The evolution of scientific thought from Newton to Einstein

by A. D'Abro

term, i.e., the equations applying to the infinite universe, namely

, had received brilliant support in having enabled him to account for the motion of the planet Mercury, and to anticipate such unsuspected phenomena as the double bending of a ray of light and the Einstein shift-effect. It was only natural, therefore, to endeavour to extract from these equations all the remaining treasures they contained. But here integrations of an extremely difficult nature awaited mathematicians.

When the equations were treated in a very approximate way, they yielded Newton’s law of gravitation and the Einstein shift. When the approximation was increased, they yielded the double bending of a ray of light. When the approximation was increased still further, we obtained the precessional advance of Mercury’s perihelion. When treated in a more general way, they enabled Einstein to predict that gravitation would be propagated with the speed of light. Thirring then attacked the equations, carrying his calculations to a still higher degree of approximation, and at last, the effects necessitated by Mach’s mechanics appeared. Thirring found that the rotation of a central body would modify the nature of its attraction on the planets or satellites circumscribing it. (These anticipations would appear to be susceptible of astronomical verification in the case of Jupiter and one of its satellites.) But, most important of all, he discovered that Mach’s belief in the gravitational influence of the stars rotating round a fixed body was finally justified. In particular, the equations proved that even in the hypothesis of an infinite universe, a rotation of matter would produce around a test-body a field of force disposed in exactly the same peculiar way as are the centrifugal and Coriolis forces in the space attached to a body in rotation. Furthermore, the mass of a test-body would be affected by the proximity of surrounding matter. This last consequence, however, can be obtained without resorting to difficult calculations.[117]

Now the gravitational equations which have yielded these interesting results have most certainly not been doctored up for the sole purpose of justifying Mach’s ideas. As explained in previous chapters, they were discovered by Einstein as a result of a totally different train of thought, long before the relativity of rotation was even considered. Hence, it cannot be denied that by vindicating, as they did, the correctness of Mach’s ideas, a fact of the utmost importance had been established. Mach’s mechanics received at last that rational justification which until then had been lacking; and the first inkling of a possible physical connection between the origin of inertia and gravitation was obtained. However, it is important to note that although from a qualitative standpoint the agreement was perfect, yet when we view the problem quantitatively, calculation shows that the increase of mass and the centrifugal forces generated by the rotation of the stars would produce but an infinitesimal fraction of the magnitudes observed in practice. In order to obtain magnitudes corresponding with observation, it would be necessary to assume that the stars were many trillion times more numerous than we believed them to be, and this supposition would lead us into other difficulties.

The reason for this discrepancy is not hard to discover. Centrifugal force, as we know, is produced when a body is torn away from the space-time geodesic which the metrical field would compel it to follow. To say, therefore, that a rotation of the firmament round a test-body would affect the field of force surrounding the body is equivalent to stating that a rotation of the stars would be capable of affecting the distribution of the metrical field round the body and of dragging the geodesics along. There is no reason to be surprised at some such effect taking place, seeing that the phenomenon of gravitation connotes that matter, whether at rest or in motion, must affect the curvature of space-time, hence the lay of the metrical field. But, on the other hand, until Einstein had discovered the cylindrical universe, the action of matter could only modify the lay of a pre-existing metrical field, and could not create such a field out of nothing. Space-time of itself was assumed to be flat, and matter could only endow it with a certain degree of curvature. The result was that the major part of centrifugal force was due to the intrinsic texture of flat space-time, and only an infinitesimal portion of this force could be attributed to the additional action of the stars. Centrifugal force thus still appeared, in the main at least, to betray absolute rotation in empty, absolute space-time.

None the less, an important point had been established. Mach’s ideas, though still refuted quantitatively, yet seemed to be acceptable qualitatively, whereas in classical science they had appeared utterly impossible from every standpoint. The thin end of the wedge had been driven in. From then on, it became permissible to assume that with an increase in our understanding of the universe the complete relativity of all motion would be established, not only qualitatively, but also quantitatively. In other words, not merely an infinitesimal portion of centrifugal force and of mass, but their totality, might be accounted for by appealing to the interaction of matter and matter. It is easy to understand how this achievement would be realised.

All we should have to do would be to assume that the totality of the metrical field, and not merely a portion of it, was created by the distribution of matter throughout the universe. As we have seen, this condition is precisely the one realised in the cylindrical universe; so that with Einstein’s universe, Mach’s ideas may be vindicated and the bugaboo of absolute rotation dispelled.[118]

Both de Sitter’s universe and the quasi-Euclidean one would be incompatible with Mach’s mechanics, since in either universe the metrical field and the geodesics of space-time would exist in the complete absence of matter. This means that in the empty world of de Sitter, and far beyond the nucleus of stars in the quasi-Euclidean universe, bodies would be submitted to forces of inertia when torn away from their natural geodesical world lines. In either case the space-time void would appear to be endowed with dynamical properties even in the absence of all matter.

The following passage quoted from Einstein’s writings expresses his views on the subject: