# Homogeneous Relativistic Cosmologies

Michael P. Ryan
Lawrence C. Shepley
Pages: 335
https://www.jstor.org/stable/j.ctt13x1fk3

1. Front Matter
(pp. i-vi)
2. PREFACE
(pp. vii-viii)
MICHAEL P. RYAN JR. and LAWRENCE C. SHEPLEY
(pp. ix-1)
4. [Illustration]
(pp. 2-2)
5. 1. COSMOLOGY: THE STUDY OF UNIVERSES
(pp. 3-10)

It is difficult to explain to a layman the fact that the universe expands. How, he asks, can the universe expand? Into what is it expanding? We tell him that he does not have the correct picture in mind. Indeed the universe can “expand” in spite of the fact that it is everything and does not in any sense develop into unoccupied space. Instead, we tell him that distances between astronomical objects are becoming larger and larger as time moves forward.

Our layman will then turn to the title of this chapter and ask: If our universe is everything, what...

6. 2. GEOMETRY IN THE LANGUAGE OF FORMS
(pp. 11-39)

Nothing is so vital to general relativity as the physical reality of an “event,” or point, in spacetime, completely separate from coordinate systems used to describe it. On the surface of the earth Moscow is Moscow no matter what latitude or longitude we assign to it. Modern mathematics recognizes this separateness in the concept of amanifold, the set of points on which is placed the geometry of spacetime.

In general relativity the manifold is spacetime. A point of the manifold is identified with a physical event. A sample event is shown in Figure 2.1. As a point in a...

7. 3. SPACETIME AND FLUID FLOW
(pp. 40-56)

In this chapter we shall concentrate on the description of a fluid in general relativity and fluid-filled cosmological models. Emphasis in this chapter will be on the use of the coordinate free language of Chapter 2 for relativistic hydrodynamics. Figure 3.1 is an outline of the chapter.

In the theory of relativity we study the behavior of a four-dimensional manifold M on which there is a metric of signature (−+++). The path of any particle in this manifold is affected by the curvature of the manifold. This matter in turn determines the geometry through Einstein’s field equations${{\text{R}}_{\mu \nu }}-\frac{1}{2}\text{R}{{\text{g}}_{\mu \nu }}=\text{k}{{\text{T}}_{\mu \nu }},\caption {(3.1)}$with Rμν...

8. 4. FRIEDMANN-ROBERTSON-WALKER MODELS: BEGIN WITH A BANG
(pp. 57-73)

The closed Friedmann-Robertson-Walker (FRW) universe (Friedmann, 1922; Robertson, 1929; Walker, 1935) is the most provocative and important cosmological model which has been devised since Bruno. It is also one of the simplest. It is isotropic, spatially homogeneous, and fluid-filled. Each spatial section is closed (compact, yet without boundary, finite in extent and volume). Compactness of the spatial sections was considered vital by Einstein (1917) in his earliest cosmological ideas and it is still an intriguing idea, if not necessary as once postulated.

The most shocking feature of this model is its expansion: The volume of the spatial sections changes with...

9. 5. SINGULARITIES IN A SPACETIME
(pp. 74-95)

Each FRW cosmological model is said to be singular. It has at least one region within which the density is unbounded. A freely falling observer in this region, travelling toward increasing density, would see the matter around him become infinitely dense in a finite amount of proper time. However, in the presently accepted viewpoint the points of infinite density (singular points) are not within the model but are treated as an additional structure, theboundaryof the manifold proper.

At present there is no fully accepted method of defining the structure of the singular boundary points of a general manifold....

10. 6. ISOMETRIES OF SPACE AND SPACETIME
(pp. 96-117)

The field equations of general relativity are a complicated set of coupled, non-linear partial differential equations. In cosmology we simplify these equations by imposing symmetries on the solution. Moreover, a symmetrical, orhomogeneous, model thus obtained will not merely be symmetrical in appearance (which might imply that a preferred coordinate frame is necessary). Rather the symmetry of the model will be expressed in a manner that is free of the encumbrance of special coordinates by the use of differential forms and vector fields. Figure 6.1 is an outline of this chapter.

A homogeneous cosmological model is a manifold M. The...

11. 7. UNIVERSES HOMOGENEOUS IN SPACE AND TIME
(pp. 118-131)

We shall call a cosmological model in which the metric is the same at all points of space and timehomogeneous in space and time(ST-homogeneous) (see Figure 7.1). Such a model is a manifold M on which a transitive (simply or multiply) group of isometries G acts. The metric of M is most easily expressed in an invariant basis where the gμνand the structure coefficients Cαβγare all constant. Einstein’s equations become purely algebraic. We shall show that all the solutions are physically meaningless in some sense.

The archtypes of these universes are E, the Einstein universe (Einstein,...

12. 8. T-NUT-M SPACE – OPEN TO CLOSED TO OPEN
(pp. 132-146)

The models of the preceding chapters have no expansion, and therefore other models must be used to describe the actual universe. We now considerspatially homogeneousuniverses, those homogeneous in space but not in time.

We might hope that a vacuum metric mocks the behavior of the actual universe, but this is not the case. Any amount of matter added to a vacuum model changes its character drastically. Nevertheless, we shall consider some vacuum models because the field equations are simpler and their features can be studied in detail.

The most important vacuum spatially-homogeneous model is T-NUT-M (Taub, 1951; a...

