1.1. From the Preface.

Cosmological theory is that branch of physics which deals with the structure of matter in its most bulky and massive state, the whole physical universe being regarded as a single system whose broad features are to be investigated. The subject is necessarily highly mathematical, but, this introductory account, attention has been concentrated on those developments most easily comparable with observation to the exclusion of others of a purely mathematical interest.

1.2. Quote from the text.

In the previous chapters the general theory of relativity provided us with a scheme of ideas which had already achieved success as a theory of gravitation. We found that this scheme, which accounted for the small-scale gravitational motions observed in nature, was equally capable of dealing with the structure of the whole universe. But a moment's reflection will convince the reader that the most striking phenomenon exhibited by the universe, the recession of the spiral nebulae, has very little resemblance to gravitational phenomena as exhibited in the motions of planetary systems, double stars, &c. It is therefore legitimate to inquire whether a theory of the universe can be constructed without an a priori appeal to a theory of gravitation. The problem which was set by E A Milne, and of which he gave one solution, was that of first building up a theory of the whole universe and then, if possible, of deducing from it the necessity of small-scale gravitational motion.

1.3. Review by: H Spencer Jones.
Science Progress (1933-) 33 (129) (1938), 151-152.

To compress an account of cosmological theory into the compass of this small volume, which forms one of Methuen's series of "Monographs on Physical Subjects," is no easy matter. Dr McVittie has concentrated mainly on those developments of the theory that are most easily comparable with observation and has excluded those that are of purely mathematical interest. The counts made by Hubble of the numbers of extra-galactic nebulae down to various limiting stellar magnitudes make comparison between theory and observation possible. It is in such a comparison that the interest lies - at any rate for the observational astronomer. Without such a check the various theoretical possibilities can tell us little about the broad features of the physical universe.

Dr McVittie gives a brief summary of the observational material and then, after two mathematical chapters dealing with tensor calculus and the principles of general relativity, he proceeds to the mathematical investigation of the expanding universe and to a comparison with observation. The conclusion is drawn that the volume of space so far surveyed is a small fraction of the whole. The astronomical evidence appears to be in favour of hyperbolic space, but some extrapolation of data is involved and, in view of the observational uncertainties, this conclusion is to be accepted with reserve. The 200-inch telescope should provide in due course data extending to a fainter limiting magnitude and will perhaps enable a more definite conclusion to be drawn. The possibility that this instrument may provide definite information on this important matter would be sufficient in itself to justify its construction.

Meanwhile, astronomers will be grateful to Dr McVittie for his careful mathematical discussion of the problem, which abounds in pitfalls. It is emphasised, for instance, that the term "distance" in an expanding universe is ambiguous as long as the method of measurement is not specified.

The volume concludes with an account of Milne's kinematical theory of the universe, developed in such a way that its resemblances to, as well as its differences from, general relativity are clearly brought out.

There is a slip on p. 3 in the definition of a parsec, which is the distance at which the radius of the earth's orbit subtends an angle of one second of arc. On p. 68, the author apparently accepts the long time scale of stellar evolution (not less than $10^{12}$ years), whereas the astronomical evidence is strongly converging in favour of the shorter time scale inferred from the dynamics of rotating galaxies (shorter than $10^{11}$ years).

