De Sitter attempted in 1917 to calculate the radius of the uni verse from the observed mean density of matter, but his estimate for this observed mean density, namely, 1 was probably far too high. In 1926 Hubble estimated the mean density of matter in space to be of the order of 1.5 x grammes per cubic centi metre, and calculated the radius of the universe to be 8.5 x 1o" centimetres or 88,000 million light years. This is about 600 times the greatest distance at which spiral nebulae are visible in the great 1 oo-inch telescope at Mount Wilson, so that we may say that the range of vision of this telescope is one six-hundredth part of the radius of the universe. The mass corresponding to this size of universe is 1.8 x 1o" grammes or about 900,00o million million million times the mass of the sun.
These calculations have been made on the supposition that, apart from local irregularities, the curvature of space is the same everywhere so that space is spherical ; this is consistent with the assumption that the mean density of matter has a uniform value throughout space.
There is no reason why space should not, to an approximation, be spherical, but it is unlikely that the whole continuum can be. The cross-section of the continuum which corresponds to a given instant of time is the spatial universe at that instant, and if the continuum were spherical, this cross-section would be of ever varying size, so that the mass of the universe would have to vary continually.
For this and other reasons Einstein imagines the continuum to be cylindrical as regards its extension in time, but spherical as regards its extension in space, so that cross-sections at different instants always give a spherical universe of constant size and so of constant mass.
In terms of this simplified scheme, the primitive cosmology of pre-relativity days would represent the continuum by a plane which extended to infinity in all directions, both of space and time. In accordance with Newton's first Law of Motion the world lines of particles are straight lines in the plane, except in so far as the particles are compelled by impressed "forces" to de part from these straight lines. It is convenient to take as unit
of length the distance travelled by light in the unit of time, so that unless the particles move with velocities comparable with that of light, their world lines are approximately parallel to the axis of time, and so are nearly vertical. The world lines of rays of light are straight lines inclined at angles of 45° to the vertical.
Einstein's cosmology replaces this plane by a right circular cylinder, whose axis is the axis of time. The world lines of particles whose velocity is small compared with that of light are vertical lines on this cylinder. The world line of a ray of light is -a helix which travels round and round the cylinder, always making an angle of 45° with the horizontal. This meets the world line of a particle an infinite number of times at intervals of time apart, each of which represents the time taken for light to travel completely round the universe. A consequence, which is of greater interest theoretically than practically, is that an observer would see an infinite number of images of every star, exhibiting its state at successive epochs in its development, these being separated by the time taken for light to travel round the universe.
De Sitter has proposed a modification of Einstein's cosmology which is free from this last feature. He replaces Einstein's cylin der by a hyperboloid of one sheet on which the rectilinear gener ators are inclined at an angle of to the vertical. These rectilinear generators now represent the paths of rays of light, while the world lines of particles are hyperbolae, which are nearly in vertical planes provided that the velocities of the particles are small in comparison with that of light. In this scheme the world lines of a particle and of a light ray only intersect once. De Sitter's scheme requires that the distances of particles from one another should increase with the progress of time, so that the relative velocities of pairs of objects in the universe ought to be mainly velocities of recession. With few exceptions the spectra of the spiral nebulae are observed to show a strong displacement of their lines towards the red. This is generally interpreted as evi dence of a velocity of recession which appears to average some 7ookm. a second, and if this is the true interpretation, it obvi ously lends some support to De Sitter's scheme of cosmology.
Modifications of the schemes of Einstein and De Sitter have been proposed by Weyl and Lanczos, but astronomical obser vation has so far provided no means either of deciding between the original schemes or between the proposed modifications.