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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  • 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

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