Foundations of classical electrodynamics - Charge, flux, and metric

This book presents the foundations of classical electrodynamics with particular attention to mathematical axioms based on physical concepts. The text is divided into 4 parts and unfolds systematically at a level suitable for advanced undergraduates and graduates in mathematics and physics.

Basic mathematical tools that motivate and supply the mathematical framework of the book are introduced patiently and carefully in the Introduction, Parts A and C: Vector spaces and their duals; tensors and their geometric significance; differential forms; exterior calculus and cohomology; integration on manifolds and the general Stokes theorem for p-forms. Three fundamental physical axioms are presented in Part B: The conservation of electric charge and magnetic flux, and the structure of the Lorentz force, which yield Maxwell's equations, derived in integral form, tensor form, and differential form. The formulation of the quantum Hall effect by means of a topological (Chern--Simons) Lagrangian is imminent in this approach to viewing electrodynamics.

More mathematical tools in Part C provide a geometric understanding of electromagnetic propagation through curved spacetime (gravitation): Linear connections, covariant differentiation, Hodge duality, torsion and curvature. Included here is a computerized matrix algebra program to help work out the practical examples presented.

Finally, a fourth axiom is introduced in Part D to describe propagation through material media in a general relativistic setting. From this viewpoint, the so-called "constitutive relations" include curved vacuum spacetime as a special case. This last sector is at the forefront of current research and includes many recent ideas that have not previously appeared in book form.

Contents

• Introduction
• Mathematics: Some Exterior Calculus
  • Algebra
  • Exterior Calculus
  • Integration on a Manifold
• Axioms of Classical Electrodynamics
  • Electric Charge Conservation
  • Lorentz Force
  • Magnetic Flux Conservation
  • Basic Classical Electrodynamics Summarized
  • Electromagnetic Energy-Momentum Current and Action
• More Mathematics
  • Linear Connection
  • Metric
• The Maxwell-Lorentz Spacetime Relation
  • A linear realtion between H and F
  • Propagation of electromagnetic waves : Quartic wave surface
  • First constraint : Electric / magnetic reciprocity
  • Second constraint : Vanishing skewon field. Emergence of the light cone
  • Extracting the metric by an alternative method
  • Fifth axiom : Maxwell-Lorentz spacetime relation
• Electrodynamics in Vacuum and in Matter
  • Standard Maxwell-Lorentz theory in vacuum
  • Electromagnetic spacetime relations beyond locality and linearity
  • Electrodynamics in matter, constitutive law
  • Electrodynamics of moving continua
• Index