11/30): Problem 1: Show that the four scalar fields defined below Eq. (5.174) satisfy the wave equation in Schwarzschild spacetime, and that this implies Eqs. (5.177) and (5.178). 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.