13. 9. THE GENERAL SPATIALLY HOMOGENEOUS MODEL IN THE SYNCHRONOUS SYSTEM
(pp. 147-162)

The T-NUT-M model shows that a spatially homogeneous model can have exciting characteristics. To extend our discussion to the general spatially homogeneous model we must first develop some useful equations. We shall compute the Ricci tensor of homogeneous models in a particularly simple basis – the synchronous system. These equations will be applicable to both matter-filled and vacuum models. In succeeding chapters we shall use the synchronous system to discuss the existence of singularities in these models. Figure 8.1 included a logical outline of Chapters 9 and 10.

Through every point in a spatially homogeneous model M passes an invariant...

14. 10. SINGULARITIES IN SPATIALLY HOMOGENEOUS MODELS
(pp. 163-181)

A cosmological model containing a perfect fluid is a manifold on which the metric obeys the field equations${{\text{R}}_{\mu \nu }}=(\text{w}+\text{p}){{\text{u}}_{\mu }}{{\text{u}}_{\nu }}+\frac{1}{2}(\text{w}-\text{p}){{\text{g}}_{\mu \nu }},\caption {(10.1)}$where w is the energy density and p the pressure of the fluid; the uμare the components of the fluid velocity field. It is accepted by most cosmologists that a perfect-fluid model can represent the real universe very well (however, see deVaucouleurs, 1970, and Ellis, 1973).

Often the additional requirement of spatial homogeneity is imposed on a fluid model. If isotropy is also required the Friedmann-Robertson-Walker (FRW) models result (see Chapter 4). If homogeneity, but not isotropy, is required, the...

15. 11. HAMILTONIAN COSMOLOGY
(pp. 182-200)

The previous chapters dealt mainly with mathematical notions. Here we begin a series of four chapters on more physical questions appropriate to the general spatially homogeneous cosmological model. The portrayal of the real universe by a homogeneous model allows very complex problems to be treated; at the same time, the high symmetry of the model makes these problems tractable. Figure 11.1 is a flow chart for Chapters 11, 12, and 13.

For the portraits we will paint of the universe (different portraits to emphasize different features), we accept the theory of general relativity without a cosmological constant. In our discussion...

16. 12. TYPE I MODELS AND TYPE IX MODELS – THE SIMPLEST AND THE MOST INTERESTING
(pp. 201-220)

Type I models ate the simplest of anisotropic models, but already illustrate some of the intriquing features of all Bianchi-type models, particularly in their Hamiltonian formulation. Type IX models illustrate the full range of problems encountered in classical and quantum cosmology.

In Chapter 11 we defined the quantitiesβ+,β, and Ω, the general Type I metric being$\text{d}{{\text{s}}^{2}}=-\text{d}{{\text{t}}^{2}}+{{\text{e}}^{-2\Omega }}\left[ {{\text{e}}^{2({{\beta }_{+}}+\sqrt{3}{{\beta }_{\_}})}}{{(\text{d}{{\text{x}}^{1}})}^{2}}+{{\text{e}}^{2({{\beta }_{+}}-\sqrt{3}{{\beta }_{\_}})}}{{(\text{d}{{\text{x}}^{2}})}^{2}}+{{\text{e}}^{-4{{\beta }_{+}}}}{{(\text{d}{{\text{x}}^{3}})}^{2}} \right].$

From (11.19) and (11.22b), with appropriate resealing ofμand R0, we find

H2= p+2+ p2+ μe−3(1+k)Ω, (12.1)

where Ω plays the role of time in H, and p+and pare momenta conjugate toβ+andβ...

17. 13. NUMERICAL TECHNIQUES
(pp. 221-236)

Except in the case of the FRW models and simple Bianchi Type I cosmologies, we cannot expect to find exact solutions for homogeneous cosmologies (even though qualitative solutions can be found by means of the Hamiltonian techniques of Chapter 11). Inhomogeneous models are even harder to handle. Numerical analysis of homogeneous cosmologies is therefore a necessity – not only to give us exact solutions (to act as checks of our qualitative solutions) but as a testing ground for general numerical studies in cosmology. In this chapter we present solutions for Bianchi Type IX universes as examples of numerical techniques in...

18. 14. ASTROPHYSICAL STUDIES IN ANISOTROPIC TYPE I MODELS
(pp. 237-259)

Spatial homogeneity without isotropy represents a compromise between simplicity and generality. The existence of anisotropy in the model allows a theoretical discussion of many vital effects, and we shall discuss three in this chapter. They are the effects of primordial magnetic fields, neutrino viscosity and kinetic theory, and perturbation theory as it pertains to the formation of galaxies. Our discussion will be brief, principally dealing with setting up equations and with basic properties, to serve as a program for further research. We will deal with Type I models in this chapter: models with anisotropy, but at the same time simple...

19. 15. FINAL REMARKS: WHAT IS, WHAT IS NOT, AND WHAT SHOULD BE
(pp. 260-272)

In a book of this kind one necessarily leaves out certain subjects. Instead of covering such subjects in detail we will here present them in outline form. Our personal preferences are clearly toward the mathematical end of this subject. What we have left out, therefore, are areas of primarily astrophysical or physical content. The general categories we will outline here include cosmogony and the physical universe as well as cosmological aspects of theories which compete with general relativity.

Relating mathematical forms given for cosmological models to the actual physical universe is one task of observation. The most distressing thing about...

20. EXERCISES AND PROBLEMS
(pp. 273-287)
21. BIBLIOGRAPHY
(pp. 288-315)
22. INDEX
(pp. 316-320)
23. Back Matter
(pp. 321-321)