Einstein's General Theory of Relativity—with Modern Applications in Cosmology

Abstract

The increasing prominence of general relativityin astrophysics and cosmology is reflected inthe growing number of texts, particularly at theundergraduate level. A natural attitude beforeopening a new one is to ask i) what makes thisdifferent from those already published? And ii) doesit follow the 'physics-first approach' as for instancethe book by Hartle where the basic physicalconcepts are introduced first with as little formalismas possible, or does it follow the more traditional'math-first approach' for which the mathematicalformalism comes first and is then applied to phyics?As announced in the title, a distinctive featureof the book by Gron and Hervik is the space(almost half the book) devoted to cosmology and inparticular to some of the most recent developmentsin this rapidly evolving field. It is also apparentthat the authors have chosen, like the majority ofcurrent books on general relativity, the 'math-firstapproach'.The book is divided into six parts, each of themsubdivided into chapters with part VI containing afew short technical appendices. The first part of thebook briefly presents in chapter I the principles ofrelativity, Newtonian mechanics and the Newtoniantheory of gravity. In chapter II, a short introductionto special relativity is given. It seems at firstsurprising that the four-dimensional structure ofspace-time is not more fully exploited so that thereader would gain familiarity early on with notionslike 4-velocity, 4-momentum and the stress–energytensor. This is in fact postponed to part II as anillustration of the mathematical formalism.The second part is devoted to those elementsof differential geometry needed in this kind ofcourse. The authors' presentation is somewhatsimilar to that of the books by Misner, Thorne andWheeler and by Straumann (2nd edition). Vectorsand forms are treated separately and the formalismof differential forms is introduced in detail. Thevarious kinds of differentiation on forms and onvectors (exterior covariant and Lie derivatives) arepresented, and emphasis is given to the Cartanformalism as it is later systematically used toderive the curvature tensor and for solutions ofthe Einstein field equations. One also finds theproperties of hypersurfaces, such as the intrinsicand extrinsic curvatures and the Gauss–Codazzirelations. This makes this part of the bookvery useful and convenient since those importantelements are gathered in one place. However thedensity of exposition in this part might appear a bitsteep to a reader without some previous knowledgeof differential geometry.Part III deals with Einstein's field equations,and their applications to gravitational waves and black holes. The field equations are derived from avariational principle, the geometrical part (Einsteintensor) from the Einstein–Hilbert action, and thematter part (stress–energy tensor) from a genericaction integral for matter. Various examples ofstress–energy tensors and in particular, for fluids,are considered, and several are used later in cosmology(for instance quintessence and Lorentz invariantvacuum energy). A short chapter on the linearapproximation and gravitational waves thenfollows and it is good to see a section on gravito-electromagnetism.This part ends with a chapterdevoted to black holes which is perhaps the weakestpart of the book as it is quite sketchy. Howeverthis is to be expected in a book with an emphasison cosmology, and such topics are extensivelydescribed in other books.The rest of the book (parts IV and V) isessentially concerned with cosmology. The authorsgive a detailed description of the applications ofthe Einstein field equations to a universe withvarious matter contents, and present in a successfulway the recent developments in this domain. Thefirst chapter of part IV describes the standardhomogeneous and isotropic cosmological model.It is followed by an interesting chapter dealing withuniverses composed of vacuum energy. There onefinds, after the description of the Einstein staticuniverse and the de Sitter solution, sections oninflation, on the Friedman–Lemaître model andon models with quintessence and dark energy.This chapter ends with sections on cosmic densityperturbations, temperature fluctations in the cosmicmicrowave background and on the history ofour universe. With an additional chapter onanisotropic and homogeneous universes, part IVappears to be a very good and complete introductionto the basic and classical (i.e. non-quantum)elements of cosmology. In part V some advancedtools, such as Lie groups and the Lagrangian andHamiltonian formalism are introduced and appliedto cosmology. Also part V contains a chapter onthe extrinsic curvature formalism for surface layersand its application to the recently introduced braneworldmodels. Finally it is a pleasant surprise tofind an introduction to the Kaluza–Klein theory asthe last chapter of part V.This book by Gron and Hervik certainly hasits place in any good library. It covers mostof the classical aspects of the theory of generalrelativity. The authors have made the effort todiscuss many observational aspects and to illustratethe different chapters with many problems. Onemight regret that the authors' style is generally rather terse and not enough space is alwaysreserved for explanation of physical concepts andfor motivations of the theory (for instance, whycurvature is so fundamental). This book would bemost appropriate for graduate students and I willdefinitely recommend it as a reference textbook aswell as a useful complement to other textbooks ongeneral relativity.