# Just relatively simple

## An Introduction to Special Relativity and its Applications

The discovery of the mathematical laws of perspective by Brunelleschi and others in the 15th century did not lead to a new understanding of the nature of solid objects. Nor should we expect it to have done so. The viewpoint from which an object is observed has little bearing on its intrinsic nature. Why should mere epistemology tell us anything about the nature of things as they are? Yet that is what happened when, early this century, a mathematical theory ostensibly concerned with the relationships between measurements made by observers in different states of motion led to a radical revision of the laws of motion and much else besides.

The analogy between the laws of perspective and the Lorentz transformation which underlies the special theory of relativity must not be pressed too far, but it is noteworthy that both are conditioned by the properties of light: rectilinear propagation in the first case, and invariance of speed in the second. The first, of course, deals with the representation of a static three-dimensional world on a two-dimensional plane; the second, with the representation of a moving three-dimensional world by a four-dimensional map in which the orientation of the gridlines depends on the motion of the real-world observer. The fact that a theory which, at root, is predominantly epistemological can lead to such profound consequences for our understanding of the physical world is remarkable.

The late Neville Robinson's book on the special theory of relativity (the general theory is not within its scope) is the product of prolonged reflection sharpened by years of teaching and examining bright undergraduates. Robinson's motivation for writing yet another text on the subject may be summarised by an extract from the preface: "I found that although most students could manipulate the formulae of special relativity, very few could explain, even qualitatively, where the notion of a variable inertial mass came from. It then seemed to me that there might be room for an elementary book on special relativity that emphasised the basic physical ideas, their consequences and their applications other than in particle physics."

The text is elementary in some respects, but I would not recommend it to a beginner. The style is terse and makes few concessions to students who are not prepared to think hard and to exercise their imagination; the diagrams are sparse and minimalist. I would recommend it, with enthusiasm, to anyone who, having completed an introductory course in relativity, wants to learn more about its practical applications or to discover the problems that arise when it is applied to situations a little more complex than those commonly addressed in elementary texts. Other valuable features are the comments on apparent paradoxes and the salutary warnings about appeals to geometrical intuition. In the latter category Robinson might have included Figure 2.2 which, like similar diagrams in other texts on relativity, purports to represent a cross-section of Minkowski space.

Of all the abstract spaces encountered in a typical undergraduate physics course, Minkowski space is the one that does most violence to intuition. Its indefinite metric defies visual imagination. Pictorial representation of a space in which the "distance" between two quite distinct points may be zero (or imaginary) is bound to be misleading, if not illusory.

It is refreshing to find so many applications drawn from areas other than particle physics. These include the free-electron laser, the elimination of the first-order Doppler effect in a wave-guide and the use of the Sagnac effect to measure "absolute" rotation. The key role of relativity in atomic physics is illustrated by the Thomas precession and by a succinct sketch of the relativistic basis of the Dirac equation.

Robinson dispels the impression that relativity is a theory of elegance and simplicity at least comparable to that of classical mechanics. The power and beauty of the four-vector formalism is undoubtedly apparent when applied to the electromagnetic field in vacuo or to the dynamics of a single particle. It is only in slightly more involved situations such as the passage of electromagnetic radiation through a material medium or to the theory of angular momentum that the complications become evident. The illusion of simplicity might have been more tellingly revealed by considering the elementary problem of two particles moving under the influence of an arbitrary central force.

The elegance and generality of the classical solution, in which the motion of the two particles in three dimensions is reducible to that of a single particle moving in a one-dimensional effective potential, is a delight. Relativity, though undoubtedly the more accurate theory, has nothing comparable to offer.

An outline of the relativistic treatment of the two-body problem would probably be more instructive for the average student than the highly abstract treatment of the principle of least action in chapter 11.

Facts commonly taken for granted in elementary texts, or for which dubious demonstrations are offered, are the linearity of the Lorentz transformation and the invariance of electric charge. Robinson provides a succinct proof of the first of these but merely asserts the second. My only other criticism, bordering perhaps on the pedantic, is the use of "velocity" where "speed" would be more appropriate.

This stimulating book includes a variety of concrete examples and applications over and above those offered in most elementary texts. It also does not conceal the subtleties and complications entailed in relativistic treatments of all but the simplest physical problems.

Malcolm McCausland was formerly honorary fellow in physics, University of Manchester.

## An Introduction to Special Relativity and its Applications

Author - F. N. H. Robinson
ISBN - 981 02 2499 0
Publisher - World Scientific
Price - £20.00
Pages - 183

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