Gravitational waves

Mathematical Relativity — 2nd Semester 2021/2022

Lecturer: José Natário
Office: Mathematics Building, 4th floor, room 4.29
Classes: Tuesdays and Thursdays from 9:00 to 11:00.
Office Hours:  Drop by my office or send me an email.


You can see the final grades after the second exam on the side bar. Drop by my office if you want to take a look at your graded exam.


Examples: Minkowski space-time, Schwarzschild solution, Einstein, de Sitter, anti-de Sitter and Friedman-Lemaitre-Robertson-Walker universes; matching and Oppenheimer-Snyder collapse; Penrose diagrams.

Causality: time orientabiliy, chronological and causal past and future, domain of dependence; chronological, stably causal and globally hyperbolic spacetimes.

Singularities: Jacobi equation and conjugate points; energy conditions; existence of maximizing geodesics; Hawking and Penrose singularity theorems.

Cauchy Problem: wave equation; Cauchy problems with constraints; Gauss-Coddazzi relations and 3+1 decomposition of the Einstein equation; Choquet-Bruhat theorem; constraint equations for the initial data.

Positive Mass Theorem: Komar mass; Einstein-Hilbert action; Lagrangian and Hamiltonian formulation of the Einstein equations; ADM mass; positive mass theorem; Penrose inequality.

Black Holes: Kerr solution; Killing horizons; surface gravity; Smarr formula; area theorem; black hole thermodynamics.




Grading Policy

Homework: Makes up 50% of the grade. Late homework will not be accepted.

Final exam: Makes up 50% of the grade. Can be retaken if necessary.


  • Homework 1 (due on March 24): Exercise 1 in Chapter 2
  • Homework 2 (due on April 5): Exercise 8 in Chapter 2
  • Homework 3 (due on April 19): Exercise 7 in Chapter 3
  • Homework 4 (due on May 17): Exercises 2 and 4 in Chapter 4
  • Homework 5 (due on May 26): Exercises 7, 8 and 10 in Chapter 4
  • Homework 6 (due on June 9): Exercises 1 and 4 in Chapter 5
  • Homework 7 (due on June 21): Exercises 5 and 6 in Chapter 5
  • Homework 8 (due on June 30): Exercises 5 and 8 in Chapter 6


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