Expanding Universes of General Relativity
Relativistic cosmology is a natural offshoot of Einstein's theory of general relativity. However, the cosmology is a superstructure, including other principles, and, if the present formulation were found to be inadequate, the failure would not necessarily affect the underlying theory. Relativity contributes the basic proposition that the geometry of space is determined by the contents of space. To this principle has been added another proposition, formulated in various ways and called by various names, but equivalent, in a sense, to the statement that all observers, regardless of their location, will see the same general picture of the universe. The second principle is a sheer assumption. It seems plausible and it appeals strongly to our sense of proportion. Nevertheless, it leads to a rather remarkable consequence, for it demands that, if we see the nebulae all receding from our position in space, then every other observer, no matter where he may be located, will see the nebulae all receding from his position. However, the assumption is adopted. There must be no favoured location in the universe, no centre, no boundary; all must see the universe alike. And, in order to ensure this situation, the cosmologist, postulates spatial isotropy and spatial homogeneity, which is his way of stating that the universe must be pretty much alike everywhere and in all directions.
The kinds of universes that would be compatible with the relativity principle and the assumption of homogeneity have been determined by intricate mathematical reasoning. A body of necessary characteristics has been derived, one of which is of exceptional interest for our immediate problem, Such a universe, if it contains matter, will be unstable. At best it could be in unstable equilibrium, like a ball balanced on a point. The slightest disturbance would upset the balance - and internal disturbances evidently must occur. The universe would then revert to its natural state of either contraction or expansion. Theory does not indicate either the direction or the rate of the change to be expected. The universe might be expanding or contracting and at a rate that is rapid or imperceptible. At this point the cosmologist seizes upon the observed red-shifts, interprets them as velocity-shifts, and presents them as visible evidence that the actual universe is now expanding, and expanding rapidly. It is for these reasons that relativistic cosmology is described as the theory of homogeneous, expanding universes which obey the relativistic laws of gravitation.
Another important aspect of such universes is the highly abstract concept of spatial curvature. The relativity principle states, that the geometry of space is determined by the contents of space. Theoretical investigators, guided by the assumption of homogeneity, adopt Riemannian geometry which operates in curved space. The curvature cannot be visualized, and will not be discussed in detail. It is sufficient to say that the nature of the curvature is indicated, and the amount is measured, by the radius of curvature (which projects, as it were, into a higher dimension). The radius in our universe might be positive, negative, or zero, and might be large or small. A positive curvature implies closed space, a universe with a definite, finite volume but with no boundary. A negative curvature implies open space, an infinite universe. The limiting case of zero curvature is `flat' Euclidean space with an infinite radius. Thus there are various types of curvature, and, in all but flat space, the amount of curvature has a wide range of possible values.
The general formulae which describe expanding universes include three arbitrary terms. These represent (a) the nature of the expansion, (b) the nature of the spatial curvature, and (c) the nature of the contents (to use a very loose conception of the so-called cosmological constant). Since the numerical values of the terms may vary through wide ranges, the formulae present an infinite array of possible worlds. The problem of the observer, if the theory is valid, is to measure the three elements in our universe and thereby identify, among the possible models, the actual world we inhabit.
The first step has already been taken. For the law of red-shifts, the deviations from linearity introduced by the recession factors, has determined one of the three elements. If the universe is expanding, we now know how it expands. The law of nebular distribution, as we shall see, determines a second element, namely, the nature of the curvature.