David Tong: General Relativity

# 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

• Lecture notes copyright © 2019 David Tong unless otherwise credited. Permission is granted to copy and distribute freely, so long as proper attribution is given, no alterations are made, and no monetary profit is gained.