2.1. From the Preface.

The present volume originated in the various courses of lectures on relativity which I have delivered during the past ten years in the Universities of London, Harvard and Illinois. I have attempted to explore the possibilities of general relativity as a method in mathematical physics and astronomy and as a means of interpreting the data supplied by observation. This work therefore continues the tradition which was set by R C Tolman in his Relativity, Thermodynamics and Cosmology and developed by O Heckmann in Theorien der Kosmologie. Investigations of a purely mathematical character, of which there are many in the field of general relativity, have therefore been omitted for the most part. The first five chapters contain an account of the basic principles of the theory and of its application to the gravitational field of the sun. Readers who are chiefly interested in classical gas-dynamics will find in Chapters VI and VII a description of the way in which general relativity can help in the solution of the problems with which they are concerned. The subject of cosmology is treated in Chapters 8 and 9, and I have there endeavoured to throw the theoretical formulae into forms suitable for the numerical computation of the parameters they contain, as and when more complete observational data become available. It remains for me to express my gratitude to Dr A H Taub for reading the manuscript and making many valuable suggestions and criticisms; to my colleagues at the Observatory, Dr Stanley P Wyatt, Jr, and Dr M H Rogers, and to a graduate student, Mr Vernon Fowler, for checking the formulae in certain sections of the book; and to the Director and astronomers of the Mount Wilson and Palomar Observatories who so readily supplied me with data and described the observational programmes having cosmological bearing on which they were engaged or which they were planning.

2.2. Review by: J L Synge.
Quarterly of Applied Mathematics 15 (3) (1957), 326-327.

This is the fourth volume of the International Astrophysics Series, which aims to provide authoritative volumes, suitable for both specialists and students, dealing with the main branches of astrophysics and radio astronomy. On leafing through the book, two things catch the eye. First, a welcome rarity in technical literature, the author explains at some length his philosophical views about science. Secondly, he presents details which one is glad to have readily available. Thus, we find Dingle's formulae for the Einstein tensor calculated for any orthogonal line element, and tables of observed results for the bending of a light ray grazing the sun and for red-shifts, solar and cosmological. The introductory paragraphs to the various chapters are excellent, and it was only when I started to study the mathematics of the book that I found myself out of tune with the author.

Perhaps this is due to two related views I hold about the general theory of relativity: (1) it is a thing of beauty, the essential simplicity of which must at all costs be made to shine through those formal manipulations which cannot be avoided in applying it, and (2) we must develop and use to the full our geometrical intuitions about space-time in the same was as mathematical physicists, through the ages, developed and used their kinematical intuitions about space and time in the Newtonian scheme.

I can best describe the book in a negative sense by saying that those views are not the author's. He has no use for geometrical intuition in space-time, but, in supporting his own scepticism, he should not mislead the reader by telling him that in a Riemannian 2-space it is impossible to write down the curvature tensor, the Ricci tensor and the curvature invariant in terms of the Gaussian curvature; the very simple formulae which do this will be found on pages 89, 96 of Synge and Schild, Tensor Calculus.

The following list of chapter headings will show the scope of the book: I. Introduction. II. The tensor calculus and Riemannian geometry. III. Newtonian mechanics and special relativity. IV. The principles of general relativity. V. The Schwarzschild space-time. VI. Approximations to Einstein's equation and Newtonian gas-dynamics. VII. Special cases in Newtonian gas-dynamics. VIII. Theory of uniform model universes. IX. Model universes and the system of galaxies. There are references and notes at the end, and a good index.

This is not, in my opinion, a book for a beginner in relativity. But for one who understands the theory already in its general outlines, and seeks additional special information, particularly in relation to cosmological problems, the last two chapters will provide much food for thought.

Here are some minor criticisms. ... It is a pity to cry down Minkowski's great achievement by referring to him as "one of the mathematicians who first employed" the idea of flat space-time; or is there another claimant to priority?

2.3. Review by: H Spencer Jones.
Science Progress (1933-) 45 (178) (1957), 345-346.

The observational data that have been accumulated in recent years about the distant galaxies, their apparent magnitudes, their line-of-sight motions, and their spatial distribution will be refined and improved by work now in progress with the 200-inch reflector of the Mount Palomar Observatory and the large astrograph of the Lick Observatory. The interpretation of such observational data, on the basis of general relativity, and their bearing on cosmological theories involve considerations of great complexity. For such investigations this volume will be of great value.

The mathematical formulation of general relativity is based on the tensor calculus and Riemannian geometry. The essential details of these techniques are given. The principles of Newtonian mechanics, special relativity and general relativity are then dealt with; the applications of general relativity to the motions of the perihelia of planets, the bending of rays of light in a gravitational field, and the gravitational red-shift of spectral lines are discussed. Two chapters are devoted to an account of the ways in which general relativity can assist in the solution of problems in gas dynamics.

