# David Tong: Lectures on General Relativity

This is a course on general relativity, given to Part III (i.e. masters level) students. It covers advanced material, but is designed to be understandable for students who haven't had a first course in the subject. Please do email me if you find any typos or mistakes.

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# Content

**1. Geodesics: PDF**

Introduction. Non-relativistic particles and the geodesic equation; Relativistic particles, Minkowski space and mortality; Electromagnetism and gravity; The equivalence principle; Time dilation. The Schwarzschild metric, planetary orbits and perihelion precession; The pull of Venus and Jupiter; Light bending.**2. Introducing Differential Geometry: PDF**

Manifolds: Topological spaces, differentiable manifolds and maps between manifolds. Tangent Spaces: tangent vectors, vector fields, integral curves and the Lie derivative. Tensors, covectors and one-forms. Differential Forms: the exterior derivative, de Rahm cohomology, integration and Stokes' theorem.**3. Introducing Riemannian Geometry: PDF**

The metric; Riemannian and Lorentzian manifolds, the volume form and the Hodge dual. The Maxwell action. Hodge theory. Connections and the covariant derivative, curvature and torsion, the Levi-Civita connection. The divergence theorem. Parallel transport, normal coordinates and the exponential map, holonomy, geodesic deviation. The Ricci tensor and Einstein tensor. Connection 1-forms and curvature 2-forms.**4. The Einstein Equations: PDF**

The Einstein-Hilbert action, the cosmological constant; diffeomorphisms and the Bianchi identity; Minkowski, de Sitter and anti-de Sitter spacetimes; Symmetries and isometries, Killing vectors, conserved quantities; Asymptotics of spacetime, conformal transformations and Penrose diagrams; Coupling matter, the energy-momentum tensor, perfect fluids, spinors, energy conditions; Cosmology.**5. When Gravity is Weak: PDF**

The Linearised theory, gauge symmetry, the Newtonian limit; Gravitational waves, de Donder gauge, transverse traceless gauge, LIGO; Gravitational wave production, binary systems, the quadrupole formula, gravitational wave sources.**6. Black Holes: PDF**

The Schwarzschild solution, Birkhoff's theorem, Eddington-Finkelstein Coordinates, Kruskal diagrams and Penrose diagrams, weak cosmic censorship; The Reissner-Nordstrom solution, Cauchy horizons and strong cosmic censorship, Extremal black holes; The Kerr solution, global structure, the ergoregion, the Penrose process and superradiance, no hair theorems.

# Problem Sheets

João Melo has put together a preparatory worksheet, based on Chapter 1 of the lectures notes, to help refresh your understanding of geodesics before the course begins. It can be downloaded here.

**Problem Sheet 1: PDF**Differential Geometry**Problem Sheet 2: PDF**Riemannian Geometry**Problem Sheet 3: PDF**Gravity**Problem Sheet 4: PDF**Linearised Gravity

# General Relativity on the Web

**Part III General Relativity**by Harvey Reall**General Relativity**by Matthias Blau-
**General Relativity**by John McGreevy **General Relativity**by Sean Carroll**Black Holes**by Fay Dowker-
**Black Holes**by Paul Townsend -
**Black Holes**by Harvey Reall