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Relativistic electrodynamics and differential geometry. (English) Zbl 0609.53045
New York etc.: Springer-Verlag. XI, 308 p. DM 110.00 (1987).
There is some truth in the statement that mathematicians often find physics texts hard to read because of frequent looseness of language and lack of careful definitions, and physicists in turn often complain that mathematics texts give too little attention to motivation and too much to generalities, while furthermore being clothed in a labouring ”definition- \(lemma\)-\(theorem\)-\(proof\)-\(corollary\)-\(definition''\) style. In the excellent book under review the author has succeeded admirably in treading a middle path between these extremes. He also fully achieves his stated aim 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. Whereas the book is intended primarily for mathematicians who want an account on the topic in a familiar language, the secondary readership it addresses would consist of physicists who are interested to see how the subject looks in terms of the viewpoint of differential geometry (which has become such an important language and tool for theoretical physics). The style of presentation is relaxed and fairly informal (but not at the cost of exactness), and the book reads well.
There are 5 chapters: 1. Special relativity; 2. Mathematical tools; 3. The electrodynamics of infinitesimal charges; 4. The electrodynamics of point charges; 5. Further difficulties and alternate approaches. Then there is a useful appendix on units and a second appendix giving a technical proof of a result in the text, plus 46 pages of ”solutions to exercises”. A bibliography of approximately 90 entries and a table with notations completes a formal inventory of the contents.
Now some more detailed comments on the book. Its central theme are the difficulties which evidently still very much exist in the determination of the correct equation to describe the motion of a classical charged particle. Without taking any one-sided position, the author presents the basic issues very clearly and rather precisely, preferring the reader to draw his or her own conclusions. More about this presently, but first something about the background required from the reader. A familiarity with the language of modern mathematics is assumed, and acquaintance with electromagnetic theory on the level of that of a good freshman’s physics course seems quite essential. In particular, a working knowledge of special relativity and abstract differential geometry, while not really a prerequisite, will have to be acquired while reading the text; for this ample and useful references are provided at appropriate places. The first two chapters in a sense are providing this service for these topics, but more in the spirit of refresher courses than as introduction (for instance at times results are described somewhat informally, such as Stokes’s theorem in 2.6); a newcomer to the field will definitely have to spend some time with additional reading.
Chapters 3 and 4 in essence give a very good expository synthesis (in a rather novel way) of fairly standard material on that part of electrodynamics which deals with continuous distributions of charge (Chapter 3), and radiation, plus a presentation of the usual motivation leading to the Lorentz-Dirac equation (Chapter 4). Section 4.4 contains some new material: in it differential-geometric ideas are applied to clarify radiation calculations usually done in less transparent ways. The author stresses and makes use of the great simplification which one obtains by using the so-called Bhabha tube (for which reference is made to [H. J. Bhasha, Proc. Indian Acad. Sci. 10, 324-332 (1939; Zbl 0023.42703)], which, the author asserts, ”surprisingly has not been generally adopted in textbooks even up to the present”). Already in Chapter 4, while the most widely accepted version of the theory of point electrodynamics is described some of the rather nontrivial problems associated with it are pointed out, for instance the somewhat disturbing fact that the Lorentz-Dirac equation does not necessarily guarantee conservation of energy in any practical sense.
The most significant original contribution of the book lies in the last chapter. In it further difficulties with the usual formulation of electrodynamics are explored and alternate approaches discussed. It rests on papers from the research literature and also contains some new results in their own right. Particularly noteworthy here is Section 5.5, entitled ”Peculiar solutions of the Lorentz-Dirac equation”, which presents a proof of Eliezer’s theorem on non-existence of physical solutions of the Lorentz-Dirac equation for one-dimensional symmetric motion of opposite charges. An implication of this theorem is that if the Lorentz-Dirac equation holds, then two oppositely charged point charges released at rest can never collide, and will eventually recede at velocities asymptotic to that of light (yielding infinite kinetic energy). This obviously should cast serious doubt on the Lorentz-Dirac equation.
It nearly goes without saying that the book raises quite a number of explicitly open problems. Perhaps it also must be noted at this point that the author stresses the fact that he is a mathematician and, at least where he begins the part on the physical evidence for the usual energy-momentum tensor in Section 5.6, he states that he is ”venturing beyond [his] domain of primary competence”. The highlight of the book (call it ”punch-line” if you will) is the final section (5.7) entitled ”Alternate energy-momentum tensors and equations of motion”, which is well worth taking note of. In fact, this part alone should make it essential reading matter for anyone actively interested in the topic, be she or he a mathematician or, in particular, a physicist.
In summary then, this book is remarkably successful in its aim of presenting the reader with a rather complete and state-of-the-art picture of the logical structure of classical electrodynamics, and it may well become a classic in its field. Another impressive feature which must be mentioned is the large set of exercises. Ther are 72 of them, very few of which can be described to be of the routine variety. They are mostly supplementary to the text and are formulated in an instructive and meaningful way, quite a few of the results to which the reader is led thus being of quite some significance in their own right. In addition, what is particularly noteworthy and regrettably seldom found in other books of this nature, is that rather detailed solutions are provided (comprising 50 pages at the end of the text). This certainly adds to the value of the book!
Whereas very few misprints were detected (the most serious one being that the reference [SPIVAK] cited on page 53 is not listed in the bibliography at the end), the appearance of the book suffers a bit from the fact that different types of fonts are used, for instance that in the list of notations differs from the rest while some obviously final alterations were added rather informally in very small print. However, these are minor matters in such a fine book.
Reviewer: H.S.P.Gräser

53C80 Applications of global differential geometry to the sciences
53B50 Applications of local differential geometry to the sciences
83C50 Electromagnetic fields in general relativity and gravitational theory
78A35 Motion of charged particles
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory