This appendix contains a list of papers which, in the opinion of this author, played a crucial role in the development of inhomogeneous cosmological models. It must be stressed that, except for a few, the papers listed below have never been properly appreciated, and many of them are virtually unknown even today. The list is thus a call for historical justice (based on a personal assessment by this author) rather than a presentation of development of the field.

Lemaître (1933a) – the pioneering paper, and probably the most underappreciated one. The author introduced the Lemaître–Tolman model, and in addition presented or solved a few problems commonly associated now with names and papers younger by a whole generation. Examples: the definition of mass for a spherically symmetric perfect fluid, a proof that the Schwarzschild horizon is not a singularity (by a coordinate transformation to a system of freely falling observers), a preliminary statement of a singularity theorem illustrated by a Bianchi I model.

McVittie (1933) – presented a superposition of the Schwarzschild and FLRW metrics which is a perfect fluid solution. A remarkably bold and early entry, but the solution has still not been satisfactorily interpreted.

Dingle (1933) – a preliminary investigation of spherically symmetric shearfree perfect fluid solutions, later completed by Kustaanheimo and Qvist (1948). The paper is remarkable for the author's strong criticism of the cosmological principle and an explicit call for inhomogeneous models (see Appendix C).

Every review article or book raises the obvious question about its completeness. In order to give the readers an idea about the degree of completeness of this review, the method used to compile the bibliography is described briefly below. These were the essential steps:

1. The author has been interested in the subject since about 1980. Until 1988, when systematic compilation was begun, I studied every newly published research or review article on inhomogeneous cosmological models, and followed each reference whose context of citation suggested that it might contain more of the relevant material. The latest publications included are those that reached my hands in September 1994.

2. I looked through the subject indexes to all volumes of Physics Abstracts, beginning with the 1915 volume, and studied the sections on cosmology, general relativity, gravitation, gravitational collapse and spacetime configurations. Whenever any keyword of title or abstract suggested that the paper might be relevant for the review, I added the reference to the list of papers to look up. The last index so surveyed was Part I of the 1994 volume.

3. While reading or looking through the papers, I added every reference that seemed relevant to the list.

Stage 3 produced more than 1000 references in addition to about 1000 found in stage 2 (but about two-thirds of the total number of papers were discarded according to the criteria listed in Section 1.1).

For most physicists, sufficient reason to consider the Universe as being inhomogeneous is the intellectual challenge to explore the unknown. A mathematical argument in favour of inhomogeneous models was given by Tavakol and Ellis (1988). The authors demonstrated on examples that including new parameters (or changing values of parameters) in a set of ordinary differential equations may drastically change the behaviour of solutions, for example, from periodic to chaotic. This shows that the set of cosmological models is most probably structurally unstable. Tavakol and Ellis emphasized the importance of studying models without symmetry.

Several physicists were already able to see in the 1930s that the “cosmological principle” was not a summary of any kind of knowledge, but just a working hypothesis leading to the simplest models that were yet acceptable. A sample of early reservations about the cosmological principle is given in Appendix C. Unfortunately, the astronomical community was largely unconvinced. The discoveries of large inhomogeneities in the 1970s and 1980s were met with surprise. By the present day, those same discoveries could be sufficient argument for considering inhomogeneous models, and the idea seems to be gaining still wider acceptance (see an interesting proposal by Melott, 1990, to measure the topology of large-scale matter distribution in the Universe). However, over the years, astrophysical and philosophical arguments have been given by several authors in favour of generalizations of the FLRW models, and they are briefly recalled here. The collection does not pretend to be complete.

Afterthoughts

In addition to justifying the two messages from Section 1.1, the main purpose of this book was to draw together and bring to the readers' attention all those fine and illuminating contributions to relativistic cosmology that do not fit into the “standard model”, while still not negating its applicability as a first approximation. Hopefully, the book has proved that these contributions are not phantasies of isolated individuals, but that, taken together, they do tell us about interesting processes that might be going on in the Universe – processes that we would not even suspect using the FLRW models alone. The results presented highlight several problems in the current system of education and research in physics.

The first problem is the highly dogmatic approach of astronomers towards the “standard model” and other “standards”. It seems that the hypothetical, provisional character of the assumptions that lead to the FLRW models has not been given sufficient emphasis in astronomy courses. As a result, the homogeneity and isotropy of the Universe are treated by many (most?) astronomers as a revealed truth, never to be questioned. This author has sometimes experienced outright aggression during seminars and conference talks, while presenting the various ideas that now appear in this book. Physicists do not seem to suffer from this problem, but astronomers would be well advised to treat the “current knowledge” in a more relaxed way, especially in view of the still highly unsatisfactory reliability and precision of most observations and the numerous changes in the apparently well-established results.

In this chapter, those papers will be described which discuss physical and astrophysical implications of the various properties of inhomogeneous cosmological models. A great majority of the papers are based on the L–T model. Those considerations which are not based on any explicit solution are described in Section 3.9. Some more cosmological considerations, based on the McVittie (1933) solution, will be presented in Section 4.7. The papers are sorted by the subjects they discuss; each section is devoted to one subject; the sections are ordered in chronological order of the earliest contributions.

Formation of voids in the Universe

The first predictions that voids should form in an inhomogeneous Universe were formulated by Tolman (1934) and Sen (1934) on the basis of the L–T model. Tolman predicted it in just a casual remark (see quotation in Section 2.12), while Sen made a thorough study of stability of the static Einstein and the FLRW models with respect to the L–T perturbation, and concluded explicitly that “the models are unstable for initial rarefaction”.

