Dr. August Kopff. Grundzüge der Einsteinschen Relativitätstheorie, pp. IV + 198. Verlag von S. Hirzel. Leipzig, 1921.

A. Kopff, professor of astronomy at the University of Heidelberg, in the winter semester of 1919–1920 and the summer of 1920 delivered a course of lectures on the theory of relativity, the result of which is the present book. Its aim, as defined by the author, is to provide an initial introduction to the theory of relativity, presupposing in the reader a knowledge of physics and mathematics such as is given in the first semesters of a higher technical school. The book is divided into two parts: the first, Chapters 1–9, devoted to the “special theory of relativity,” and the second, Chapters 10–14, in which the author expounds the “general principle” of relativity. In the first five chapters the author sets forth the foundations of the “special principle,” the Lorentz transformation and its geometrical representation, given by H. Minkowski, and then derives the geometrical and mechanical consequences. Proceeding further to the electrodynamics of empty space, the mechanics of the “special principle,” and the question of matter and energy (Chs. 8 and 9), the author first gives a very brief account (Ch. 6) of vector analysis and the equations of electrodynamics, concluding the sixth chapter with an exposition of the fundamental concepts concerning tensors. In the seventh chapter a general tensor analysis is presented, which the author also uses subsequently.

The exposition is distinguished by simplicity and clarity, but we think that chapters (6–9) are presented somewhat briefly, and that the information given in the first semesters of higher school is by no means sufficient for a reader satisfying this requirement to be able fully to master the material presented to him.

Among the merits of the exposition one should include its impartiality. The author writes: “The circumstance that the special theory of relativity, expounded with the aid of the Lorentz transformation, just like electron theory, is able to encompass the phenomena, has a deep basis in the fact that in principle they are both identical. The fundamental equations of electrodynamics in the Lorentz–Maxwell form correspond to the equations of the special principle; thus all phenomena explained by electron theory can also be represented by the theory of relativity.”

Of course, the author points out that what is essential in electron theory is the assumption that for its fundamental equations there exists a preferred system of coordinates, which is provided by the immobile ether.

The second part (ch. 10) begins with an exposition of the principle of equivalence: “Every observed uniform acceleration of a material point presented by itself can be described either as the accelerated motion of a heavy mass in a resting homogeneous gravitational field, or as the uniform motion of an inertial mass in a coordinate system devoid of a gravitational field and having acceleration.”

Proceeding from this principle, the author derives the shift of spectral lines toward the red end of the spectrum as a function of the increase of the potential of the gravitational field, and sets out the possibilities that would be available to astronomers if this explanation satisfied the systematic shifts of spectral lines toward the red end in stellar spectra.

Next (ch. 11) the connection of the “general principle” with Riemannian geometry is presented, after which (ch. 12) the author again turns to an exposition of tensor analysis, applying it especially to the derivation of the fundamental equations of the general principle.

Chapter 13 is devoted to Einstein’s theory of gravitation, and in it the equations of the gravitational field are derived in the form of the so-called equations of the first, second, and third kind.

In chapter 14, from the equations of the first kind, the author derives the motion of planets around a central body, the path of a light ray in a gravitational field, and investigates the motion of Mercury’s perihelion. The conclusion of chapter 14 is devoted to the most interesting question of rotational motion.

We believe that, despite the already mentioned brevity of exposition, this book is an excellent introduction for one studying the theory of relativity, and one should wish for its appearance in Russian translation. The auxiliary material that the author sets out in the chapters,

devoted to vector analysis and especially to tensor analysis, in our view nevertheless requires a whole series of additions, since a number of the most important equations are given without derivations, which are probably inaccessible to the majority of readers.

M. Shuleikin.