Introduction to Einstein’s Theory of Relativity

Preface to the Second Edition
Preface to the First Edition
Contents
List of Figures
List of Definitions
List of Examples
List of Exercises
1 Newton’s Theory of Gravitation
	1.1 The Force Law of Gravitation
	1.2 Newton’s Law of Gravitation in Local Form
	1.3 Newtonian Incompressible Star
	1.4 Tidal Forces
	1.5 The Principle of Equivalence
	1.6 The General Principle of Relativity
	1.7 The Covariance Principle
	1.8 Mach’s Principle
	1.9 Exercises
	References
2 The Special Theory of Relativity
	2.1 Coordinate Systems and Minkowski Diagrams
	2.2 Synchronization of Clocks
	2.3 The Doppler Effect
	2.4 Relativistic Time Dilation
	2.5 The Relativity of Simultaneity
	2.6 The Lorentz Contraction
	2.7 The Lorentz Transformation
	2.8 Lorentz Invariant Interval
	2.9 The Twin Paradox
	2.10 Hyperbolic Motion
	2.11 Energy and Mass
	2.12 Relativistic Increase of Mass
	2.13 Lorentz Transformation of Velocity, Momentum, Energy and Force
	2.14 Tachyons
	2.15 Magnetism as a Relativistic Second-Order Effect
	Exercises
	Reference
3 Vectors, Tensors and Forms
	3.1 Vectors
		3.1.1 Four-Vectors
		3.1.2 Tangent Vector Fields and Coordinate Vectors
		3.1.3 Coordinate Transformations
		3.1.4 Structure Coefficients
	3.2 Tensors
		3.2.1 Transformation of Tensor Components
		3.2.2 Transformation of Basis One-Forms
		3.2.3 The Metric Tensor
	3.3 The Causal Structure of Spacetime
	3.4 Forms
		3.4.1 The Volume Form
		3.4.2 Dual Forms
	Exercises
4 Accelerated Reference Frames
	4.1 The Spatial Metric Tensor
	4.2 Einstein Synchronization of Clocks in a Rotating Reference Frame
	4.3 Angular Acceleration in the Rotating Frame
	4.4 Gravitational Time Dilation
	4.5 Path of Photons Emitted from the Axis in a Rotating Reference Frame
	4.6 The Sagnac Effect
	4.7 Non-integrability of a Simultaneity Curve in a Rotating Frame
	4.8 Orthonormal Basis Field in a Rotating Frame
	4.9 Uniformly Accelerated Reference Frame
	4.10 The Projection Tensor
	Exercises
5 Covariant Differentiation
	5.1 Differentiation of Forms
		5.1.1 Exterior Differentiation
		5.1.2 Covariant Derivative
	5.2 The Christoffel Symbols
	5.3 Geodesic Curves
	5.4 The Covariant Euler–Lagrange Equations
	5.5 Application of the Lagrange Formalism to Free Particles
		5.5.1 Equation of Motion from Lagrange’s Equations
		5.5.2 Geodesic World Lines in Spacetime
		5.5.3 Acceleration of Gravity
		5.5.4 Gravitational Shift of Wavelength
	5.6 Connection Coefficients
		5.6.1 Structure Coefficients
	5.7 Covariant Differentiation of Vectors, Forms and Tensors
		5.7.1 Covariant Differentiation of Vectors
		5.7.2 Covariant Differentiation of Forms
		5.7.3 Covariant Differentiation of Tensors of Arbitrary Rank
	5.8 The Cartan Connection
	5.9 Covariant Decomposition of a Velocity Field
		5.9.1 Newtonian 3-Velocity
		5.9.2 Relativistic 4-Velocity
	5.10 Killing Vectors and Symmetries
	5.11 Covariant Expressions for Gradient, Divergence, Curl, Laplacian and D’Alembert’s Wave Operator
	5.12 Electromagnetism in Form Language
	Exercises
6 Curvature
	6.1 The Riemann Curvature Tensor
	6.2 Differential Geometry of Surfaces
		6.2.1 Surface Curvature Using the Cartan Formalism
	6.3 The Ricci Identity
	6.4 Bianchi’s 1. Identity
	6.5 Bianchi’s 2. Identity
	6.6 Torsion
	6.7 The Equation of Geodesic Deviation
	6.8 Tidal Acceleration and Spacetime Curvature
	6.9 The Newtonian Tidal Tensor
	6.10 The Tidal and Non-tidal Components of a Gravitational Field
	Exercises
7 Einstein’s Field Equations
	7.1 Newtonian Fluid
	7.2 Perfect Fluids
		7.2.1 Lorentz Invariant Vacuum Energy—LIVE
		7.2.2 Energy–Momentum Tensor of an Electromagnetic Field
	7.3 Einstein’s Curvature Tensor
	7.4 Einstein’s Field Equations
	7.5 The “Geodesic Postulate” as a Consequence of the Field Equations
	7.6 Einstein’s Field Equations Deduced from a Variational Principle
	Exercises
8 Schwarzschild Spacetime
	8.1 Schwarzschild’s Exterior Solution
	8.2 Radial Free Fall in Schwarzschild Spacetime
	8.3 Light Cones in Schwarzschild Spacetime
	8.4 Analytical Extension of the Curvature Coordinates