In a follow-up paper, Sen (1935) considered the influence of pressure on the stability. That investigation is based on the Einstein equations in a spherically symmetric perfect fluid spacetime, without invoking any explicit solution. Depending on the spatial distribution of pressure, stability may be restored or instability enhanced, but this observation is not developed further.

Bondi (1947) predicted the formation of voids in just one phrase: “… if originally there is a small empty region round O and if the matter nearest to O does not move inward at first … then it will never move inwards”.

This work is a review of inhomogeneous cosmological exact solutions of the Einstein equations and their properties. A solution is, by definition, cosmological if it can reproduce any metric of the Friedmann (1922, 1924)–Lemaître (1927, 1931)– Robertson (1929, 1933)–Walker (1935) class (abbreviated FLRW) by taking limiting values of arbitrary constants or functions. The book is intended to attain two objectives: (i) to list all independently derived cosmological solutions and reveal all interconnections between them; (ii) to collect in one text all discussions of the physical properties of our Universe based on inhomogeneous models. Objective (i) is attained by showing that the more than 300 independently published solutions can be derived by limiting transitions from about 60 parent solutions. The solutions are arranged in relatively few disjoint families, and a few of them have been rediscovered up to 20 times. In attaining objective (ii) it is shown that exact inhomogeneous solutions can describe several features of the Universe in agreement with observations. Examples are the presence of voids and high-density membranes in the distribution of mass or the fluctuations of temperature of the microwave background radiation. Among the effects predicted are: (i) cosmological expansion of planetary orbits (unmeasureably small, but nonzero); (ii) prevention of the Big Bang singularity by an arbitrarily small charge anywhere in the Universe. In addition, papers that have discussed averaging the small-scale Einstein equations to obtain large-scale cosmological field equations have been reviewed.

This appendix is meant to pay justice to those authors who could see earlier than others that the FLRW models are an oversimplification of Nature. It contains a short (and very possibly incomplete) selection of quotations.

General remarks

In this chapter, we shall study a family of solutions which were constructed with the explicit aim of superposing the FLRW models with various important vacuum or electrovacuum solutions. They become the FLRW models in the homogeneous perfect fluid limit, and they reduce to the Kerr or related solutions in the stationary (electro-) vacuum limit. They were guessed rather than derived by integration of the Einstein equations. The null radiation in them was not introduced as an additional physical component of matter, but appeared ex post as a device to interpret those components of the Einstein tensor that do not belong to the perfect fluid or electromagnetic field. As a result, the various components of the source (the fluid, the electromagnetic field and the null radiation) are coupled through the parameters and functions that they all contain, and cannot be set to zero separately. Usually, setting the null radiation component to zero results in trivializing the other components automatically (for example, it may result in reducing the solution to a FLRW model or to a vacuum). This is a disadvantage, of course, but otherwise the solutions constitute a very interesting experiment in combining different models that has already reached remarkable sophistication and provided new insights into the properties of known solutions. The papers from this family do not contain sufficient information to assign the solutions to the Wainwright (1979 and 1981) classes. The solutions are displayed in Figure 5.1.

Underlying what has been said above are some theoretical issues that remain unresolved, despite their importance for understanding the geometry of realistic universe models. The central feature is that the models on which the analysis above is based are the standard homogeneous and isotropic Friedman–Robertson–Walker (FRW) universe models; but the actual universe – from which our observations derive – is in fact inhomogeneous on small and intermediate scales.

Lumpy universe models

A series of related problems arise. The first point is that most methods of estimating Ω0 cannot determine the background model without simultaneously solving for its perturbations (for example, estimating velocity flows; calculating lensing effects; considering the effects of inhomogeneities on the CMB; and so on). Furthermore, there is a feedback from these inhomogeneities to the background model dynamics (and so to the equations whereby one estimates Ω0). But that raises the issue of gauge freedom in describing such perturbations, and how one chooses the background model when faced with a ‘lumpy’ real universe, which is the second main point.

The gauge issue has been intensively studied in recent years by Bardeen (1980), Ellis & Bruni (1989) and many others; see, e.g., Bruni, Dunsby & Ellis (1992) for a summary and references. The question here relates to the task of finding a description of inhomogeneities that does not depend on the way the background model is mapped to the real universe. The converse problem of the optimal way to fit a background FRW model to the real lumpy universe (Ellis & Stoeger 1987) has been the subject of much less thought.

What we have done in this book is to look at those aspects of cosmology in which the density of the universe plays a role as either a prediction or a parameter of a model, and compared them with the data. It is now our task to weigh up the arguments we have described, and try to make a reasoned assessment of their implications. This will be done by a forensic approach: some of the evidence is quite reliable, but some of it is purely circumstantial, some unreliable, and some contradictory. In view of this we shall not adopt the criterion of proof that applies in the criminal court (‘beyond all reasonable doubt’). Rather we look at the ‘balance of the evidence’, as in a civil case. Doing this, we believe that despite some counter-indications, it is possible to discern a strong case for a low-density universe having negatively curved spatial sections, i.e. to conclude on the balance of evidence and argument that we live in an open universe.

The starting point for this conclusion is that, as should be clear on reviewing the various considerations laid out in the previous chapters, there is no convincing observational case to be made for a critical density universe: the strongest motivation for the supposition that Ω0 is very close to unity comes from theory rather than observation. We consider the theoretical arguments most commonly advanced, which rely on presuppositions about physics which are not amenable to direct test, to be inconclusive for reasons discussed at length in Chapter 2.