The Physical and Mathematical Foundations of the Theory of Relativity : A Critical Analysis

This unique textbook offers a mathematically rigorous presentation of the theory of relativity, emphasizing the need for a critical analysis of the foundations of general relativity in order to best study the theory and its implications. The transitions from classical mechanics to special relativity and then to general relativity are explored in detail as well, helping readers to gain a more profound and nuanced understanding of the theory as a whole. After reviewing the fundamentals of differential geometry and classical mechanics, the text introduces special relativity, first using the physical approach proposed by Einstein and then via Minkowski’s mathematical model. The authors then address the relativistic thermodynamics of continua and electromagnetic fields in matter – topics which are normally covered only very briefly in other treatments – in the next two chapters. The text then turns to a discussion of general relativity by means of the authors’ unique critical approach, underlining the difficulty of recognizing the physical meaning of some statements, such as the physical meaning of coordinates and the derivation of physical quantities from those of space-time. Chapters in this section cover the model of space-time proposed by Schwarzschild; black holes; the Friedman equations and the different cosmological models they describe; and the Fermi-Walker derivative. Well-suited for graduate students in physics and mathematics who have a strong foundation in real analysis, classical mechanics, and general physics, this textbook is appropriate for a variety of graduate-level courses that cover topics in relativity. Additionally, it will interest physicists and other researchers who wish to further study the subtleties of these theories and understand the contemporary scholarly discussions surrounding them. Preface......Page 5 Contents......Page 9 List of Figures......Page 15 Elements of Differential Geometry......Page 17 1.1 Linear Forms and Dual Vector Spaces......Page 18 1.2 Biduality......Page 20 1.3 Covariant 2-Tensors......Page 21 1.4 (r,s)-Tensors......Page 25 1.5 Contraction and Contracted Multiplication......Page 26 1.6 Skew-Symmetric (0, 2)-Tensors......Page 28 1.7 Skew-Symmetric (0,r)-Tensors......Page 31 1.8 Exterior Algebra......Page 34 1.9 Oriented Vector Spaces......Page 35 1.10 Representation Theorems for Symmetric and Skew-Symmetric (0,2)-Tensors......Page 37 1.11 Degenerate and Nondegenerate (0.2)-Tensors......Page 40 1.12 Pseudo-Euclidean Vector Spaces......Page 43 1.13 Euclidean Vector Spaces......Page 45 1.14 Eigenvectors of Euclidean 2-Tensors......Page 48 1.16 Exercises......Page 50 2.1 Historical Introduction......Page 54 2.2 Elements of the Geometry of Curves......Page 56 2.3 Elements of Geometry of Surfaces......Page 59 2.4 The Second Fundamental Form......Page 61 2.5 Parallel Transport and Geodesics......Page 65 2.6 An Example......Page 68 2.7 Riemann's Tensor and the Theorema Egregium......Page 72 2.8 Curvilinear Coordinates......Page 74 2.9 Differentiable Manifolds......Page 75 2.10 Differentiable Functions and Curves on Manifolds......Page 78 2.11 Tangent Vector Space......Page 80 2.12 Cotangent Vector Space......Page 82 2.13 Differential and Codifferential of a Map......Page 85 2.14 Tangent and Cotangent Fiber Bundles......Page 88 2.15 Riemannian Manifolds......Page 89 2.16 Geodesics over Riemannian Manifolds......Page 92 2.17 Exercises......Page 96 3.1 Global and Local One-Parameter Groups......Page 98 3.2 Lie Derivative......Page 101 3.3 Exterior Derivative......Page 105 3.4 Closed and Exact Differential Forms......Page 107 3.5 Properties of the Exterior Derivative......Page 109 3.6 An Introduction to the Integration of r-Forms......Page 110 3.7 Exercises......Page 114 4.1 Preliminary Considerations......Page 117 4.2 Affine Connection on Manifolds......Page 120 4.3 Parallel Transport and Autoparallel Curves......Page 122 4.4 Covariant Differential of Tensor Fields......Page 