A Relativist's Toolkit
The Mathematics of BlackHole 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.
Contents

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 LeviCivita tensor. 9
Curvature. 10 Geodesic deviation. 11 Fermi normal coordinates. 12
Bibliographical notes. 13 Problems.

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.

Hypersurfaces
1 Description of hypersurfaces. 2 Integration on hypersurfaces.
3 GaussStokes theorem. 4 Differentiation of tangent tensor fields.
5 GaussCodazzi equations. 6 Initialvalue problem. 7 Junction
conditions and thin shells. 8 OppenheimerSnyder collapse. 9
Thinshell collapse. 10 Slowly rotating shell. 11 Null shells. 12
Bibliographical notes. 13 Problems.

Lagrangian and Hamiltonian formulations of general
relativity
1 Lagrangian formulation. 2 Hamiltonian formulation. 3 Mass
and angular momentum. 4 Bibliographical notes. 5 Problems.

Black holes
1 Schwarzschild black hole. 2 ReissnerNordstrom black hole. 3
Kerr black hole. 4 General properties of black holes. 5 The laws of
blackhole mechanics. 6 Bibliographical notes. 7 Problems.