The evolution of scientific thought from Newton to Einstein

by A. D'Abro

’s. So here, again, we see that owing to the low velocities of the stars, all the

’s vary but slightly from place to place throughout the universe (when computed in a Galilean frame). Thus, with Einstein’s theory, just as with Newton’s, there appears to be nothing to prevent stars and rays of light from leaving the nucleus; the mutual gravitational attractions would be insufficient to retain them.

In short, if there were to be a self-contained universe, something would have to be modified somewhere.

We now come to Einstein’s final solution, that of the cylindrical universe. He remarked that the low velocities of the stars, together with their more or less uniform distribution throughout space, suggested that they were in a state of statistical equilibrium. A few lines will explain what is meant by this.

Suppose that on a billiard table containing a very large number of identical balls we impart a rapid motion to one of them. Our ball will eventually collide with some of the others. These others, in turn, will collide with yet others, and finally all the balls will be set in motion, if there is no friction of any kind, and no loss of momentum when the balls rebound from the cushions, they will go on moving forever. But as time progresses a gradual change will take place. Little by little we shall find that the energy we imparted to the first ball appears to have split itself up among all the balls on the table, so that finally, on an average, all the balls will be moving with approximately the same velocity. When this condition of equipartition of energy is attained we shall have reached stability of configuration, or statistical equilibrium. This state of equilibrium will correspond to maximum entropy, and will endure forever.[110] We see that it is characterised by the fact that all the balls possess more or less the same energies or velocities.

Now, as we have said, astronomical observation would suggest that the stellar universe has reached this condition of statistical equilibrium. It is permissible, therefore, to neglect the trifling irregularities and agglomerations in the distribution of star-matter, and to assimilate the entire stellar universe to a vast cloud of cosmic dust at relative rest, everywhere of the same density. It is then found that for this cloud of dust to maintain a permanent, stable configuration, the time direction of the universe as a whole, for an observer who is non-rotating with respect to the stars, must be straight. It is easy to see why this condition should arise.

If the stars can be considered at relative rest, inasmuch as their world lines constitute geodesics of space-time, it is necessary that these geodesics should not taper together but should lie along parallel time directions through space-time. This is possible only provided the structure of space-time is such that one same time direction endures at every point of space for an observer at rest with respect to the star distribution. We thus obtain one same straight time direction for the universe. Of course, this does not mean that around each individual star the space-time curvature, and hence the time direction, do not suffer modifications. Far from it; since it is these local curvatures which are responsible for gravitational attraction between matter and matter. In the present case, however, we are concerned solely with the structure of space-time as a whole, on a grand universal scale, so that the star distribution can be treated as uniform and likened to a gigantic cloud of dust. Stability, therefore, requires that this ultra-macroscopic structure should be conditioned by a straight time direction. This eliminates de Sitter’s universe, since in his model the direction of time was curved, and his universe, in consequence, not stable.

The net result of Einstein’s calculations was to show that if the universe were a closed three-dimensional spherical (or elliptical) space, not space-time, on which a straight time axis was erected, stability would be possible, the star velocities would remain low, and their distribution uniform. We should thus obtain the Cylindrical Universe, where the time direction proceeding from minus to plus infinity would represent the axis of the cylinder.