Relativity: An Introduction to Special and General Relativity, 3rd edn

Abstract

This new edition of Hans Stephani's book is a welcome addition to the many texts now available to the student wishing to study relativity. A new feature, as compared with earlier editions, is a substantial introductory section on special relativity amounting to almost a quarter of the total length of the book. This forms part I and covers all the usual topics one might expect, including particle mechanics, the formulae for aberration and the Doppler effect, tensors in Minkowski spacetime, Maxwell's equations and the energy-momentum tensor for the electromagnetic field, the equations of motion for charged point particles and their fields (with a brief discussion of radiation reaction and runaway solutions) and the energy-momentum tensor for a perfect fluid. But there is also a useful section on the algebraic classification of the electromagnetic field using the apparatus of null tetrads and self-dual bivectors. An unusual feature is a section on pole-dipole charged particles.General relativity is introduced and developed in parts II to VII of the book, which follow closely the text of the 1982 English edition. In part II, the reasons for regarding spacetime as a four-dimensional Riemannian manifold with a metric of Lorentz signature are clearly explained. There follows a concise treatment of the tensor calculus including the covariant derivative, parallel and Fermi-Walker transport, the Lie derivative and the properties of the Riemann tensor. Invariant forms for line, surface and volume integrals are discussed and Stokes' theorem is stated without proof. There is a final section on how physical laws within special relativity may be generalized to curved spacetime.Part III introduces the reader to the Einstein field equations. Schwarzschild's exterior solution and Birkoff's theorem are derived, as are the usual formulae for the perihelion advance for planetary orbits, the deflection of light rays, the gravitational red shift and the geodesic precession of a top. Gravitational lensing is briefly discussed. This part ends with a derivation of the Schwarzschild interior metric for a source with constant rest mass density.The linearized field equations and the question as to whether their solution for a time varying system gives reliable information about solutions to the full nonlinear equations are discussed in part IV. The Landau-Lifshitz pseudo energy-momentum tensor is used to obtain the standard formula for the outward flow of energy due to quadrupole radiation from a bounded system. While the limitations of this approach are emphasized by the author, particularly in regard to the question of whether gravitational waves lead to energy transfer, it is surprising that no mention is made of the important work by Bondi and others in the 1960s which used the nonlinear equations to show that such radiating systems must lose mass. The final sections contain standard treatments of plane wave solutions of both the linearized and nonlinear theory, and a discussion of the Cauchy problem for the vacuum equations.Part V contains a clear account of the Petrov classification of the Weyl tensor (using the same technique as that used earlier in the algebraic classification of the Maxwell field), and of Killing vectors (including a brief introduction to the Bianchi classification of groups of motion) and their relation to conservation laws. Gravitational collapse and black holes are discussed in part VI, which covers such topics as the Kruskal extension of the Schwarzschild metric, the critical mass of a star, the gravitational collapse of a sphere of dust and a brief discussion of the properties of the Kerr metric. Also included in this new edition is a short but useful section on the problem of quantization, of quantum field theory in a given curved spacetime and on the Hawking effect, and on the conformal structure of infinity leading to a definition of asymptotic flatness. In part VII on cosmology, after showing how the cosmological principle leads to the Robertson-Walker metric, the various Friedmann models are derived and discussed. The final sections discuss a Bianchi type I model and the Gödel universe.For its size (at about four hundred pages) the book covers a great deal of material, and it does this very clearly and well. There are some misprints, but almost all of these are of an obvious kind and easily corrected. At the end of most sections there are some well chosen exercises (perhaps there could have been more of these) and useful suggestions for further reading. Students who work through the book carefully, completing the details for those calculations and proofs where these are outlined in the text, will learn a great deal about relativity and be well placed for further study in the subject. While it is probably too advanced in its scope for the average final year (UK) undergraduate course, I would recommend it as a reference book for further reading at this level or as a text for study by a beginning graduate student.