Introduction to Tensor Calculus, Relativity and Cosmology

First published in 1982 and now a reissue by the much-appreciated Dover Publications, the book under review was offered by its author as a supporting text for university and early graduate courses in mathematical physics at British universities. Lawden based his work on some two decades of experience teaching this material. His goal was to provide his charges with a solid approach to modern relativistic physics and its methodologies, free of unnecessary and outdated scaffolding (he refers to the Michelson-Morley experiment as a “historical accident,” with special relativity being in the works regardless). The compact book is replete with a wealth of exercises to bring the reader into the thick of things effectively and does so, pardon the play on words, relatively swiftly.

Lawden clearly wants to provide fledgling researchers in this wonderful area of mathematical physics with a very solid and relevant background for delving deeper and deeper into the material, and of course eventually to get to research. I can’t help but note that this particular boot-camp approach is idiosyncratic: I think it is something of a characteristic of the British system, and this is certainly a virtue, given that the student is given a heavy responsibility for doing his own learning — do the reading, write all over the book (or, if you’re not so inclined, all over note books and tablets), and do the exercises — the text is fast-paced and no-nonsense. After all the kids are to be prepared for sporty examinations.

The author gets to the point right away: special relativity is featured in chapters 1 and 3, with a solid start in chapter 2 on tensors (in the sense geometers and physicists — don’t go looking for \(\otimes\)s and universality properties: it’s all about doing funky things with indices. In the afterlife, Einstein and Dirac are smiling…). Lawden downshifts in chapter 4: it’s about special relativity electrodynamics; thereafter it’s on to general relativity. Chapter 5 does a load of Riemannian geometry and general tensor calculus, and in Chapter 6 the big cat is let out of the bag: we hit general relativity properly so-called with extreme prejudice.

Lawden starts with the principle of equivalence and the business of a metric in a gravitational field, does such marvelous things as analysis of particles’ behavior in a gravitational field and the effect of gravitation on light rays and on spectral lines (hubba hubba!), and ends with black holes and gravitational waves. Finally, with the stage having been set in no uncertain terms in the previous discussions, Chapter 7 is all on cosmology. We go from cosmical time and spaces of constant curvature through model universes à la Einstein and de Sitter, and then to particle and event horizons. Even though the material dates to over thirty years ago, there is, as Feynman would put it, a great deal, or even most, of “the good stuff” to be had here.

Again, the book is compact and dense (about 200 pages), laden with pretty beefy and sporty exercise sets that need to be taken very seriously, ambitious and even a bit polemical: for instance, Lawden complains about the fact that many other authors, including authors of very serious books and papers appearing in serious journals, suggest that there is still trouble about paradoxes of relativistic physics that in truth have been completely resolved. His book hits this mendacity square on, and thus intends his reader to be free of such misconceptions and therefore more effective and honest as a future scholar. Feisty stuff, and pedagogically ambitious. But I think Lawden’s book is on target: if you want to learn relativity of both flavors in the proper modern context, with boots on the ground, go for it here.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.