Introduction to Einstein’s Theory of Relativity: From Newton’s Attractive Gravity to the Repulsive Gravity of Vacuum Energy (Undergraduate Texts in Physics)

The revised and updated 2 nd edition of this established textbook provides a self-contained introduction to the general theory of relativity, describing not only the physical principles and applications of the theory, but also the mathematics needed, in particular the calculus of differential forms. Updated throughout, the book contains more detailed explanations and extended discussions of several conceptual points, and strengthened mathematical deductions where required. It includes examples of work conducted in the ten years since the first edition of the book was published, for example the pedagogically helpful concept of a "river of space" and a more detailed discussion of how far the principle of relativity is contained in the general theory of relativity. Also presented is a discussion of the concept of the 'gravitational field' in Einstein's theory, and some new material concerning the 'twin paradox' in the theory of relativity. Finally, the book contains a new section about gravitational waves, exploring the dramatic progress in this field following the LIGO observations. Based on a long-established masters course, the book serves advanced undergraduate and graduate level students, and also provides a useful reference for researchers. Preface to the Second Edition 6 Preface to the First Edition 7 Contents 8 List of Figures 14 List of Definitions 17 List of Examples 19 List of Exercises 21 1 Newton’s Theory of Gravitation 23 1.1 The Force Law of Gravitation 24 1.2 Newton’s Law of Gravitation in Local Form 26 1.3 Newtonian Incompressible Star 29 1.4 Tidal Forces 32 1.5 The Principle of Equivalence 36 1.6 The General Principle of Relativity 39 1.7 The Covariance Principle 39 1.8 Mach’s Principle 40 1.9 Exercises 41 References 44 2 The Special Theory of Relativity 45 2.1 Coordinate Systems and Minkowski Diagrams 45 2.2 Synchronization of Clocks 47 2.3 The Doppler Effect 48 2.4 Relativistic Time Dilation 50 2.5 The Relativity of Simultaneity 52 2.6 The Lorentz Contraction 55 2.7 The Lorentz Transformation 56 2.8 Lorentz Invariant Interval 59 2.9 The Twin Paradox 62 2.10 Hyperbolic Motion 63 2.11 Energy and Mass 66 2.12 Relativistic Increase of Mass 67 2.13 Lorentz Transformation of Velocity, Momentum, Energy and Force 69 2.14 Tachyons 72 2.15 Magnetism as a Relativistic Second-Order Effect 73 Exercises 76 Reference 80 3 Vectors, Tensors and Forms 81 3.1 Vectors 81 3.1.1 Four-Vectors 82 3.1.2 Tangent Vector Fields and Coordinate Vectors 84 3.1.3 Coordinate Transformations 87 3.1.4 Structure Coefficients 90 3.2 Tensors 91 3.2.1 Transformation of Tensor Components 93 3.2.2 Transformation of Basis One-Forms 93 3.2.3 The Metric Tensor 94 3.3 The Causal Structure of Spacetime 98 3.4 Forms 100 3.4.1 The Volume Form 102 3.4.2 Dual Forms 104 Exercises 107 4 Accelerated Reference Frames 110 4.1 The Spatial Metric Tensor 110 4.2 Einstein Synchronization of Clocks in a Rotating Reference Frame 113 4.3 Angular Acceleration in the Rotating Frame 116 4.4 Gravitational Time Dilation 119 4.5 Path of Photons Emitted from the Axis in a Rotating Reference Frame 120 4.6 The Sagnac Effect 120 4.7 Non-integrability of a Simultaneity Curve in a Rotating