	8.5 Embedding of the Schwarzschild Metric
	8.6 The Shapiro Experiment
	8.7 Particle Trajectories in Schwarzschild 3-Space
		8.7.1 Motion in the Equatorial Plane
	8.8 Classical Tests of Einstein’s General Theory of Relativity
		8.8.1 The Hafele–Keating Experiment
		8.8.2 Mercury’s Perihelion Precession
		8.8.3 Deflection of Light
	8.9 The Reissner–Nordström Spacetime
	Exercises
	References
9 The Linear Field Approximation and Gravitational Waves
	9.1 The Linear Field Approximation
	9.2 Solutions of the Linearized Field Equations
		9.2.1 The Gravitational Potential of a Point Mass
		9.2.2 Spacetime Inside and Outside a Rotating Spherical Shell
	9.3 Inertial Dragging
	9.4 Gravitoelectromagnetism
	9.5 Gravitational Waves
		9.5.1 What Sort of Gravitational Waves Is Predicted by Einstein’s Theory?
		9.5.2 Polarization of the Gravitational Waves
	9.6 The Effect of Gravitational Waves upon Matter
	9.7 The LIGO-Detection of Gravitational Waves
		9.7.1 Kepler’s Third Law and the Strain of the Detector
		9.7.2 Newtonian Description of a Binary System
		9.7.3 Gravitational Radiation Emission
		9.7.4 The Chirp
	References
10 Black Holes
	10.1 “Surface Gravity”: Acceleration of Gravity at the Horizon of a Black Hole
	10.2 Hawking Radiation: Radiation from a Black Hole
	10.3 Rotating Black Holes: The Kerr Metric
		10.3.1 Zero-Angular Momentum Observers
		10.3.2 Does the Kerr Spacetime Have a Horizon?
	Exercises
11 Sources of Gravitational Fields
	11.1 The Pressure Contribution to the Gravitational Mass of a Static, Spherically Symmetric System
	11.2 The Tolman–Oppenheimer–Volkoff Equation
	11.3 An Exact Solution for Incompressible Stars—Schwarzschild’s Interior Solution
	11.4 The Israel Formalism for Describing Singular Mass Shells in the General Theory of Relativity
	11.5 The Levi-Civita—Bertotti—Robinson Solution of Einstein’s Field Equations
	11.6 The Source of the Levi-Civita—Bertotti—Robinson Spacetime
	11.7 A Source of the Kerr–Newman Spacetime
	11.8 Physical Interpretation of the Components of the Energy–Momentum Tensor by Means of the Eigenvalues of the Tensor
	11.9 The River of Space
	Exercises
	References
12 Cosmology
	12.1 Co-moving Coordinate System
	12.2 Curvature Isotropy—The Robertson–Walker Metric
	12.3 Cosmic Kinematics and Dynamics
		12.3.1 The Hubble–Lemaître Law
		12.3.2 Cosmological Redshift of Light
		12.3.3 Cosmic Fluids
		12.3.4 Isotropic and Homogeneous Universe Models
		12.3.5 Cosmic Redshift
		12.3.6 Energy–Momentum Conservation
	12.4 Some LFRW Cosmological Models
		12.4.1 Radiation-Dominated Universe Model
		12.4.2 Dust-Dominated Universe Model
		12.4.3 Transition from Radiation-Dominated to Matter-Dominated Universe
		12.4.4 The de Sitter Universe Models
		12.4.5 The Friedmann–Lemaître Model
		12.4.6 Flat Universe with Dust and Phantom Energy
	12.5 Flat Anisotropic Universe Models
	12.6 Inhomogeneous Universe Models
		12.6.1 Dust-Dominated Model
		12.6.2 Inhomogeneous Universe Model with Dust and LIVE
	12.7 The Horizon and Flatness Problems
		12.7.1 The Horizon Problem
		12.7.2 The Flatness Problem
	12.8 Inflationary Universe Models
		12.8.1 Spontaneous Symmetry Breaking and the Higgs Mechanism
		12.8.2 Guth’s Inflationary Model [24]
		12.8.3 The Inflationary Models’ Answers to the Problems of the Friedmann Models
		12.8.4 Dynamics of the Inflationary Era
		12.8.5 Testing Observable Consequences of the Inflationary Era
	12.9 The Significance of Inertial Dragging for the Relativity of Rotation
		12.9.1 The Cosmic Causal Mass in the Einstein-de Sitter Universe
		12.9.2 The Cosmic Causal Mass in the Flat ΛCDM Universe
	Exercises
	References
Appendix  Kaluza–Klein Theory
A.1 The Structure of the Kaluza–Klein Theory
A.2 Calculation of the 5-dimensional Curvature Scalar
A.3 Field Equations for Kaluza–Klein Theory with  g55  = 1
A.4 The 5-dimensional Counterpart of Electric Charge
A.5 Quantization of Charge as a Consequence of Quantization of Momentum Along a Closed Path Around the Fifth Cylinder Dimension
A.6 Electric Field from Inertial Dragging in the Fifth Dimension
Solutions to the Exercises
Index