Relativistic Electrodynamics and Differential Geometry

1. Special Relativity.- 1.1 Coordinatizations of spacetime.- 1.2 Lorentz coordinatizations.- 1.3 Minkowski space.- 1.4 Lorentz transformations.- 1.5 Orientations.- 1.6 Spacetime diagrams and the metric tensor.- 1.7 Proper time and four-velocity.- 1.8 Mass and relativistic momentum.- Exercises 1.- 2. Mathematical Tools.- 2.1 Multilinear algebra.- 2.2 Alternating forms.- 2.3 Manifolds.- 2.4 Tangent spaces and vector fields.- 2.5 Covariant derivatives.- 2.6 Stokes' Theorem.- 2.7 The metric tensor.- 2.8 The covariant divergence.- 2.9 The equation d ? = ? and ?? = ?.- Exercises 2.- 3. The Electrodynamics of Infinitesimal Charges.- 3.1 Introduction.- 3.2 The Lorentz force law.- 3.3 The electromagnetic field tensor.- 3.4 The electric and magnetic fields.- 3.5 The first Maxwell equation.- 3.6 The second Maxwell equation.- 3.7 Potentials.- 3.8 The energy-momentum tensor.- Exercises 3.- 4. The Electrodynamics of Point Charges.- 4.1 Introduction.- 4.2 The retarded potentials and fields of a point particle.- 4.3 Radiation reaction and the Lorentz-Dirac equation.- 4.4 Calculation of the energy-momentum radiated by a point particle.- 4.5 Summary of the logical structure of electrodynamics.- Exercises 4.- 5. Further Difficulties and Alternate Approaches.- 5.1 The Cauchy problem for the Maxwell-Lorentz system.- 5.2 Spherically symmetric solutions of the Maxwell-Lorentz system.- 5.3 Nonexistence of global solutions of the Maxwell-Lorentz system.- 5.4 An alternate fluid model.- 5.5 Peculiar solutions of the Lorentz-Dirac equation.- 5.6 Evidence for the usual energy-momentum tensor.- 5.7 Alternate energy-momentum tensors and equations of motion.- Appendix on Units.- Solutions to Exercises.- Appendix 2.- Table of Notations.