Eric Poisson

Department of Physics

University of Guelph

Guelph, Ontario, N1G 2W1

(519) 824-4120 x53991

Eric Poisson
Department of Physics
University of Guelph

A Relativist's Toolkit

The Mathematics of Black-Hole Mechanics

This textbook, published by Cambridge University Press, fills a gap in the existing literature on general relativity by providing the advanced student with practical tools for the computation of many physically interesting quantities. The context is provided by the mathematical theory of black holes, one of the most successful and relevant applications of general relativity. Topics covered include congruences of timelike and null geodesics, the embedding of spacelike, timelike and null hypersurfaces in spacetime, and the Lagrangian and Hamiltonian formulations of general relativity.

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Read Bernard Whiting's review in Classical and Quantum Gravity.
Read Kayll Lake's review in Physics in Canada.

Errors, typographical and otherwise

A small number of errors have been reported. They are listed in this postscript document. This list was last compiled on October 12, 2006.


  1. Fundamentals

    1- Vectors, dual vectors, and tensors. 2- Covariant differentiation. 3- Geodesics. 4- Lie differentiation. 5- Killing vectors. 6- Local flatness. 7- Metric determinant. 8- Levi-Civita tensor. 9- Curvature. 10- Geodesic deviation. 11- Fermi normal coordinates. 12- Bibliographical notes. 13- Problems.

  2. Geodesic congruences

    1- Energy conditions. 2- Kinematics of a deformable medium. 3- Congruence of timelike geodesics. 4- Congruence of null geodesics. 5- Bibliographical notes. 6- Problems.

  3. Hypersurfaces

    1- Description of hypersurfaces. 2- Integration on hypersurfaces. 3- Gauss-Stokes theorem. 4- Differentiation of tangent tensor fields. 5- Gauss-Codazzi equations. 6- Initial-value problem. 7- Junction conditions and thin shells. 8- Oppenheimer-Snyder collapse. 9- Thin-shell collapse. 10- Slowly rotating shell. 11- Null shells. 12- Bibliographical notes. 13- Problems.

  4. Lagrangian and Hamiltonian formulations of general relativity

    1- Lagrangian formulation. 2- Hamiltonian formulation. 3- Mass and angular momentum. 4- Bibliographical notes. 5- Problems.

  5. Black holes

    1- Schwarzschild black hole. 2- Reissner-Nordstrom black hole. 3- Kerr black hole. 4- General properties of black holes. 5- The laws of black-hole mechanics. 6- Bibliographical notes. 7- Problems.