The last two chapters are concerned with model universes, constructed in accordance with general relativity, and the system of galaxies. Observational data are not at present sufficiently complete or accurate to decide whether the expansion of the Universe is constant or whether the rate of expansion is slowing down; whether space is hyperbolic and infinite or finite but unbounded; and whether Einstein's cosmical constant is zero or not. The observational data are discussed and theoretical formulae are given from which the various parameters involved can be computed when more complete and more accurate observational data become available.

The subject is from its nature highly mathematical and uses a difficult technique; it also bristles with pitfalls. The author is a sure guide through these difficulties. The volume will be of much interest both to theoretical investigators and to observational astronomers. It is a worthy addition to the International Astrophysics Series.

3.1. Review by: R H Dicke.
Science, New Series 149 (3691) (1965), 1493.

Professor McVtttie's book is in refreshing contrast to most recent publications on general relativity. For the past quarter century, work in this field has grown ever more formal, with little attempt to relate the theory to the observed world. In marked contrast, McVittie continues the tradition of Tolman and Robertson, drawing from the observational data, meagre as they are, the vitality needed to convert formal mathematics in to theory.

The past decade has witnessed a great expansion of our knowledge of the Universe and its cosmological setting, and in this second edition of his book McVittie has completely rewritten the last two chapters dealing with this problem. The experts on relativity will find this the most interesting and important part of the book.

While most modern relativists have dropped the cosmological term from Einstein's equation, McVittie has retained it and based his discussion of cosmology on an assumed nonzero value of the "cosmological constant." His argument that the constant appears naturally as a "constant of integration" is not completely convincing after one recognises that this term does not appear in the Euler equation derived from a variational principle without the introduction, ad hoc, of an added term in the variational equation. Most relativists find distasteful the arbitrary introduction into the theory of a large characteristic constant length. They would prefer to drop this term until the observations clearly demand it. In my opinion the observations are not yet complete enough to demand a nonzero "cosmological constant."

McVittie has an easy flowing style that makes his book easy to read. While several important parts of the traditional presentation of general relativity are omitted from his development, the parts he treats are covered with care, and this part of the book can be recommended to the student who wants an introduction to the subject. The section on cosmology should be of interest to a very wide range of readers.

3.2. Review by: F E Kaempffer.
Publications of the Astronomical Society of the Pacific 78 (461) (1966), 181-182.

One of the unresolved questions in general relativity is whether the equations governing the metric field should or should not contain the so-called cosmical constant A. The quantum physicist, viewing gravitation as mediated by massless gravitons can argue persuasively against inclusion of A, whereas the field theorist, adhering to the criterion of letting Einstein's tensor have vanishing divergence, can argue the opposite case with equal conviction.

Under these circumstances it become meritorious to develop the cosmological consequences of general relativity for arbitrary A, so that the reader who strongly favours the case A = 0 can extract the relevant information without much effort. This has been done with great skill by Professor McVittie in this book, now available in a second edition containing a completely new comparison of various cosmological models with the latest observations produced by radio astronomy.

The introduction to general relativity and its classic tests, given in the first half of this work, is so excellent that it can be recommended even to readers who are not particularly interested in the applications to gas dynamics and cosmology making up the second half.

4.1. From the Preface.

I have attempted in this book to weld together the astronomical observations relevant to cosmology with cosmological theory without entering into detailed mathematical proofs. In the nature of things, the work must necessarily be an interim report because new data are continually flowing in. The main draft of this book was completed in 1960 when J G Bolton and R Minkowski were discovering, through the interplay of optical and radio astronomy, a cluster of galaxies immensely more remote than any previously studied. It proved feasible to rewrite certain portions of the book to take account of this new result. I have not attempted to describe every theory of cosmology that is to be found in the technical literature of today. My hope is that, by concentrating on general relativity, the steady-state theory and, to some extent, kinematical relativity, I shall have given the reader an adequate picture of the methods employed. That the book contains defects and limitations must be, I fear, only too true. They would have been more numerous than they are had it not been for the helpful criticisms of the General Editor of the series, Mr Colin Ronan. I owe him a great debt of gratitude for his careful reading of the original manuscript, for pointing out obscurities and for his suggestions regarding their elimination.

