**3.1.1 The cosmological principle**

``. . . The character of space curvature is, depending on the distribution of matter, variable locally and with time, but on a large scale it can be approximated by spherical space. This is at least logically without contradiction and it is the most obvious from the point of view of the general theory of relativity; whether it is tenable from the point of view of our present astronomical knowledge shall not be discussed here. In order to reach this non-contradictory conclusion we had to introduce, however, a new addition

^{(4)}to the field equations, which is not justified by our actual knowledge about gravitation. It must, however, be pointed out, that a positive curvature of space also results from the matter contained in it when this additional parameter is not introduced; the latter is only required to ascertain a quasistatical distribution of matter, as corresponds to the fact of small stellar velocities.''

``Its density (in natural measure) is constant when sufficiently large units of space are used to measure it. Locally its distribution may be very inhomogeneous.''

``In our world matter is, however, not uniformly distributed but concentrated in individual celestial bodies, not at rest, but in slow relative motion (compared to the velocity of light). However, it is well possible, that the mean (``naturally measured'') space density of matter taken for spaces which contain very many fixed stars, is a nearly constant quantity.''

``. . . by Einstein and also by de Sitter certain assumptions are made concerning the matter tensor, which correspond to the incoherence of matter and its relative rest, that is the velocity of matter is assumed to be sufficiently small compared to the basic velocity - the velocity of light.``

Lemaître (1927) - see Sect. 3.2.1.

``The general theory of relativity attributes the particular metrical properties of the space-time universe . . . . directly to the distribution of matter within it, and has naturally led to speculations concerning the structure of the universe as a whole, in which local irregularities caused by the agglomeration of matter into stars and stellar systems are disregarded. Chief among the resulting relativistic cosmologies are those based on the cylindrical world of Einstein and the spherical world of de Sitter; the line elements on which these interpretations are based have not, however, been derived from the intrinsic properties of homogeneity and isotropy attributable a priori to such an idealized universe, but rather are presented as defining manifolds which do possess the desired uniformity. It is the purpose of the present note to formulate explicitly an assumption embodying the uniformity demanded by such a cosmology and deduce all line elements satisfying it . . .

Space-time shall be spatially homogeneous and isotropic in the sense that it shall admit a transformation which sends an arbitrary configuration in any 3-spaces t = const, . . . into any other such configuration in the same 3-space in such a way that all intrinsic properties of space-time are left unaltered be the transformation. That is, any such configuration shall be fully equivalent to any other in the same 3-space in the sense that it shall be impossible to distinguish between them by any intrinsic property of space-time . . . .

If we wish to require in addition that their intrinsic properties be independent of timet, we may amend the above assumption to state thatany configuration, as there described,in any 3-space t = const. is fully equivalent to any such in any 3-space of the family.''

``Einstein's [1931] postulate that all places in the universe must be equivalent is modified to read:

The universe must appear the same to all observers. As the view of any particular observer depends on the space-time frame he adopts, this requires to be made more precise. We therefore posit:not only the laws of nature, but also the events occurring in nature, the world itself, must appear the same to all observers, wherever they be, provided their space-frames and time-scales are similarly oriented with respect to the events which are the subject of observation. By `the world' I do not mean `the world at an instant' but the totality of the flux of events. This postulate is referred to as the `extended principle of relativity'.''

Here the understanding of the terms *homogeneity* and *isotropy* in
cosmology seems to have reached its full meaning. Evolved has what has
been termed the *cosmological principle* by Milne.