“The distribution of the visible stars is extremely irregular, so that we on no account may venture to set down the mean density of star-matter in the universe as equal, let us say, to the mean density in the Milky Way. In any case, however great the space examined may be, we could not feel convinced that there were no more stars beyond that space. So it seems impossible to estimate the mean density.”
Einstein then suggests an indirect method for determining the value of
, hence of
and of the size of the universe. We shall discuss this method on a later page; it refers to the motions of the stars in the Milky Way.
It is easy to understand how it comes that in the cylindrical world no nucleus of stars need exist. The two-dimensional analogue of a star distribution spread out uniformly throughout the entire volume of a three-dimensional spherical space would be given by a homogeneous distribution of stars over the two-dimensional surface of an ordinary sphere situated in three-dimensional Euclidean space.[111] It would be the actual surface of the sphere which would be the analogue of the space of the universe, and not the volume enclosed within the sphere. Reasons of symmetry would show us at first sight that a uniform distribution of matter on the spherical surface would be stable, there being no reason for a nucleus to form here rather than there.
In the same way, an observer moving through the three-dimensional space of the universe would be represented by a perfectly flat being moving over the surface of the two-dimensional sphere. Wherever this imaginary being might move, the star distribution on the sphere’s surface would appear to be the same. Likewise, in the spherical space of the universe, wherever we might move, the distribution of the stars around us would remain homogeneous.[112] Thus the cylindrical universe obviates the displeasing necessity of believing in the existence of an island of star-matter in an ocean of emptiness.
In Einstein’s universe, a ray of light, following a geodesic, would circle round the universe and return to its starting point. This constitutes a difference as against de Sitter’s universe in which the ray of light would be arrested at the passive horizon. In the cylindrical universe, owing to the straightness of the time direction, there is no horizon. For the same reason there will be no slowing down of time in the remote regions of space, so that the general reddening of the light emitted by the spiral nebulæ would appear difficult to account for in Einstein’s universe. De Sitter’s scheme, as we have noted, affords an immediate explanation of this remarkable effect.
It has sometimes been held that the light rays emitted from a star would circle round the universe of space in all directions, and then meet again at the antipodes of space, forming the ghost of a star. It would be well to point out that though theoretically this would be possible, its chances of being realised in practice would be extremely slight, to say the least. The fact is that the perfect sphericity of the universal space can hold only provided matter is homogeneously distributed and everywhere at rest. As this is only approximately the case in the real universe, the spherical spatial universe will be rippled and will deviate more or less from this simple form. As a result, the rays of light emitted from a star will not converge accurately at the antipodes; hence, ghosts of stars need not be considered.