ln"troduc:"tion "to TE!n!ior l:alc:ulu!i, RE!Ia"tivi"ty and I:O§IllDiogy D. Lawden This elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. Its use of relatively unsophisticated mathematics in the e arly chapters allows readers to develop their confidence within the framework of Cartesian coordinates before undertaking the theory of tensors in curved spaces and its application to general relativity theory. Topics include the special principle of relativity and Lorentz transformations; orthogonal transformations and Cartesian tensors; special relativity mechanics and electrodynamics; general tensor calculus and Riemannian space; and the general theory of relativity, including a focus on black holes and gravitational waves.
The text concludes with a chapter offering a sound background in applying the principles of general relativity to cosmology. Numerous exercises advance the theoretical developments of the main text, thus enhancing this volume's appeal to students of applied mathematics and physics at both undergraduate and postgraduate levels. Dover (2002) unabridged republication of the third edition, originally published by John Wiley & Sons, New York, 1982. List of Constants.
ALSO AVAILABLE MATHEMATICS FOR PHYSICISTS, Philippe Dennery and Andre Krzywicki. Synge and A Schild. 63612-7 TENSOR ANALYSIS FOR PHYSICISTS, J. 65582-2 For current price information write to Dover Publications, or log on to www.com-and see every Dover book in print.
Free Dover Mathe matics and Science Catalog (59065-8) available upon request.com Introduction to TENSOR CALCULUS, RELATIVITY AND COSMOLOGY Third Edition D. Lawden Emeritus Professor Unit)ersity of Aston in Binningham, U.K DOVER PUBLICATIONS, INC. Mineola, New York www.com Copyright Copyright~ 1962. Lawden Copyright t 1982 by John Wiley & Sons, Ltd.
All rights reserved under Pan American and International Copyright Conventions. Bibliographical :Vote This Dover edition, first published in 2002. is an unabridged republication of the third edition of the work, originally published by John Wiley & Sons, l\:ew York. Readers of this book who would like to receive the solutions to the exercises may request them from the publisher at the following e-mail address: editors@doverpublications.
Library of Congress Cataloging-in-Publication Data Lawden, Derek F. Introduction to tensor calculus. Originally published: 3rd ed. Chichester [Sussex] : :"\ew York : Wile:;.
Includes bibliographical references and index. Calculus of tensors. ll--dc21 200::!03Tl:i3 Manufactured in the Cnited Swtes of America Dover Publications. 31 East 2nd Street.com Contents Preface.
IX List of Constants. X111 Chapter 1 Special Principle of Relath·ity. Newton's laws of motion. Covariance of the laws of motion.
Special principle of relativity. Minkowski space- time 6 5. The special Lorentz transformation. Spacelike and timelike intervals.
Light cone 14 Exercises I. I7 Chapter 2 Orthogonal Transformations. Repeated-index summation convention. Rectangular Cartesian tensors.
Derivatives of tensors. 31 Chapter 3 Special Relath·ity Mechanics 39 I 5. The velocity vector. Mass and momentum.
The force vector. Lorentz transformation equations for force. Photon and neutrino. Lagrange's and Hamilton's equations.
Energy-momentum tensor. Energy-momentum tensor for a fluid. Angular momentum 57 Exercises 3 .com VI Chapter 4 Special Relathity F:lectrod~namics. The field tensor.
Lorentz transformations of electric and magnetic vectors 77 28. The Lorentz force. The energy-momentum tensor for an electromagnetic field 79 Exercises 4. !12 Chapter 5 General Tensor Calculus.
Generalized N-dimensional spaces. Contravariant and covariant tensors. The quotient theorem. Transformation of an affinity.
Covariant derivatives of tensors. The Riemann--Christoffel curvature tensor 102 37. Raising and lowering indices. Magnitudes of vectors.
The covariant curvature tensor. II 7 Chapter 6 General Theory of Relathity 127 44. Principle of equivalence. Metric in a gra,itational field.
Motion of a free particle in a gravitational field. Einstein's law of gravitation. Acceleration of a particle in a weak gravitational field 137 49. Newton's law of gravitation.
Freely falling dust cloud. Metrics with spherical symmetry. Gravitational deflection of a light ray. Gravitational displacement of spectral lines.
Maxwell's equations in a gravitational field.com vii Chapter 7 Cosmology. Spaces of constant curvature. The Robertson-Walker metric. Hubble's constant and the deceleration parameter.
Red shift of galaxies 182 64. Model universes of Einstein and de Sitter 188 67. Particle and event horizons .com Preface The revolt against the ancient world view of a universe centred upon the earth, which was initiated by Copernicus and further developed by Kepler, Galileo and Newton, reached its natural termination in Einstein's theories of relativity. Starting from the concept that there exists a unique privileged observer of the cosmos, namely man himself, natural philosophy has journeyed to the opposite pole and now accepts as a fundamental principle that all observers are equivalent, in the sense that each can explain the behaviour of the cosmos by application of the same set of natural laws.
