Essential Relativity

In retrospect, the first edition of this book now seems like a mere sketch for a book. The present version is, if not the final product, at least a closer approximation to it. The table of contents may show little change. But that is simply because the original organization of the material has been found satisfactory. Also the basic purpose of the book remains the same, and that is to make relativity come alive conceptually. I have always felt much sym­ pathy with Richard Courant's maxim (as reported and exemplified by Pascual Jordan) that, ideally, proofs should be reached by comprehension rather than computation. Where computations are necessary, I have tried to make them as transparent as possible, so as not to hinder the progress of comprehension. Among the more obvious changes, this edition contains a new section on Kruskal space, another on the plane gravitational wave, and a third on linearized general relativity; it also contains many new exercises, and two appendices: one listing the curvature components for the diagonal metric (in a little more generality than the old" Dingle formulas "), and one syn­ thesizing Maxwell's theory in tensor form. But the most significant changes and additions have occurred throughout the text. Many sections have been completely rewritten, many arguments tightened, many "asides" added, and, of course, recent developments taken into account.

1 The Rise and Fall of Absolute Space.- 1.1 Definition of Relativity.- 1.2 Newton’s Laws.- 1.3 The Galilean Transformation.- 1.4 The Set of All Inertial Frames.- 1.5 Newtonian Relativity.- 1.6 Newton’s Absolute Space.- 1.7 Objections to Newton’s Absolute Space.- 1.8 Maxwell’s Ether.- 1.9 Where is Maxwell’s Ether?.- 1.10 Lorentz’s Ether Theory.- 1.11 The Relativity Principle.- 1.12 Arguments for the Relativity Principle.- 1.13 Maxwellian Relativity.- 1.14 Origins of General Relativity.- 1.15 Mach’s Principle.- 1.16 Consequences of Mach’s Principle.- 1.17 Cosmology.- 1.18 Inertial and Gravitational Mass.- 1.19 The Equivalence Principle.- 1.20 The Semistrong Equivalence Principle.- 1.21 Consequences of the Equivalence Principle.- 2 Einsteinian Kinematics.- 2.1 Basic Features of Special Relativity.- 2.2 On the Nature of Physical Laws.- 2.3 An Archetypal Relativistic Argument.- 2.4 The Relativity of Simultaneity.- 2.5 The Coordinate Lattice.- 2.6 The Lorentz Transformation.- 2.7 Properties of the Lorentz Transformation.- 2.8 Hyperbolic Forms of the Lorentz Transformation.- 2.9 Graphical Representation of the Lorentz Transformation.- 2.10 World-picture and World-map.- 2.11 Length Contraction.- 2.12 Length Contraction Paradoxes.- 2.13 Time Dilation.- 2.14 The Twin Paradox.- 2.15 Velocity Transformation.- 2.16 Proper Acceleration.- 2.17 Special Relativity without the Second Postulate.- 3 Einsteinian Optics.- 3.1 The Drag Effect.- 3.2 The Doppler Effect.- 3.3 Aberration and the Visual Appearance of Moving Objects.- 4 Spacetime and Four-Vectors.- 4.1 Spacetime.- 4.2 Three-Vectors.- 4.3 Four-Vectors.- 4.4 Four-Tensors.- 4.5 The Three-Dimensional Minkowski Diagram.- 4.6 Wave Motion.- 5 Relativistic Particle Mechanics.- 5.1 Domain of Sufficient Validity of Newton’s Laws.- 5.2 Why Gravity Does not Fit Naturally into Special Relativity.- 5.3 Relativistic Inertial Mass.- 5.4 Four-Vector Formulation of Relativistic Mechanics.- 5.5 A Note on Galilean Four-Vectors.- 5.6 Equivalence of Mass and Energy.- 5.7 The Center of Momentum Frame.- 5.8 Relativistic Billiards.- 5.9 Threshold Energies.- 5.10 Three-Force and Four-Force.- 5.11 De Broglie Waves.- 5.12 Photons. The Compton Effect.- 5.13 The Energy Tensor of Dust.- 6 Relativity and Electrodynamics.- 6.1 Transformation of the Field Vectors.- 6.2 Magnetic Deflection of Charged Particles.- 6.3 The Field of a Uniformly Moving Charge.- 6.4 The Field of an Infinite Straight Current.- 7 Basic Ideas of General Relativity.- 7.1 Curved Surfaces.- 7.2 Curved Spaces of Higher Dimensions.- 7.3 Riemannian Spaces.- 7.4 A Plan for General Relativity.- 7.5 The Gravitational Doppler Effect.- 7.6 Metric of Static Fields.- 7.7 Geodesics in Static Fields.- 8 Formal Development of General Relativity.- 8.1 Tensors in General Relativity.- 8.2 The Vacuum Field Equations of General Relativity.- 8.3 The Schwarzschild Solution.- 8.4 Rays and Orbits in Schwarzschild Space.- 8.5 The Schwarzschild Horizon, Gravitational Collapse, and Black Holes.- 8.6 Kruskal Space and the Uniform Acceleration Field.- 8.7 A General-Relativistic “Proof” of E = mc2.- 8.8 A Plane-Fronted Gravity Wave.- 8.9 The Laws of Physics in Curved Spacetime.- 8.10 The Field Equations in the Presence of Matter.- 8.11 From Modified Schwarzschild to de Sitter Space.- 8.12 The Linear Approximation to GR.- 9 Cosmology.- 9.1 The Basic Facts.- 9.2 Apparent Difficulties of Prerelativistic Cosmology.- 9.3 Cosmological Relativity: The Cosmological Principle.- 9.4 Milne’s Model.- 9.5 The Robertson-Walker Metric.- 9.6 Rubber Models, Red Shifts, and Horizons.- 9.7 Comparison with Observation.- 9.8 Cosmic Dynamics According to Pseudo-Newtonian Theory.- 9.9 Cosmic Dynamics According to General Relativity.- 9.10 The Friedmann Models.- 9.11 Once Again, Comparison with Observation.- 9.12 Mach’s Principle Reexamined.- Appendices.- Appendix I: Curvature Tensor Components for the Diagonal Metric.- Appendix II: How to “Invent” Maxwell’s Theory.- Exercises.