Foundations of Classical Electrodynamics
by: F. W. Hehl, Yuri N. Obukhov
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This book presents a fresh, original exposition of the foundations of classical electrodynamics in the tradition of the so-called metric-free approach. The fundamental structure of classical electrodynamics is described in the form of six axioms: (1) electric charge conservation, (2) existence of the Lorentz force, (3) magnetic flux conservation, (4) localization of electromagnetic energy-momentum, (5) existence of an electromagnetic spacetime relation, and (6) splitting of the electric current into material and external pieces.The first four axioms require an arbitrary 4-dimensional differentiable manifold. The fifth axiom characterizes spacetime as the environment in which the electromagnetic field propagates ? a research topic of considerable interest ? and in which the metric tensor of spacetime makes its appearance, thus coupling electromagnetism and gravitation. Repeated emphasis is placed on interweaving the mathematical definitions of physical notions and the actual physical measurement procedures.The tool for formulating the theory is the calculus of exterior differential forms, which is explained in sufficient detail, along with the corresponding computer algebra programs. Prerequisites for the reader include a knowledge of elementary electrodynamics (with Maxwell's equations), linear algebra and elementary vector analysis; some knowledge of differential geometry would help. Foundations of Classical Electrodynamics unfolds systematically at a level suitable for graduate students and researchers in mathematics, physics, and electrical engineering.
Text displays the fundamental structure underlying classical electrodynamics; the phenomenological theory of electric and magnetic effects. Suitable as a text for an advanced course in theoretical electrodynamics for physics and mathematics students; or as a reference for researchers. DLC: Electrodynamics--Mathematics.
Reviews:
New view on Classical Electrodynamics: The differential geometric method has been one of the most fundamental tools for theoretical physicists since its first introduction into special relativity (general relativity) by Albert Einstein in 1905 (1915). Later it has been applied to many research areas, such as fluid mechanics, elastomechanics, thermodynamics, solid state physics, optics, electromagnetism, quantum field theory, etc. As a distinctive feature of traditional classical electrodynamics, this book rests on the metric-free integral formulation of the conservation laws of electrodynamics as represented by exterior differential forms. Therefore the book will be of great interest to graduate students and researchers in mathematics and theoretical physics who work in field theory and general relativity. The book consists of five parts; a short list of references and an author and a subject index are included. Every part ends with a list of references. The authors begin in Part A, as an introductory section, with an elementary presentation of exterior differential forms. The necessary geometric concepts, needed to formulate classical electrodynamics and gravitational theory in the language of differential forms, are explained in Part A and in Part C, too. The axioms of classical electrodynamics, the integral formulations of electric charge and magnetic flux conservation, are presented in Part B. Subsequently, the linear connection and the metric are introduced in Part C. The general framework is completed in Part D by a specific electrodynamic spacetime relation and in Part E by applying electrodynamics to moving continua and to rotating and accelerating observers, for instance. Moreover, a computer algebra program is introduced in the book in a simple way, and some cartoon drawings will add to the tedious mathematics some humor. As to the exposition of the book, we are impressed by illustrations and diagrams, which support our geometrical insight. The mathematical abstraction and physical relevance are displayed neatly and appropriately. It is concise and comprehensive as an introductory textbook for graduate students and a complete reference book for researchers. Thus, there is no doubt that many specialists will be interested in the book under review. The book proves to be a good scientific resource for university libraries as well as for graduate students and researchers working in mathematical physics, field theory, and general relativity.
A metric-free approach to Classical Electrodynamics: This is, in my opinion, the best book available on the foundations of Classical Electrodynamics. Using differential forms, the authors derive the two Maxwell equations (dF=0, dH=J) from four basic axioms in a metric-free approach. Only when two additional axioms are presented, the standard Maxwell-Lorentz theory in vacuum and in matter is developed by taking into account the metric structure of spacetime. Therefore, this framework allows for an almost trivial transition to the curved spacetime of general relativity. Moreover, the electromagnetic excitation H is considered as a microscopic field - whereas, conventionally, only the electromagnetic field strength F is considered as a truly microscopic field.