124 4.5 Torsion Tensor and Curvature Tensor......Page 125 4.6 Properties of the Riemann Tensor......Page 129 4.7 Geodesic Deviation......Page 131 4.8 Levi-Civita Connection......Page 132 4.9 Ricci Decomposition......Page 135 4.10 Differential Operators on a Riemannian Manifold......Page 138 4.11 Riemann's Theorem......Page 139 Newtonian Dynamics, Gravitation, and Cosmology......Page 143 5.1 Introduction......Page 144 5.2 Foundations of Classical Kinematics......Page 145 5.2.1 Change of the Frame of Reference......Page 146 5.2.2 Absolute and Relative Velocity and Acceleration......Page 149 5.3 Laws of Newtonian Mechanics......Page 150 5.3.1 Force Laws and the Action–Reaction Principle......Page 152 5.3.2 Newton's Second Law......Page 153 5.3.3 Dynamics in Noninertial Frames......Page 154 5.3.4 Restrictions on the Force Laws......Page 155 5.4 Collision Between Two Particles......Page 158 5.5 Galilean Principle of Relativity......Page 159 5.6 Comments About the Galilean Principle of Relativity......Page 160 5.7 Developments of Newtonian Mechanics......Page 162 5.8 Classical Thermodynamics of Continua......Page 163 5.9 Electromagnetic Fields and the Theory of Light......Page 169 5.10 Incompatibility Between Newtonian Mechanics and Electromagnetism......Page 171 6.1 Newton's Gravitational Law......Page 174 6.2 Newton's Theory of Gravitation of an Extended Body......Page 175 6.3 Asymptotic Behavior of the Gravitational Potential......Page 179 6.4 Local Inertial Frames and Tidal Forces......Page 182 6.5 Equilibrium of Self-gravitating Bodies......Page 185 6.6 Evolution of a Spherically Symmetric Self-gravitating Body......Page 190 6.7 Polytropic Transformations......Page 192 6.8 Lane–Emden Equation for Polytropic Gases......Page 194 6.9 Evolution of a Spherically Symmetric Perfect Gas......Page 196 6.10 Difficulties of Newtonian Gravitation......Page 202 6.11 Newtonian Cosmology: Kinematics......Page 203 6.12 Mass Balance and Motion of a Substratum......Page 207 Special Relativity......Page 211 7.1 The Optical Isotropy Principle......Page 212 7.2 The Lorentz Transformations......Page 213 7.2.1 The Special Lorentz Transformations......Page 214 7.2.2 The General Lorentz Transformations......Page 217 7.3 Relativistic Composition of Velocities and Accelerations......Page 221 7.4 A Different Approach to Lorentz Transformations......Page 223 7.5 Some Consequences of the Lorentz Transformations......Page 225 7.6 The Principle of Relativity......Page 229 7.7 Maxwell's Equations in Vacuum......Page 231 7.8 Relativistic Dynamics......Page 233 7.9 Transformation Formulas of Momentum and Energy......Page 236 7.10 Two Examples of Relativistic Dynamics......Page 238 7.11 Proper Time......Page 240 7.12.1 Approach to Relativistic Dynamics Based on Particle Collision......Page 241 7.12.2 Another Mechanical Formulation of Relativistic Dynamics......Page 242 7.13 Collision of Two Particles......Page 246 7.14 Experimental Verification of Relativistic Dynamics......Page 247 8.1 Minkowski Spacetime......Page 249 8.2 Physical Meaning of Minkowski Spacetime......Page 253 8.3 Classification of Lorentz Transformations......Page 255 8.4 Four-Dimensional Equation of Motion......Page 256 8.5 Tensor Formulation of Electromagnetism in a Vacuum......Page 258 8.6 Electromagnetic Potentials......Page 260 8.7 The Electromagnetic Momentum–Energy Tensor in Vacuum......Page 262 8.8 Exterior Algebra and Maxwell's Equations......Page 263 8.9 Spacetime Decomposition of 4-Tensors......Page 265 8.10 Infinitesimal Lorentz Transformations......Page 268 8.11 Fermi's Transport and Fermi's Derivative of a 4-Vector......Page 271 8.12 The Thomas Precession......Page 274 9.1 Relativistic Equations for Incoherent Matter......Page 277 9.2 Integral Laws of Balance......Page 279 9.3 The Momentum–Energy Tensor......Page 284 9.4 Intrinsic Deformation Gradient......Page 287 9.5 Relativistic Dissipation Inequality......Page 292 9.6 Thermoelastic Materials in Relativity......Page 295 9.7 On the Physical Meaning of Relative Quantities......Page 299 10.1 Maxwell's