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This book has three main goals. First, it explores a selection of topics from the early period of the theory of relativity, focusing on particular aspects that are interesting or unusual. These include the twin paradox relativistic mechanics and its interaction with Maxwell's laws the earliest triumphs of general relativity relating to the orbit of Mercury and the deflection of light passing near the sun and the surprising bizarre metric of Kurt Godel, in which time travel is possible. Second, it provides an exposition of the differential geometry needed to understand these topics on a level that is intended to be accessible to those with just two years of university-level mathematics as background. Third, it reflects on the historical development of the subject and its significance for our understanding of what reality is and how we can know about the physical universe. The book also takes note of historical prefigurations of relativity, such as Euler's 1744 result that a particle moving on a surface and subject to no tangential acceleration will move along a geodesic, and the work of Lorentz and Poincare on space-time coordinate transformations between two observers in motion at constant relative velocity. The book is aimed at advanced undergraduate mathematics, science, and engineering majors (and, of course, at any interested person who knows a little university-level mathematics). The reader is assumed to know the rudiments of advanced calculus, a few techniques for solving differential equations, some linear algebra, and basics of set theory and groups. Cover 1 Title page 2 Contents 4 Preface 8 Pedagogical Aims 10 Humanistic Aims 13 Special Features of This Book 14 Other Works on the Subject 17 Background Necessary to Read This Book 18 Plan of the Work 19 Acknowledgments 20 Part 1 . The Special Theory 22 Chapter 1. Time, Space, and Space-Time 24 1. Simultaneity and Sequentiality 24 2. Synchronization in Newtonian Mechanics 27 3. An Asymmetry in Newtonian Mechanics: Electromagnetic Forces 40 4. The Lorentz Transformation 41 5. Contraction of Length and Time 47 6. Composition of Parallel Velocities 51 7. The Twin Paradox 53 8. Relativistic Triangles 56 9. Composition of Relativistic Velocities as a Binary Operation* 60 10. Plane Trigonometry* 66 11. The Lorentz Group* 69 12. Closure of Lorentz Transformations under Composition* 73 13. Rotational Motion and a Non-Euclidean Geometry* 78 14. Problems 85 Chapter 2. Relativistic Mechanics 92 1. The Kinematics of a Particle 92 2. From Kinematics to Dynamics: Mass and Momentum 96 3. Relativistic Force 100 4. Work, Energy, and the Famous E=mc2 106 5. Newtonian Potential Energy 108 6. Hamilton’s Principle 113 7. The Newtonian Lagrangian 114 8. The Relativistic Lagrangian 117 9. Angular Momentum and Torque 119 10. Four-Vectors and Tensors* 122 11. Problems 134 Chapter 3. Electromagnetic Theory* 136 1. Charge and Charge Density 137 2. Current and Current Density 139 3. Transformation of Electric and Magnetic Fields 140 4. Derivation of the Curl Equations from the Divergence Equations 143 5. Problems 145 Part 2 . The General Theory 146 Introduction to Part 2 148 Chapter 4. Precession and Deflection 150 1. Gravitation as Curvature of Space 152 2. First Analysis: Newtonian Orbits 153 3. Second Analysis: Newton’s Law with Relativistic Force 158 4. Third Analysis: Newtonian Orbits as Geodesics 161 5. Fourth Analysis: General Relativity 176 6. Einstein’s Law of Gravity 182 7. Computation of the Relativistic Orbit 187 8. The Speed of Light 198 9. Deflection of Light Near the Sun 200 10. Problems 204 Chapter 5. Concepts of Curvature, 1700–1850 210 1. Differential Geometry 211 2. Curvature, Phase 1: Euler 218 3. Curvature, Phase 2: Gauss 233 4. Problems 244 Chapter 6. Concepts of Curvature, 1850–1950 246 1. Second-Order Derivations 247 2. Curvature, Phase 3: Riemann 253 3. Parallel Transport 260 4. The Exponential Mapping and Normal Coordinates 269 5. Sectional Curvature 281 6. The Laplace–Beltrami Operator 285 7. Curvature, Phase 4: Ricci 308 8. Problems 318 Chapter 7. The Geometrization of Gravity 324 1. The Einstein Field Equations 325 2. Further Developments 335 3. “Temporonautics” and the Gödel Rotating Universe 336 4. Black Holes 341 5. Problems 346 Part 3 . Historical and Philosophical Context 350 Chapter 8. Experiments, Chronology, Metaphysics 352 1. Experimental Tests of General Relativity 353 2. Chronology 357 3. Space and Time 371 4. The Reality of Physical Concepts 382 5. The Harmony Between Mathematics and the Physical World 387 6. Knowledge of Hypothetical Objects: An Example 397 7. Knowledge of the Physical World 401 8. A Few Words from the Discoverers 405 9. Epilogue: The Reception of Relativity 407 Bibliography 410 Subject Index 414 Name Index 422 Back Cover 426 This book has three main goals. First, it explores a selection of topics from the early period of the theory of relativity, focusing on particular aspects that are interesting or unusual. These include the twin paradox; relativistic mechanics and its interaction with Maxwell's laws; the earliest triumphs of general relativity relating to the orbit of Mercury and the deflection of light passing near the sun; and the surprising bizarre metric of Kurt Gödel, in which time travel is possible. Second, it provides an exposition of the differential geometry needed to understand these topics on a level that is intended to be accessible to those with just two years of university-level mathematics as background. Third, it reflects on the historical development of the subject and its significance for our understanding of what reality is and how we can know about the physical universe. The book also takes note of historical prefigurations of relativity, such as Euler's 1744 result that a particle moving on a surface and subject to no tangential acceleration will move along a geodesic, and the work of Lorentz and Poincaré on space-time coordinate transformations between two observers in motion at constant relative velocity. The book is aimed at advanced undergraduate mathematics, science, and engineering majors (and, of course, at any interested person who knows a little university-level mathematics). The reader is assumed to know the rudiments of advanced calculus, a few techniques for solving differential equations, some linear algebra, and basics of set theory and groups.