Part I. Fundamentals --
1. Introduction --
1.1 Introduction --
1.2 Space and Time in Prerelativity Physics and in Special Relativity --
1.3 The Spacetime Metric --
1.4 General Relativity --
2. Manifolds and Tensor Fields --
2.1 Manifolds --
2.2 Vectors --
2.3 Tensors the Metric Tensor --
2.4 The Abstract Index Notation --
3. Curvature --
3.1 Derivative Operators and Parallel Transport --
3.2 Curvature --
3.3 Geodesics --
3.4 Methods for Computing Curvature --
4. Einstein's Equation --
4.1 The Geometry of Space in Prerelativity Physics --
General and Special Covariance --
4.2 Special Relativity --
4.3 General Relativity --
4.4 Linearized Gravity: The Newtonian Limit and Gravitational Radiation --
5. Homogeneous, Isotropic Cosmology --
5.1 Homogeneity and Isotrophy --
5.2 Dynamics of a Homogeneous, Isotropic Universe --
5.3 The Cosmological Redshift --
Horizons --
5.4 The Evolution of Our Universe --
6. The Schwartzschild Solution. 6.1 Derivation of the Schwartzschild Solution --
6.2 Interior Solutions --
6.3 Geodesics of Schwartzschild: Gravitation Redshift, Perihelion Precession, Bending of Light, and Time Delay --
6.4 The Kruskal Extension --
Part II. Advanced Topics --
7. Methods for Solving Einstein's Equation --
7.1 Stationary, Axisymmetric Solutions --
7.2 Spatially Homogeneous Cosmologies --
7.3 Algebraically Special Solutions --
7.4 Methods for Generating Solutions --
7.5 Perturbations --
8. Casual Structure --
8.1 Futures and Pasts: Basic Definitions and Results --
8.2 Causality Conditions --
8.3 Domains of Dependence --
Global Hyperbolicity --
9. Singularities --
9.1 What is a Singularity? --
9.2 Timelike and Null Geodesic Congruences --
9.3 Conjugate Points --
9.4 Existence of Maximum Length Curves --
9.5 Singularity Theorems --
10. The Initial Value Formulation --
10.1 Initial Value Formulation for Particles and Fields --
10.2 Initial Value Formulation of General Relativity. 11. Asymptotic Flatness --
11.1 Conformal Infinity --
11.2 Energy --
12. Black Holes --
12.1 Black Holes and the Cosmic Censor Conjecture --
12.2 General Properties of Black Holes --
12.3 The Charged Kerr Black Holes --
12.4 Energy Extraction from Black Holes --
12.5 Black Holes and Thermodynamics --
13. Spinors --
13.1 Spinors in Minkowski Spacetime --
13.2 Spinors in Curved Spacetime --
14. Quantum Effects in Strong Gravitational Fields --
14.1 Quantum Gravity --
14.2 Quantum Fields in Curved Spacetime --
14.3 Particle Creation near Black Holes --
14.4 Black Hold Thermodynamics --
Appendices --
A. Topological Spaces --
B. Differential Forms, Integration, and Frobenius's Theorem --
B.1 Differential Forms --
B.2 Integration --
B.3 Frobenius's Theorem --
C. Maps of Manifolds, Lie Derivatives, and Killing Fields --
C. 1 Maps of Manifolds --
C. 2 Lie Derivatives --
C. 3 Killing Vector Fields --
D. Conformal Transformations --
E. Lagrangian and Hamiltonian Formulations of Einstein's Equation --
E.1 Lagrangian Formulation --
E.2 Hamiltonian Formulation --
F. Units and Dimensions.