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Full Description
The aim of this book is to provide a short but complete exposition of the logical structure of classical relativistic electrodynamics written in the language and spirit of coordinate-free differential geometry. The intended audience is primarily mathematicians who want a bare-bones account of the foundations of electrodynamics written in language with which they are familiar and secondarily physicists who may be curious how their old friend looks in the new clothes of the differential-geometric viewpoint which in recent years has become an important language and tool for theoretical physics. This work is not intended to be a textbook in electrodynamics in the usual sense; in particular no applications are treated, and the focus is exclusively the equations of motion of charged particles. Rather, it is hoped that it may serve as a bridge between mathemat- ics and physics. Many non-physicists are surprised to learn that the correct equation to describe the motion of a classical charged particle is still a matter of some controversy. The most mentioned candidate is the Lorentz-Dirac equation t .
However, it is experimentally unverified, is known to have no physically reasonable solutions in certain circumstances, and its usual derivations raise serious foundational issues. Such difficulties are not extensively discussed in most electrodynamics texts, which quite naturally are oriented toward applying the well-verified part of the subject to con- crete problems.
Contents
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.
