Gravitation and Gauge Symmetries

Abstract

The purpose of this book (I quote verbatim from the back cover) is to `shed light upon the intrinsic structure of gravity and the principle of gauge invariance, which may lead to a consistent unified field theory', a very laudable aim. The content divides fairly clearly into four sections (and origins). After a brief introduction, chapters 2-6 review the `Structure of gravity as a theory based on spacetime gauge symmetries'. This is fairly straightforward material, apparently based on a one-semester graduate course taught at the University of Belgrade for about two decades, and, by implication, this is a reasonably accurate description of its level and assumed knowledge. There follow two chapters of new material entitled `Gravity in flat spacetime' and `Nonlinear effects in gravity'. The final three chapters, entitled `Supersymmetry and supergravity', `Kaluza-Klein theory' and `String theory' have been used for the basis of a one-semester graduate course on the unification of fundamental interactions. The book concludes with thirteen appendices, covering mainly technical issues but also including Ashtekhar variables and Chern-Simons theory. Irritatingly, there is a separate bibliography for each chapter (which leads to much duplication) but commendably, the author highlights selected sources for suitable further reading. Also to be welcomed wholeheartedly are well-chosen worked examples and exercises, ranging from easy to fairly challenging, in each chapter. This is a properly bound paperback, hardly inexpensive, but well up to IOP's impeccable production standards. Could this be the ideal textbook for a research student or even an established researcher from another field to pick up the latest developments in field theory?Unfortunately this is not the whole story. The back cover also asserts, while talking about gauge invariance, that `It is less known that the principle of equivalence, one of the basic dynamical properties of the gravitational interaction, can be expressed as a (spacetime) gauge symmetry'. On page 10 this is qualified to be a local symmetry. On pages 62-3 this hidden result is revealed. As every relativist knows, the principle of equivalence implies that for each spacetime point p a chart (normal coordinates at p) can be chosen so that the metric tensor takes its Minkowski value and its (partial) derivatives vanish at p. In other words every (pseudo-)Riemannian manifold is locally flat, the `less known' result. So what else can the author tell us about gravity?There is little to fault in the first two sections of the book, trying to express gravity as a nonlinear spin-2 theory on Minkowski spacetime, apart from the obvious objection. A well-known and highly-recommendable textbook by S Weinberg 1972 Gravitation and Cosmology (New York: Wiley), carries out, at a more elementary level, a similar programme with the same defect. Such local theories cannot predict global changes whereby spacetime has a different topology to the Minkowski one. (The unconvinced reader should investigate references to black holes in Weinberg's text.) As speaker after speaker at the S W Hawking 60th Birthday Conference last week emphasized, when trying to unify gravity with quantum theory, it is perilous to neglect geometry.Because the later chapters on supersymmetry, supergravity, Kaluza-Klein and string theories are more fashionable, I need to point out an important caveat. Excluding textbooks and conference volumes, all (English language) citations in the supersymmetry/supergravity chapter are at least 16 years old. Apart from one 2000 article (on teleparallel theory) this age gap drops by three years for the chapter on Kaluza-Klein theory, and it does not change any further for the string theory chapter. There have been more recent significant advances in our understanding and interpretation of these theories but, alas, they are not chronicled here.I raised a question as to the value of this book at the end of the first paragraph, and I want to answer it within the context of the conference mentioned above, which tried to draw together all active research aimed at unifying quantum theory and gravitation. A dozen years ago I could have been very enthusiastic about this book (and libraries should hold a copy, it is very convenient for checking details) but the world has moved on a great deal since then.