A Relativist's Toolkit

The Mathematics of Black-Hole Mechanics

Eric Poisson

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.

You can order the book from Amazon.com.
Read Bernard Whiting's review in Classical and Quantum Gravity.
Read Eric Sheldon's review in Contemporary Physics.
Read Kayll Lake's review in Physics in Canada.

Contents

  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.

Errors, typographical and otherwise

A number of errors were reported by readers. They have my gratitude.