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Preface

Notation and Conventions

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**

**References**

**Index**

Notation and Conventions

PART I. FUNDAMENTALS

1.1 Introduction

1.2 Space and Time in Prerelativity Physics and in Special Relativity

1.3 The Spacetime Metric

1.4 General Relativity

2.1 Manifolds

2.2 Vectors

2.3 Tensors the Metric Tensor

2.4 The Abstract Index Notation

3.1 Derivative Operators and Parallel Transport

3.2 Curvature

3.3 Geodesics

3.4 Methods for Computing Curvature

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

5.1 Homogeneity and Isotrophy

5.2 Dynamics of a Homogeneous, Isotropic Universe

5.3 The Cosmological Redshift Horizons

5.4 The Evolution of Our Universe

6.1 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

7.1 Stationary, Axisymmetric Solutions

7.2 Spatially Homogeneous Cosmologies

7.3 Algebraically Special Solutions

7.4 Methods for Generating Solutions

7.5 Perturbations

8.1 Futures and Pasts: Basic Definitions and Results

8.2 Causality Conditions

8.3 Domains of Dependence Global Hyperbolicity

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

10.1 Initial Value Formulation for Particles and Fields

10.2 Initial Value Formulation of General Relativity

11.1 Conformal Infinity

11.2 Energy

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

13.1 Spinors in Minkowski Spacetime

13.2 Spinors in Curved Spacetime

14.1 Quantum Gravity

14.2 Quantum Fields in Curved Spacetime

14.3 Particle Creation near Black Holes

14.4 Black Hold Thermodynamics

APPENDICES

B.1 Differential Forms

B.2 Integration

B.3 Frobenius's Theorem

C.1 Maps of Manifolds

C.2 Lie Derivatives

C.3 Killing Vector Fields

E.1 Lagrangian Formulation

E.2 Hamiltonian Formulation

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