4.2. Review by: John B Irwin.
Science, New Series 139 (3550) (1963), 101-102.

G C McVittie's Fact and Theory in Cosmology is the third in a series of books edited by Colin A Ronan and designed to fill the gap between the many elementary astronomy books, on the one hand, and the numerous advanced monographs, on the other. McVittie, an expert in the fields of cosmology and relativity, effectively presents this difficult material on an understandable level. He bases his discussion on observations rather than on airy bubbles of pure speculation, and one gets the impression that he, like Herbert Dingle, prefers "calling a spade a spade and not a perfect agricultural principle." The observables in question are: the red-shift in the lines of the spectra of galaxies; the optical apparent magnitudes of galaxies; the flux-densities of those galaxies which are radio sources; the numbers of galaxies; the diameters of extragalactic radio sources; and the characteristics of clusters of galaxies. The observable data, however, are often all too scanty or imprecise and frequently subject to unknown systematic errors and to errors of interpretation. But McVittie is director of the most powerful radio telescope in this country (the recently dedicated instrument at the University of Illinois), and this instrument, which consists of a parabolic cylindrical reflector 400 by 600 feet, should soon provide accurate new observations of thousands of distant radio galaxies.

An introductory chapter on the nature of cosmology is followed by a discussion of distance in the universe and then by a chapter on the system of galaxies. The next three chapters, which are the most difficult, deal with cosmological theories, model universe and the red-shift, and the selection of a model universe. The final chapter is a short summary and conclusion. The author concentrates on general relativity, the steady-state theory, and, briefly, kinematical relativity. He is not a devotee of the latter two theories, but he correctly points out that their controversial natures have forced all cosmologists to refine and to make precise their own ideas.

The book would be improved by illustrations and by a more complete discussion of distance determination technique, such as moving cluster parallaxes and the use of the zero-age main sequence. A second edition should necessarily include a discussion of the exciting new radio data on the numbers, diameters, and duplicity of these strange, incredibly distant sources.

4.3. Review by: G J Whitrow.
Nature 194 (1962), 48.

Prof McVittie, formerly of the University of London and since 1952 professor of astronomy in the University of Illinois at Urbana, is one of the leading authorities on the interpretation of the empirical data bearing on the cosmological problem. In his latest book, to quote his own words, he has attempted "to weld together the astronomical observations relevant to cosmology with cosmological theory without entering into detailed mathematical proofs". He is an unashamed empiricist in his outlook, and there is a refreshing air of 'realism' about his approach to a subject in which the inadequacy of the data tends to encourage dogmatic theorizing from intellectually seductive 'principles' and postulates. In his opening chapter on the nature of cosmology, he exposes the variant of the 'argument from ignorance' fallacy whereby it is asserted that a statement is to be taken seriously if no observational evidence 'against' it can be adduced. Nevertheless, although McVittie lays his emphasis on the empirical data, he recognizes the necessity of theory for their interpretation.

Chapter 2 is devoted to an introductory account of the fundamental problem of distance determination, including a careful description of the use of Cepheid variables to determine distance moduli of local galaxies. Prof McVittie brings out very clearly the uncertainty in our knowledge of the distance of so familiar an object as the great nebula in Andromeda. Chapter 3 is on the galaxies in general, and is useful for bringing together both visual and radio astronomy.

The next chapter, on theories of the universe and links with observation, is followed by one in which the effects of the red-shift are described, in particular its relation to distance. McVittie emphasizes that this relation is, in general, non-linear and that the linear form holds only for small shifts.

