دانلود کتاب Differential Geometry and General Relativity

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Foreword
Preface to the English Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Outline of Volume II
Outline of Volume III
1 Topological Spaces in Brief
1.1 The ABCs of Set Theory
1.2 Topological Spaces
1.3 Compactness [Optional Reading]
Exercises
Reference
2 Manifolds and Tensor Fields
2.1 Differentiable Manifolds
2.2 Tangent Vectors and Tangent Vector Fields
2.2.1 Tangent Vectors
2.2.2 Tangent Vector Fields on Manifolds
2.3 Dual Vector Fields
2.4 Tensor Fields
2.5 Metric Tensor Fields
2.6 The Abstract Index Notation
Exercises
References
3 The Riemann (Intrinsic) Curvature Tensor
3.1 Derivative Operators
3.2 Derivative and Parallel Transport of a Vector Field Along a Curve
3.2.1 Parallel Transport of a Vector Field Along a Curve
3.2.2 The Derivative Operator Associated with a Metric
3.2.3 Relationship Between the Derivative and Parallel Transport of a Vector Field Along a Curve
3.3 Geodesics
3.4 The Riemann Curvature Tensor
3.4.1 Definition and Properties of the Riemann Curvature
3.4.2 Computing Riemann Curvature from a Metric
3.5 The Intrinsic Curvature and the Extrinsic Curvature
Exercises
References
4 Lie Derivatives, Killing Fields and Hypersurfaces
4.1 Maps of Manifolds
4.2 Lie Derivatives
4.3 Killing Vector Fields
4.4 Hypersurfaces
Exercises
References
5 Differential Forms and Their Integrals
5.1 Differential Forms
5.2 Integration on Manifolds
5.3 Stokes\'s Theorem
5.4 Volume Elements
5.5 Integrating Functions on Manifolds, Gauss\'s Theorem
5.6 Dual Differential Forms
5.7 Computing the Riemann Curvature Using the Tetrad Method [Optional Reading]
Exercises
References
6 Special Relativity
6.1 Foundations of the 4-Dimensional Formulation
6.1.1 Preliminaries
6.1.2 The Background Spacetime of Special Relativity
6.1.3 Inertial Observers and Inertial Frames
6.1.4 Proper Time and Coordinate Time
6.1.5 Spacetime Diagrams
6.1.6 Spacetime Structure: Special Relativity Versus Pre-Relativity Physics
6.2 Interesting Typical Effects
6.2.1 Length Contraction
6.2.2 Time Dilation
6.2.3 The Twin ``Paradox\'\'
6.2.4 The Garage ``Paradox\'\'
6.3 Kinematics and Dynamics of a Point Mass
6.4 The Energy-Momentum Tensor of Continuous Media
6.5 Perfect Fluid Dynamics
6.6 Electrodynamics
6.6.1 Electromagnetic Fields and 4-Current Densities
6.6.2 Maxwell\'s Equations
6.6.3 Lorentz 4-Force
6.6.4 The Energy-Momentum Tensor of an Electromagnetic Field
6.6.5 Electromagnetic 4-Potential and Its Equation of Motion, Electromagnetic Waves
6.6.6 The Doppler Effect on a Light Wave
Exercises
References
7 Foundations of General Relativity
7.1 Gravity and Spacetime Geometry
7.2 Physical Laws in Curved Spacetime
7.3 Fermi-Walker Transport and Non-Rotating Observers
7.4 The Proper Coordinate System of an Arbitrary Observer
7.5 Equivalence Principles and Local Inertial Frames
7.6 Tidal Forces and the Geodesic Deviation Equation
7.7 The Einstein Field Equation
7.8 Linear Approximation and the Newtonian Limit
7.8.1 Linearized Theory of Gravity
7.8.2 The Newtonian Limit
7.9 Gravitational Radiation
7.9.1 Gauge Conditions of the Linearized Theory of Gravity
7.9.2 Gravitational Plane Waves
7.9.3 Emission of Gravitational Waves
7.9.4 Detection of Gravitational Waves
Exercises
References
8 Solving Einstein\'s Equation
8.1 Stationary Spacetimes and Static Spacetimes
8.2 Spherically Symmetric Spacetimes
8.3 The Vacuum Schwarzschild Solution
8.3.1 Static Spherically Symmetric Metrics
8.3.2 The Vacuum Schwarzschild Solution
8.3.3 Birkhoff\'s Theorem
8.4 The Reissner-Nordström Solution
8.4.1 Electrovacuum Spacetimes and the Einstein-Maxwell Equations
8.4.2 The Reissner-Nordström Solution
8.5 Axisymmetric Metrics [Optional Reading]
8.6 Plane Symmetric Metrics [Optional Reading]
8.7 The Newman-Penrose (NP) Formalism [Optional Reading]
8.8 Solving the Einstein-Maxwell Equations Using the NP …
8.8.1 Maxwell\'s Equations and Einstein\'s Equation in the NP Formalism
8.8.2 An Example of Solving the Einstein-Maxwell Equations Under the Axisymmetric Condition
8.9 The Vaidya Metric and the Kinnersley Metric
8.9.1 From the Schwarzschild Metric to the Vaidya Metric
8.9.2 The Kinnersley Metric
8.9.3 The Kinnersley Metric (Detailed Discussions)
8.10 Coordinate Conditions, the Gauge Freedom of General Relativity
8.10.1 Coordinate Conditions
8.10.2 The Gauge Freedom of General Relativity
Exercises
References
9 Schwarzschild Spacetimes
9.1 Geodesics in Schwarzschild Spacetimes
9.2 Classical Experimental Tests of General Relativity
9.2.1 Gravitational Redshift
9.2.2 Perihelion Precession of Mercury
9.2.3 Light Deflection
9.3 Spherical Stars and Their Evolution
9.3.1 Interior Solutions for Static Spherical Stars
9.3.2 Stellar Evolution
9.4 The Kruskal Extension and Schwarzschild Black Holes
9.4.1 The Definition of a Spacetime Singularity
9.4.2 Coordinate Singularities of Rindler Metrics
9.4.3 The Kruskal Extension of Schwarzschild Spacetimes
9.4.4 Surfaces of Infinite Redshift in Schwarzschild Spacetimes
9.4.5 Embedding Diagrams [Optional Reading]
9.4.6 The Gravitational Collapse of a Spherical Star and Schwarzschild Black Holes
Exercises
References
10 Cosmology I
10.1 Kinematics of the Universe
10.1.1 Cosmological Principle
10.1.2 Spacial Geometries of the Universe
10.1.3 The Robertson-Walker Metric
10.2 Dynamics of the Universe
10.2.1 The Hubble-Lemaître Law
10.2.2 Cosmological Redshift
10.2.3 Evolution of the Scale Factor
10.2.4 The Cosmological Constant and Einstein\'s Static Universe
10.3 The Thermal History of Our Universe
10.3.1 A Brief History of the Universe
10.3.2 The Dark Matter Problem
10.3.3 The Cosmological Constant Problem and the ΛCDM Model
References
Appendix The Conversion Between Geometrized and Nongeometrized Unit Systems
Exercises
Exercises
Reference
Appendix Conventions and Notation
Note on Conventions
Notation List
Reference
Index