Frame 122 4.8 Orthonormal Basis Field in a Rotating Frame 123 4.9 Uniformly Accelerated Reference Frame 126 4.10 The Projection Tensor 134 Exercises 136 5 Covariant Differentiation 140 5.1 Differentiation of Forms 140 5.1.1 Exterior Differentiation 140 5.1.2 Covariant Derivative 143 5.2 The Christoffel Symbols 143 5.3 Geodesic Curves 146 5.4 The Covariant Euler–Lagrange Equations 148 5.5 Application of the Lagrange Formalism to Free Particles 150 5.5.1 Equation of Motion from Lagrange’s Equations 150 5.5.2 Geodesic World Lines in Spacetime 154 5.5.3 Acceleration of Gravity 156 5.5.4 Gravitational Shift of Wavelength 159 5.6 Connection Coefficients 161 5.6.1 Structure Coefficients 164 5.7 Covariant Differentiation of Vectors, Forms and Tensors 165 5.7.1 Covariant Differentiation of Vectors 165 5.7.2 Covariant Differentiation of Forms 166 5.7.3 Covariant Differentiation of Tensors of Arbitrary Rank 167 5.8 The Cartan Connection 168 5.9 Covariant Decomposition of a Velocity Field 172 5.9.1 Newtonian 3-Velocity 172 5.9.2 Relativistic 4-Velocity 174 5.10 Killing Vectors and Symmetries 176 5.11 Covariant Expressions for Gradient, Divergence, Curl, Laplacian and D’Alembert’s Wave Operator 178 5.12 Electromagnetism in Form Language 184 Exercises 190 6 Curvature 194 6.1 The Riemann Curvature Tensor 194 6.2 Differential Geometry of Surfaces 200 6.2.1 Surface Curvature Using the Cartan Formalism 204 6.3 The Ricci Identity 205 6.4 Bianchi’s 1. Identity 206 6.5 Bianchi’s 2. Identity 207 6.6 Torsion 208 6.7 The Equation of Geodesic Deviation 209 6.8 Tidal Acceleration and Spacetime Curvature 211 6.9 The Newtonian Tidal Tensor 212 6.10 The Tidal and Non-tidal Components of a Gravitational Field 213 Exercises 216 7 Einstein’s Field Equations 218 7.1 Newtonian Fluid 218 7.2 Perfect Fluids 220 7.2.1 Lorentz Invariant Vacuum Energy—LIVE 221 7.2.2 Energy–Momentum Tensor of an Electromagnetic Field 222 7.3 Einstein’s Curvature Tensor 222 7.4 Einstein’s Field Equations 223 7.5 The “Geodesic Postulate” as a Consequence of the Field Equations 225 7.6 Einstein’s Field Equations Deduced from a Variational Principle 227 Exercises 231 8 Schwarzschild Spacetime 232 8.1 Schwarzschild’s Exterior Solution 232 8.2 Radial Free Fall in Schwarzschild Spacetime 238 8.3 Light Cones in Schwarzschild Spacetime 239 8.4 Analytical Extension of the Curvature Coordinates 243 8.5 Embedding of the Schwarzschild Metric 246 8.6 The Shapiro Experiment 247 8.7 Particle Trajectories in Schwarzschild 3-Space 249 8.7.1 Motion in the Equatorial Plane 250 8.8 Classical Tests of Einstein’s General Theory of Relativity 253 8.8.1 The Hafele–Keating Experiment 253 8.8.2 Mercury’s Perihelion Precession 254 8.8.3 Deflection of Light 257 8.9 The Reissner–Nordström Spacetime 259 Exercises 261 References 263 9 The Linear Field Approximation and Gravitational Waves 264 9.1 The Linear Field Approximation 264 9.2 Solutions of the Linearized Field Equations 267 9.2.1 The Gravitational Potential of a Point Mass 267 9.2.2 Spacetime Inside and Outside a Rotating Spherical Shell 268 9.3 Inertial Dragging 271 9.4 Gravitoelectromagnetism 272 9.5 Gravitational Waves 274 9.5.1 What Sort of Gravitational Waves Is Predicted by Einstein’s Theory? 