Lecture Notes on General Relativity

This set of lecture notes on general relativity has been expanded into a textbook, Spacetime and Geometry: An Introduction to General Relativity, available for purchase online or at finer bookstores everywhere. About 50% of the book is completely new; I’ve also polished and improved many of the explanations, and made the organization more flexible and user-friendly. The notes as they are will always be here for free.


These lecture notes are a lightly edited version of the ones I handed out while teaching Physics 8.962, the graduate course in General Relativity at MIT, during Spring 1996. Each of the chapters is available here as PDF. Constructive comments and general flattery may be sent to me via the address below. The notes as a whole are available as gr-qc/9712019, and in html from “Level 5” at Caltech. What is even more amazing, the notes have been translated into French by Jacques Fric. Je ne parle pas francais, mais cette traduction devrait etre bonne.

Dates refer to the last nontrivial modification of the corresponding file (fixing typos doesn’t count). Note that, unlike the book, no real effort has been made to fix errata in these notes, so be sure to check your equations.

In a hurry? Can’t be bothered to slog through lovingly detailed descriptions of subtle features of curved spacetime? Try the No-Nonsense Introduction to General Relativity, a 24-page condensation of the full-blown lecture notes (PDF).

While you are here check out the Spacetime and Geometry page — including the annotated bibilography of technical and popular books, many available for purchase online.

Lecture Notes

1. Special Relativity and Flat Spacetime
(22 Nov 1997; 37 pages)
the spacetime interval — the metric — Lorentz transformations — spacetime diagrams — vectors — the tangent space — dual vectors — tensors — tensor products — the Levi-Civita tensor — index manipulation — electromagnetism — differential forms — Hodge duality — worldlines — proper time — energy-momentum vector — energy-momentum tensor — perfect fluids — energy-momentum conservation

2. Manifolds
(22 Nov 1997; 24 pages)
examples — non-examples — maps — continuity — the chain rule — open sets — charts and atlases — manifolds — examples of charts — differentiation — vectors as derivatives — coordinate bases — the tensor transformation law — partial derivatives are not tensors — the metric again — canonical form of the metric — Riemann normal coordinates — tensor densities — volume forms and integration

3. Curvature
(23 Nov 1997; 42 pages)
covariant derivatives and connections — connection coefficients — transformation properties — the Christoffel connection — structures on manifolds — parallel transport — the parallel propagator — geodesics — affine parameters — the exponential map — the Riemann curvature tensor — symmetries of the Riemann tensor — the Bianchi identity — Ricci and Einstein tensors — Weyl tensor — simple examples — geodesic deviation — tetrads and non-coordinate bases — the spin connection — Maurer-Cartan structure equations — fiber bundles and gauge transformations

4. Gravitation
(25 Nov 1997; 32 pages)
the Principle of Equivalence — gravitational redshift — gravitation as spacetime curvature — the Newtonian limit — physics in curved spacetime — Einstein’s equations — the Hilbert action — the energy-momentum tensor again — the Weak Energy Condition — alternative theories — the initial value problem — gauge invariance and harmonic gauge — domains of dependence — causality

5. More Geometry
(26 Nov 1997; 13 pages)
pullbacks and pushforwards — diffeomorphisms — integral curves — Lie derivatives — the energy-momentum tensor one more time — isometries and Killing vectors

6. Weak Fields and Gravitational Radiation
(26 Nov 1997; 22 pages)
the weak-field limit defined — gauge transformations — linearized Einstein equations — gravitational plane waves — transverse traceless gauge — polarizations — gravitational radiation by sources — energy loss

7. The Schwarzschild Solution and Black Holes
(29 Nov 1997; 53 pages)
spherical symmetry — the Schwarzschild metric — Birkhoff’s theorem — geodesics of Schwarzschild — Newtonian vs. relativistic orbits — perihelion precession — the event horizon — black holes — Kruskal coordinates — formation of black holes — Penrose diagrams — conformal infinity — no hair — charged black holes — cosmic censorship — extremal black holes — rotating black holes — Killing tensors — the Penrose process — irreducible mass — black hole thermodynamics

8. Cosmology
(1 Dec 1997; 15 pages)
homogeneity and isotropy — the Robertson-Walker metric — forms of energy-momentum — Friedmann equations — cosmological parameters — evolution of the scale factor — redshift — Hubble’s law

Related sets of notes or tutorials

Links to GR resources