Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- Notation and conventions -- 1 Fundamentals -- 1.1 Vectors, dual vectors, and tensors -- 1.2 Covariant differentiation -- 1.3 Geodesics -- 1.4 Lie differentiation -- 1.5 Killing vectors -- 1.6 Local flatness -- 1.7 Metric determinant -- 1.8 Levi-Civita tensor -- 1.9 Curvature -- 1.10 Geodesic deviation -- 1.11 Fermi normal coordinates -- 1.11.1 Geometric construction -- 1.11.2 Coordinate transformation -- 1.11.3 Deviation vectors -- 1.11.4 Metric on γ -- 1.11.5 First derivatives of the metric on γ -- 1.11.6 Second derivatives of the metric on γ -- 1.11.7 Riemann tensor in Fermi normal coordinates -- 1.12 Bibliographical notes -- 1.13 Problems -- 2 Geodesic congruences -- 2.1 Energy conditions -- 2.1.1 Introduction and summary -- 2.1.2 Weak energy condition -- 2.1.3 Null energy condition -- 2.1.4 Strong energy condition -- 2.1.5 Dominant energy condition -- 2.1.6 Violations of the energy conditions -- 2.2 Kinematics of a deformable medium -- 2.2.1 Two-dimensional medium -- 2.2.2 Expansion -- 2.2.3 Shear -- 2.2.4 Rotation -- 2.2.5 General case -- 2.2.6 Three-dimensional medium -- 2.3 Congruence of timelike geodesics -- 2.3.1 Transverse metric -- 2.3.2 Kinematics -- 2.3.3 Frobenius' theorem -- 2.3.4 Raychaudhuri's equation -- 2.3.5 Focusing theorem -- 2.3.6 Example -- 2.3.7 Another example -- 2.3.8 Interpretation of θ -- 2.4 Congruence of null geodesics -- 2.4.1 Transverse metric -- 2.4.2 Kinematics -- 2.4.3 Frobenius' theorem -- 2.4.4 Raychaudhuri's equation -- 2.4.5 Focusing theorem -- 2.4.6 Example -- 2.4.7 Another example -- 2.4.8 Interpretation of θ -- 2.5 Bibliographical notes -- 2.6 Problems -- 3 Hypersurfaces -- 3.1 Description of hypersurfaces -- 3.1.1 Defining equations -- 3.1.2 Normal vector -- 3.1.3 Induced metric -- 3.1.4 Light cone in flat spacetime.

3.2 Integration on hypersurfaces -- 3.2.1 Surface element (non-null case) -- 3.2.2 Surface element (null case) -- 3.2.3 Element of two-surface -- 3.3 Gauss-Stokes theorem -- 3.3.1 First version -- 3.3.2 Conservation -- 3.3.3 Second version -- 3.4 Differentiation of tangent vector fields -- 3.4.1 Tangent tensor fields -- 3.4.2 Intrinsic covariant derivative -- 3.4.3 Extrinsic curvature -- 3.5 Gauss-Codazzi equations -- 3.5.1 General form -- 3.5.2 Contracted form -- 3.5.3 Ricci scalar -- 3.6 Initial-value problem -- 3.6.1 Constraints -- 3.6.2 Cosmological initial values -- 3.6.3 Moment of time symmetry -- 3.6.4 Stationary and static spacetimes -- 3.6.5 Spherical space, moment of time symmetry -- 3.6.6 Spherical space, empty and flat -- 3.6.7 Conformally-flat space -- 3.7 Junction conditions and thin shells -- 3.7.1 Notation and assumptions -- 3.7.2 First junction condition -- 3.7.3 Riemann tensor -- 3.7.4 Surface stress-energy tensor -- 3.7.5 Second junction condition -- 3.7.6 Summary -- 3.8 Oppenheimer-Snyder collapse -- 3.9 Thin-shell collapse -- 3.10 Slowly rotating shell -- 3.11 Null shells -- 3.11.1 Geometry -- 3.11.2 Surface stress-energy tensor -- 3.11.3 Intrinsic formulation -- 3.11.4 Summary -- 3.11.5 Parameterization of the null generators -- 3.11.6 Imploding spherical shell -- 3.11.7 Accreting black hole -- 3.11.8 Cosmological phase transition -- 3.12 Bibliographical notes -- 3.13 Problems -- 4 Lagrangian and Hamiltonian formulations of general relativity -- 4.1 Lagrangian formulation -- 4.1.1 Mechanics -- 4.1.2 Field theory -- 4.1.3 General relativity -- 4.1.4 Variation of the Hilbert term -- 4.1.5 Variation of the boundary term -- 4.1.6 Variation of the matter action -- 4.1.7 Nondynamical term -- 4.1.8 Bianchi identities -- 4.2 Hamiltonian formulation -- 4.2.1 Mechanics -- 4.2.2 3 + 1 decomposition -- 4.2.3 Field theory.

4.2.4 Foliation of the boundary -- 4.2.5 Gravitational action -- 4.2.6 Gravitational Hamiltonian -- 4.2.7 Variation of the Hamiltonian -- 4.2.8 Hamilton's equations -- 4.2.9 Value of the Hamiltonian for solutions -- 4.3 Mass and angular momentum -- 4.3.1 Hamiltonian definitions -- 4.3.2 Mass and angular momentum for stationary, axially symmetric spacetimes -- 4.3.3 Komar formulae -- 4.3.4 Bondi-Sachs mass -- 4.3.5 Distinction between ADM and Bondi-Sachs masses: Vaidya spacetime -- 4.3.6 Transfer of mass and angular momentum -- 4.4 Bibliographical notes -- 4.5 Problems -- 5 Black holes -- 5.1 Schwarzschild black hole -- 5.1.1 Birkhoff's theorem -- 5.1.2 Kruskal coordinates -- 5.1.3 Eddington-Finkelstein coordinates -- 5.1.4 Painlevé-Gullstrand coordinates -- 5.1.5 Penrose-Carter diagram -- 5.1.6 Event horizon -- 5.1.7 Apparent horizon -- 5.1.8 Distinction between event and apparent horizons: Vaidya spacetime -- 5.1.9 Killing horizon -- 5.1.10 Bifurcation two-sphere -- 5.2 Reissner-Nordström black hole -- 5.2.1 Derivation of the Reissner-Nordström solution -- 5.2.2 Kruskal coordinates -- 5.2.3 Radial observers in Reissner-Nordström spacetime -- 5.2.4 Surface gravity -- 5.3 Kerr black hole -- 5.3.1 The Kerr metric -- 5.3.2 Dragging of inertial frames: ZAMOs -- 5.3.3 Static limit: static observers -- 5.3.4 Event horizon: stationary observers -- 5.3.5 The Penrose process -- 5.3.6 Principal null congruences -- 5.3.7 Kerr-Schild coordinates -- 5.3.8 The nature of the singularity -- 5.3.9 Maximal extension of the Kerr spacetime -- 5.3.10 Surface gravity -- 5.3.11 Bifurcation two-sphere -- 5.3.12 Smarr's formula -- 5.3.13 Variation law -- 5.4 General properties of black holes -- 5.4.1 General black holes -- 5.4.2 Stationary black holes -- 5.4.3 Stationary black holes in vacuum -- 5.5 The laws of black-hole mechanics -- 5.5.1 Preliminaries -- 5.5.2 Zeroth law.

5.5.3 Generalized Smarr formula -- 5.5.4 First law -- 5.5.5 Second law -- 5.5.6 Third law -- 5.5.7 Black-hole thermodynamics -- 5.6 Bibliographical notes -- 5.7 Problems -- References -- Index.