In Chapter 6 we encounter the central problem of using the observational data to select a particular world-model. McVittie shows that the red-shift data are at present sufficient only to narrow the class of admissible models by restricting the range of possibilities for the expansion factor. To determine the space-curvature index, appeal must be made to estimates of the mean density of matter in the universe. The principal snag is that, unlike their redshifts, the distances of remote galaxies can be determined only by prior choice of a model universe and cannot be assigned uniquely on the basis of the relevant empirical data (apparent magnitude determinations).

Hopes have been expressed in recent years that these obstacles might be by-passed by studying the distribution of class II radio sources. McVittie indicates the difficulties of this approach, and the restrictive assumptions underlying the belief of most radioastronomers that, if these sources are uniformly distributed in space, the law expressing their number as a function of limiting flux-density must involve the exponent $- \large\frac{3}{2}\normalsize$.

The final chapters are devoted to the steady-state theory and to a brief summary in conclusion. There is an appendix of useful data relating to clusters of galaxies and references to literature.

The book can be warmly recommended as an authoritative introduction to the subject at a level intermediate between the popular essay and the advanced mathematical treatise. The print is very clear and the price is extremely reasonable.

4.4. Review by: W Davidson.
Nature 198 (1963), 58.

This book provides a very readable and sound introduction to a difficult subject. Although cosmology by its enigmatic character and generality has attracted a plethora of cranks and wishful thinkers in the past, it has also some of the greatest names in science, such as Newton, Mach, Einstein and Eddington, written on its cornerstones. Prof McVittie, who has been active in modern cosmology since the early days when Hubble was making his dramatic discovery of the expanding universe, is a sure guide for the beginner to the difference between fact and fiction in this vast field.

In his lucid and entertaining style the author first provides a selective grounding in the astronomy of local space and time. The reader is then educated to a healthy scepticism of scientific formulae which are established under local conditions and applied to the cosmological scene. He is shown that the challenging phenomena revealed by the great optical and radio telescopes out to distances of the order of thousands of millions of light years have no connected meaning without a cosmological theory.

Throughout the book, however, as if to warn against undisciplined flights of fancy, the point is driven home that it is the observational tests that have in their turn the last word on any such theory.

In Chapter 1 the author describes the general nature of the cosmological problem. In Chapter 2 he discusses very carefully how distance is estimated in the solar system and in the Galaxy, from there to nearby galaxies and hence to any observed galaxy in the universe. Chapter 3 introduces the systems of galaxies, clusters, and radio sources, and their distribution and motion are examined in terms of observables such as optical brightness, spectral red-shift, radio flux density and number counts. Chapter 4 contains a description of the most important cosmological theories. This is followed in Chapter 5 by an appraisal of the properties of such model universes in terms of the red-shift. How to match a model universe against all the available observational data is the subject of detailed discussion in Chapter 6. The steady-state model gets separate scrutiny in Chapter 7, and is here subjected to strong criticism of its logical basis and its ability to satisfy the observational requirements. The book ends with a summary and conclusion in Chapter 8. There is also an appendix giving the tabulated results of the latest red- shift and apparent brightness measurements up to the end of 1960. Throughout the book the mathematics used is such that any undergraduate physicist, say, would understand with ease. The references given are adequate generally although some important omissions were noted.

Highlights of the book are the valuable discussions of the red-shift apparent magnitude data and the number counts of radio sources. Here, however, the treatment invites criticism on three points. First, the author makes the assessment on somewhat slender evidence that any conceivable evolution in intrinsic brightness of galaxies can be neglected. This is despite the fact that the periods involved are as long as 4,000 million years, and indeed Sandage has since shown that plausible evolution could significantly affect the interpretation of the red-shift apparent magnitude data. Secondly, he seems to place too much faith in an identification of the mean density of matter in the universe with that of Oort's estimated mean density of observed matter in the form of galaxies. Lastly, on the question of the number counts of radio sources the author again overlooks the possibility of significant evolution, this time in the intrinsic power of the sources.

However, these considerations are incidental in one's appreciation of the sterling qualities of a book that is of inestimable value to the student and necessary reading for the expert.