276 9.5.2 Polarization of the Gravitational Waves 277 9.6 The Effect of Gravitational Waves upon Matter 278 9.7 The LIGO-Detection of Gravitational Waves 281 9.7.1 Kepler’s Third Law and the Strain of the Detector 283 9.7.2 Newtonian Description of a Binary System 286 9.7.3 Gravitational Radiation Emission 287 9.7.4 The Chirp 288 References 290 10 Black Holes 291 10.1 “Surface Gravity”: Acceleration of Gravity at the Horizon of a Black Hole 291 10.2 Hawking Radiation: Radiation from a Black Hole 293 10.3 Rotating Black Holes: The Kerr Metric 295 10.3.1 Zero-Angular Momentum Observers 296 10.3.2 Does the Kerr Spacetime Have a Horizon? 297 Exercises 299 11 Sources of Gravitational Fields 303 11.1 The Pressure Contribution to the Gravitational Mass of a Static, Spherically Symmetric System 303 11.2 The Tolman–Oppenheimer–Volkoff Equation 305 11.3 An Exact Solution for Incompressible Stars—Schwarzschild’s Interior Solution 307 11.4 The Israel Formalism for Describing Singular Mass Shells in the General Theory of Relativity 310 11.5 The Levi-Civita—Bertotti—Robinson Solution of Einstein’s Field Equations 315 11.6 The Source of the Levi-Civita—Bertotti—Robinson Spacetime 317 11.7 A Source of the Kerr–Newman Spacetime 319 11.8 Physical Interpretation of the Components of the Energy–Momentum Tensor by Means of the Eigenvalues of the Tensor 322 11.9 The River of Space 325 Exercises 329 References 329 12 Cosmology 330 12.1 Co-moving Coordinate System 330 12.2 Curvature Isotropy—The Robertson–Walker Metric 331 12.3 Cosmic Kinematics and Dynamics 333 12.3.1 The Hubble–Lemaître Law 333 12.3.2 Cosmological Redshift of Light 334 12.3.3 Cosmic Fluids 336 12.3.4 Isotropic and Homogeneous Universe Models 337 12.3.5 Cosmic Redshift 342 12.3.6 Energy–Momentum Conservation 345 12.4 Some LFRW Cosmological Models 349 12.4.1 Radiation-Dominated Universe Model 349 12.4.2 Dust-Dominated Universe Model 350 12.4.3 Transition from Radiation-Dominated to Matter-Dominated Universe 354 12.4.4 The de Sitter Universe Models 355 12.4.5 The Friedmann–Lemaître Model 356 12.4.6 Flat Universe with Dust and Phantom Energy 367 12.5 Flat Anisotropic Universe Models 370 12.6 Inhomogeneous Universe Models 374 12.6.1 Dust-Dominated Model 375 12.6.2 Inhomogeneous Universe Model with Dust and LIVE 376 12.7 The Horizon and Flatness Problems 377 12.7.1 The Horizon Problem 377 12.7.2 The Flatness Problem 379 12.8 Inflationary Universe Models 380 12.8.1 Spontaneous Symmetry Breaking and the Higgs Mechanism 380 12.8.2 Guth’s Inflationary Model [24] 382 12.8.3 The Inflationary Models’ Answers to the Problems of the Friedmann Models 383 12.8.4 Dynamics of the Inflationary Era 385 12.8.5 Testing Observable Consequences of the Inflationary Era 391 12.9 The Significance of Inertial Dragging for the Relativity of Rotation 396 12.9.1 The Cosmic Causal Mass in the Einstein-de Sitter Universe 397 12.9.2 The Cosmic Causal Mass in the Flat ΛCDM Universe 399 Exercises 401 References 409 Appendix Kaluza–Klein Theory 411 A.1 The Structure of the Kaluza–Klein Theory 411 A.2 Calculation of the 5-dimensional Curvature Scalar 413 A.3 Field Equations for Kaluza–Klein Theory with g55 = 1 418 A.4 The 5-dimensional Counterpart of Electric Charge 419 A.5 Quantization of Charge as a Consequence of Quantization of Momentum Along a Closed Path Around the Fifth Cylinder Dimension 422 A.6 Electric Field from Inertial Dragging in the Fifth Dimension 423 Solutions to the Exercises 426 Index 527