本书介绍了相对论的数学基础的相关理论知识。目标读者是数学、其他科学和工程专业的高年级本科生和研究生。读者应了解高等微积分的基础知识、一些解微分方程的技巧、一些线性代数知识以及集合论和群论的基础知识。本书适合对相对论的数学方面感兴趣的高年级本科生、研究生以及数学研究人员阅读参考。 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 G?del, 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.

前辅文
Preface
  Pedagogical Aims
  Humanistic Aims
  Special Features of This Book
  Other Works on the Subject
  Background Necessary to Read This Book
  Plan of the Work
  Acknowledgments
Part 1. The Special Theory
  Chapter 1. Time, Space, and Space-Time
   1. Simultaneity and Sequentiality
   2. Synchronization in Newtonian Mechanics
   3. An Asymmetry in Newtonian Mechanics: Electromagnetic Forces
   4. The Lorentz Transformation
   5. Contraction of Length and Time
   6. Composition of Parallel Velocities
   7. The Twin Paradox
   8. Relativistic Triangles
   9. Composition of Relativistic Velocities as a Binary Operation*
   10. Plane Trigonometry*
   11. The Lorentz Group*
   12. Closure of Lorentz Transformations under Composition*
   13. Rotational Motion and a Non-Euclidean Geometry*
   14. Problems
  Chapter 2. Relativistic Mechanics
   1. The Kinematics of a Particle
   2. From Kinematics to Dynamics: Mass and Momentum
   3. Relativistic Force
   4. Work, Energy, and the Famous E = mc2
   5. Newtonian Potential Energy
   6. Hamilton’s Principle
   7. The Newtonian Lagrangian
   8. The Relativistic Lagrangian
   9. Angular Momentum and Torque
   10. Four-Vectors and Tensors*
   11. Problems
  Chapter 3. Electromagnetic Theory*
   1. Charge and Charge Density
   2. Current and Current Density
   3. Transformation of Electric and Magnetic Fields
   4. Derivation of the Curl Equations from the Divergence Equations
   5. Problems
Part 2. The General Theory
Introduction to Part 2
  Chapter 4. Precession and Deflection
   1. Gravitation as Curvature of Space
   2. First Analysis: Newtonian Orbits
   3. Second Analysis: Newton’s Law with Relativistic Force
   4. Third Analysis: Newtonian Orbits as Geodesics
   5. Fourth Analysis: General Relativity
   6. Einstein’s Law of Gravity
   7. Computation of the Relativistic Orbit
   8. The Speed of Light
   9. Deflection of Light Near the Sun
   10. Problems
  Chapter 5. Concepts of Curvature, 1700–1850
   1. Differential Geometry
   2. Curvature, Phase 1: Euler
   3. Curvature, Phase 2: Gauss
   4. Problems
  Chapter 6. Concepts of Curvature, 1850–1950
   1. Second-Order Derivations
   2. Curvature, Phase 3: Riemann
   3. Parallel Transport
   4. The Exponential Mapping and Normal Coordinates
   5. Sectional Curvature
   6. The Laplace–Beltrami Operator
   7. Curvature, Phase 4: Ricci
   8. Problems
  Chapter 7. The Geometrization of Gravity
   1. The Einstein Field Equations
   2. Further Developments
   3. “Temporonautics” and the G¨odel Rotating Universe
   4. Black Holes
   5. Problems
Part 3. Historical and Philosophical Context
  Chapter 8. Experiments, Chronology, Metaphysics
   1. Experimental Tests of General Relativity
   2. Chronology
   3. Space and Time
   4. The Reality of Physical Concepts
   5. The Harmony Between Mathematics and the Physical World
   6. Knowledge of Hypothetical Objects: An Example
   7. Knowledge of the Physical World
   8. A Few Words from the Discoverers
   9. Epilogue: The Reception of Relativity
Bibliography
Subject Index
Name Index