Another line of thought whose complete development takes place within the context of special relativity is that pioneered by Maxwell, electromagnetic field theory. Indeed, since the Lorentz transform- ation equations upon which the special theory is based constitute none other than the transformation group under which Maxwell's equations remain of invariant form, the relativistic expression of these equations discovered by Minkowski is more natural than Maxwell's. In the history of natural philosophy, therefore, relativity theory represents the culmination of three centuries of mathematical modelling of the macroscopic physical world; it stands at the end of an era and is a magnificent and fitting memorial to the golden age of mathematical physics which came to an end at the time of the First World War. Einstein's triumph was also his tragedy; although he was inspired to create a masterpiece, this proved to be a monument to the past and its very perfection a barrier to future development.
Thus, although all the implications of the general theory have not yet been uncovered, the barrenness of Einstein's later explorations indicates that the growth areas of mathematical physics lie elsewhere, presumably in the fecund soil of quantum and elementary-particle theory. Nevertheless, relativity theory, especially the special form, provides a found- ation upon which all later developments have been constructed and it seems destined to continue in this role for a long time yet. A thorough knowledge of its elements is accordingly a prerequisite for all students who wish to understand contemporary theories of the physical world and possibly to contribute to their expansion. This being universally recognized, university courses in applied mathematics and mathematical physics commonly include an introductory course in the subject at the undergraduate level, usually in the second and third years, but occasionally even in the first year.
This book has been written to provide a suitable supporting text for such courses. The author has taught this type of class for the past twenty-five years and has become very familiar with the difficulties regularly experienced by students when they first study this subject; ix www.com X the identification of these perplexities and their careful resolution has therefore been one of my main aims when preparing this account. To assist the student further in mastering the subject, I have collected together a large number of exercises and these will be found at the end of each chapter; most have been set as course work or in examinations for my own classes and, I think, cover almost all aspects normally treated at this level. It is hoped, therefore, that the book will also prove helpful to lecturers as a source of problems for setting in exercise classes.
When preparing my plan for the development of the subject, I decided to disregard completely the historical order of evolution of the ideas and to present these in the most natural logical and didactic manner possible. In the case of a fully established (and, indeed, venerable) theory, any other arrangement for an introductory text is unjustifiable. many facets of the subject which were at the centre of attention during the early years of its evolution have been relegated to the exercises or omitted entirely. For example, details of the seminal Michelson- Morley experiment and its associated calculations have not been included.
Although this event was the spark which ignited the relativistic tinder, it is now apparent that this was an historical accident and that, being implicit in Maxwetrs principles of electromagnetism. it was inevitable that the special theory would be formulated near the turn of the century. Neither is the experiment any longer to be regarded as a crucial test of the theory. since the theory's manifold implications for all branches of physics have provided countless other checks, all of which have told in its favour.
The early controversies attending the birth of relativity theory are, however, of great human interest and students who wish to follow these are referred to the books by Clark, HolTmann and Lanczos listed in the Bibliography at the end of this book. A curious feature of the history of the special theory is the persistence of certain paradoxes which arose shortly after it was first propounded by Einstein and which were largely disposed of at that time. In spite of this, they are rediscovered every decade or so and editors of popular scientific periodicals (and occasionally, and more reprehensibly, serious research journals) seem happy to provide space in which these old battles can be refought, thus generating a good deal of acrimony on all sides (and, presumably, improving circulation). The source of the paradoxes is invariably a failure to appreciate that the special theory is restricted in its validity to inertial frames of reference or an inability to jettison the Newtonian concept of a unique ordering of events in time.
Complete books based on these misconceptions have been published by authors who should know better, thus giving students the unfortunate impression that the consistency of this system of ideas is still in doubt. I have therefore felt it necessary to mention some of these 'paradoxes' at appropriate points in the text and to indicate how they are resolved; others have been used as a basis for exercises, providing excellent practice for the student to train himself to think relativistically. Much of the text was originally published in 1962 under the title An Introduction to Tensor Calculus and Relativity. All these sections have been thoroughly revised in the light of my teaching experience, one or two sections www.com xi have been discarded as containing material which has proved to be of little importance for an understanding of the basics (e.
relative tensors) and a number of new sections have been added (e. equations of motion of an elastic fluid, black holes. gravitational waves, and a more detailed account of the relationship between the metric and affine connections). But the main improvement is the addition of a chapter covering the application of the general theory to cosmology.
As a result of the great strides made in the development of optical and, particularly, radio astronomy during the last twenty years, cosmological science has moved towards the centre of interest for physics and very few university courses in the general theory now fail to include lectures in this area. It is a common (and desirable) practice to provide separate courses in the special and general theories, the special being covered in the second or third under- graduate year and the general in the final year of the undergraduate course or the first year of a postgraduate course. The book has been arranged with this in mind and the first four chapters form a complete unit, suitable for reading by students who may not progress to the general theory. Such students need not be burdened with the general theory of tensors and Riemannian spaces, but can acquire a mastery of the principles of the special theory using only the unsophisticated tool of Cartesian tensors in Euclidean (or quasi-Euclidean) space.
In my experience, even students who intend to take a course in the general theory also benefit from exposure to the special theory in this form, since it enables them to concentrate upon the difficulties of the relativity principles and not to be distracted by avoidable complexities of notation.