Equations in Matter......Page 302 10.2 About the Equivalence of Formulations of the Electrodynamics of Moving Bodies......Page 304 10.3 Minkowski's Description......Page 305 10.4 Ampère's Model......Page 310 10.5 Boffi's Formulation......Page 315 10.6 Chu's Formulation......Page 316 10.7 Final Remarks......Page 318 General Relativity and Cosmology......Page 320 11 Introduction to General Relativity......Page 321 11.1 Difficulties of Newtonian Gravitational Theory......Page 322 11.2 Attempts to Overcome the Difficulties of Newtonian Gravitational Theory......Page 323 11.3 Principles of General Relativity and General Covariance......Page 325 11.4 Principle of Equivalence......Page 330 11.5 The Spacetime of General Relativity......Page 331 11.6 Einstein's Gravitational Equations......Page 333 11.7 Experimental Determination of gαβ......Page 336 11.8 The Rotating Frame......Page 338 11.9 Variational Formulation of Gravitation......Page 343 11.10 Palatini's Variational Principle......Page 346 11.11 Conclusions and Perspectives......Page 349 12.1 Quasi-Minkowskian Spacetime......Page 351 12.2 Einstein's Linearized Equations......Page 355 12.3 Momentum–Energy Tensor for Weak Fields......Page 357 12.4 Static Matter Distribution......Page 359 12.5 Plane Waves......Page 361 12.6 Gravitational Wave Detection......Page 364 12.7 Gravitoelectromagnetism......Page 366 13.1 Cauchy's Problem and First Considerations......Page 368 13.2 About the Uniqueness of the Solution of Cauchy's Problem......Page 371 13.3 Mathematical Preliminaries......Page 372 13.4 Leray's Theorem......Page 374 13.5 Harmonic Coordinates......Page 375 13.6 Einstein's Equations in Harmonic Coordinates......Page 377 14.1 Gaussian Coordinates......Page 379 14.2 Matching Conditions......Page 381 14.3 Static and Stationary Spacetime......Page 382 14.4 Isometries and Killing's Vector Fields......Page 384 14.5 Three-Dimensional Spherically Symmetric Manifolds......Page 385 14.6 Schwarzschild's Exterior Solution......Page 389 14.7 Schwarzschild's Interior Solution......Page 393 14.8 Matching Interior and Exterior Solutions......Page 396 14.9 Physical Remarks About Schwarzschild's Solution......Page 399 14.10 Planetary Orbits in a Schwarzschild Field......Page 402 14.11 Gravitational Deflection of Light......Page 408 14.12 Gravitational Shift of Spectral Lines......Page 412 15.1 On the Singularity of Schwarzschild's Exterior Solution......Page 414 15.2 Physical Interpretation of the Event Horizon......Page 419 15.3 Gravitational Collapse......Page 422 15.4 Summary of Schwarzschild Spacetime and Black Holes......Page 425 15.5 Heuristic Derivation of the Kerr Metric......Page 427 15.6 Kerr Metric and Its Properties......Page 428 15.7 The Schwarzschild and Kerr Solutions......Page 429 15.8 Transformation of Ellipsoid Symmetric Orthogonal Coordinate......Page 430 15.9 A Solution of Einstein's Equations in Vacuum......Page 432 15.10 Consequences of Kerr's Solutions......Page 434 16.1 Global Properties of Spacetime......Page 435 16.2 On the Geometry of Space Sections......Page 437 16.3 Conservation Laws......Page 441 16.4 Friedmann's Equations......Page 442 16.5 Models of Universe for Λ=p=0......Page 444 16.6 Qualitative Analysis of Friedmann's Equations for Λ= p = 0......Page 446 16.7 Models of the Universe for p=0 and Λneq0......Page 448 17.1 Introduction......Page 451 17.2 Timelike Congruences......Page 457 17.3 The Fermi–Walker Derivative......Page 460 17.4 The Fermi–Walker Covariant Derivative......Page 462 17.5 F–W Derivation of 2-Tensors......Page 465 17.6 Frames of Reference......Page 466 17.7 Kinematic Characteristics of a Frame of Reference......Page 468 17.8 Relative Momentum Equation......Page 471 17.9 Relative Energy Equation......Page 475 17.10 Continuity Equation......Page 476 17.11 Divergence of a Skew-Symmetric Tensor......Page 478 17.12 Relative Maxwell's Equations......Page 482 17.13 Divergence of a Symmetric Tensor......Page 483 17.14 Momentum–Energy Tensor of Dust Matter......Page 484 BookmarkTitle:......Page 486 Index......Page 493