nicholas m. j. woodhouse: general relativity

Gen Relativ Gravit (2007) 39:1967–1968DOI 10.1007/s10714-007-0513-4

BOOK REVIEW

Nicholas M. J. Woodhouse: General RelativitySpringer Undergraduate Mathematics Series, Springer,Heidelberg, 2007, 219 p., EUR35.26, ISBN: 978-1-84628-486-1

David A. Burton

Received: 20 August 2007 / Accepted: 22 August 2007 / Published online: 9 September 2007© Springer Science+Business Media, LLC 2007

I wholeheartedly agree with the author’s statement (in the preface) that teachinggeneral relativity to undergraduates in the UK is a rewarding but challenging task.For many students, a first course in general relativity will be their first exposure totensor calculus and geometry and a compromise must be sought between mathemati-cal niceties and physical application. Clearly, where the line is drawn depends on theaudience and the present text is principally aimed at undergraduate applied mathema-ticians but is also suitable for mathematically oriented physicists in their final year.The book is an outgrowth of a lecture course given over many years by the author andhis colleagues to final-year applied mathematicians at the Mathematical Institute inOxford, UK.

The book is well-written and easy to follow because the author constructs the neces-sary apparatus layer-by-layer, from the bottom up, carefully motivating and justifyingevery new concept. Exercises are given at the end of every chapter (answers to manyof the exercises are found in an appendix) and numerous examples appear throughoutthe text. Despite its short length (219 pages), the book is essentially self-contained.

Chapter 1 explains the need for general relativity from the perspectives of specialrelativity and Newtonian gravity and Chaps. 2–3 present tensor calculus on Minkowskispacetime, with application to electromagnetism and fluids. To give the uninitiatedreader confidence, contact is made between manifestly Lorentz covariant formulae andtheir translation into Gibbs 3-vector notation. However, the author seems to assume thatmost readers will have some familiarity with tensorial methods on Minkowski space-time and the content of the early chapters is sometimes rather terse [this approach isnot unreasonable given that the book is a sequel to Woodhouse’s Special Relativity(Springer, Corrected 2nd printing, 2007)]. Chapter 4 introduces curvature and nicely

D. A. Burton (B)Physics Department, Lancaster University, Lancaster LA1 4YB, UKe-mail: d.burton@lancaster.ac.uk

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illustrates the idea with a derivation of the Gauss–Bonnet theorem on surfaces inEuclidean 3-space without tensor calculus. Chapters 5–6 introduce the Levi–Civitaconnection (torsion is briefly mentioned) and Newtonian gravity is used to motivatethe Einstein equations. The discussion of geodesic deviation and the interpretation ofthe Riemann tensor refers back to the section in Chap. 4 on the geometry of surfaces.Chapters 7–9 derive the Schwarzschild solution to the vacuum Einstein equationsand solve the geodesic equation for the orbits of massive particles and photons onSchwarzschild spacetime. Black holes and the Kruskal extension are discussed and thereader’s understanding should be greatly assisted by the considerable detail afforded tothe calculations in these chapters. Chapter 10 develops the weak-field approximationto the Einstein equations, studies the behaviour of gyroscopes outside rotating bodiesand briefly discusses the Kerr metric. Chapter 11 is devoted entirely to the genera-tion and propagation of gravitational waves and Chap. 12 focusses on horizons andthe Robertson–Walker metric. Again, many of the calculations are worked through indetail.

A comparison of this book may be drawn with Hughston and Tod’s An Introduc-tion to General Relativity (Cambridge UP, 1991). Both books are of a similar sizeand aimed at a similar audience, but Woodhouse devotes considerably more spaceto applications, examples and exercises. In particular, Woodhouse’s book contains amuch more thorough discussion of linearised gravity including the Lense-Thirringeffect and gravitational wave production. However, Woodhouse spends less time onthe formal definition of manifolds and dispenses with index-free notions such as exte-rior differential calculus. As such, Woodhouse’s book is generally less mathematicallydemanding than Hughston and Tod’s book and advanced physics students should findit easier to follow.

As a whole, the book is well-written and its expository style is very appealing.It should serve as a useful introduction to more detailed and advanced texts suchas Wald’s General Relativity (University of Chicago Press, 1984) and Stephani’s AnIntroduction to Special and General Relativity (Cambridge UP, 2004). The numer-ous exercises are to be commended although some of the more “thought-provoking”problems are a little ambiguous. The book concludes with an appendix containing aselection of past examination questions that should prove useful.

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