The evolution of scientific thought from Newton to Einstein

by A. D'Abro

Could we but annihilate all the stars, we should have a means of solving the problem; for in Einstein’s universe the result would be the total disappearance of mass and centrifugal force. In fact, complete chaos would result; for the metrical field of space-time would have vanished, so that light rays and free bodies would no longer know how to move. But as an annihilation of the stars is beyond the range of possibility, we must have recourse to other methods. So we are led to examine the question of stability. Is it true that the stars are in a state of statistical equilibrium? Is it true that their velocities are always small compared to that of light? Eddington casts some doubt on this stability by remarking that the spiral nebulæ appear to be animated with enormous velocities. This fact, coupled with the apparent reddening of their light, leads him to manifest a preference for de Sitter’s universe. But there exists still another method of settling the question.

We remember that Einstein’s original law of gravitation in the case of the infinite universe differed in certain respects from Newton’s, and that these differences were important enough to be detected within the limits of the solar system through their effects upon Mercury’s motion and the double bending of a ray of starlight. But if now we assume the universe to be finite, and hence include the

term in the gravitational equations, calculation shows that a further modification will ensue. It will be as though all space round a star were filled with negative or repulsive matter of very low density, of the same order as the curvature of the universe. The influence of this factitious negative mass distribution would be to decrease the normal value of the star’s attraction; the decrease, owing to the cumulative effect produced by the distribution, becoming more and more apparent as we considered points farther removed from the star. Needless to say, this additional modification in the law of gravitation would be incredibly small, owing to the minuteness of the world-curvature and hence of the density of the negative mass distribution. It would be idle to hope to detect it unless we studied gravitational phenomena occurring over tremendous astronomical extensions. But the effect might be observed in the following way:

Consider, for instance, gravitating masses such as the sun and its planets. We know that the reason the solar system does not collapse and all the planets fall on to the sun is because the planets are in motion (as referred to an inertial frame attached to the sun). There exists a definite mathematical relationship between the velocity of a planet along a definite orbit and the force of gravitation to which it must be subjected by the sun. The greater the velocity, the greater must be the sun’s attraction in order to maintain the planet in its orbit. Now the globular clusters of stars and the Milky Way constitute precisely such permanent gravitating systems of enormous extent. The individual stars attract one another, and their mean velocities allow us to deduce the resultant gravitational forces to which they must be subjected. If, therefore, on applying Einstein’s original law of gravitation (that corresponding to the infinite universe),[119] we found that the mean velocities of the stars in the Milky Way were smaller than would be expected, we could consider it proved that the law of attraction must fall off more rapidly than Einstein’s original law indicated. This discrepancy would then justify our belief in the finiteness of the universe. The value of

, the cause of this discrepancy, could be ascertained, and with it, of course, the value of the universal curvature. Knowing the curvature, we could determine therefrom the size of the universe, the average density of matter, and finally the total mass of the universe. Einstein is of the opinion that astronomical observations of this type will eventually settle this question.

We have exhausted the principal points which prompted Einstein to believe in the finiteness of the universe, but there are yet others connected with the possibility of reducing matter to electricity and gravitation. We shall touch on this additional aspect of the question in a later chapter.[120] For the present, we may pass to a totally different type of problem, also connected with the finiteness of the universe. We refer to the standard of gauge.

As Eddington points out, when our equations prove to us that the universe of space must be spherical and have a constant curvature, what else can it mean but that if we measured the radius of curvature with a material rod, we should obtain the same magnitude in every direction? But then it follows that the radius of the universe in any direction constitutes the gauge of length which nature imposes upon us; and that all bodies in equilibrium adjust themselves automatically so as to maintain some definite fraction of the length of this radius, in whatever direction they be placed. This view suggests that for material bodies to be on a definite scale of size, there must be a curvature of the world. Eddington also suggests that inasmuch as the finite universe is open in a time-like direction, whereas it is limited in space, a possible explanation may be given of the apparent permanency of existence of certain entities such as electrons and atoms, which are limited in spatial extension but are everlasting in time. Weyl’s theory throws further light on these rather hypothetical problems.

The finite universe introduces us to the difficult conception of a spherical space. We have already discussed this type of space in a previous chapter; but in view of its additional importance, now that the real universe is suspected of being one, it may be worth while to examine the problem afresh.