It is surprising that Newton and Einstein agree in expressing the movements of gravitating stars in an almost identical form, because their starting-points are very different.
Newton starts from the hypothesis of absolute space, the empirical laws of the motions of the planets expressed in Kepler’s laws, and the belief that gravitational attraction is a force proportional to mass. Einstein, on the other hand, in making his calculations starts from the conditions of invariance which we indicated. He starts, in a sense, from the philosophical principle or postulate or impulse to hold that the laws of nature are invariant and independent of the point of view—irrelative, if I may use the word.
Einstein even abandons the hypothesis which ascribed the curving of gravitational paths to a distinct force of attraction. Yet, starting from a point of view so different from that of Newton, and one that seems at first less overloaded with hypotheses, Einstein reaches a law of gravitation which is almost identical with Newton’s.
This “almost” is of immense interest, because it enables us to test which is the accurate law, that of Newton or that of Einstein. They give the same results when there is question of velocities that are feeble in comparison with that of light, but their results differ a little when there is question of very high velocities. We have already seen that, near the sun, light itself is bent out of its course in exact conformity with Einstein’s law, and in a way that Newton’s law did not predict as such.
But there is another divergence between the two laws. According to the Newtonian law the planets revolving round the sun describe ellipses which—neglecting the small perturbations due to the other planets—have a rigorously fixed position.
Suppose we put on a table a slice of lemon cut through the longer diameter of the fruit, and imagine that the chief stars, the northern constellations, are painted on the vaulted roof of the vast hemispherical room in the middle of which we place our table. The slice of lemon has very nearly the form of an ellipse, and, if we take one of the pips to represent the sun, it will stand for the orbit of one of our planets. Newton’s law says that—after making due corrections—the planetary orbit keeps a fixed position relatively to the stars as long as the planet continues to revolve. This means that the slice of lemon remains stationary.
Einstein’s law says, on the contrary, that the orbital ellipse turns very slowly amongst the stars while the planet traverses it. This means that our slice of lemon must turn slightly on the table, in such wise that the two ends of the lemon do not remain opposite the same stars painted on the wall.
If we calculate, in virtue of Einstein’s law, the extent to which the elliptical orbits of the planets must thus turn, we find it so small as to be impossible of observation except in the case of one planet, the swiftest of all, Mercury.
Mercury revolves completely round the sun in about eighty-eight days, and Einstein’s law shows that its orbit must at the same time turn by a small angle which amounts to forty-three seconds of an arc (43″) at the end of a century. Small as this quantity is, the refined methods of the modern astronomer can easily measure it.
As a matter of fact, it had been noticed during the last century that Mercury was the only one of the planets to show a slight anomaly in its movements, which could not be explained by Newton’s law. Le Verrier made prodigious calculations in connection with it, as he thought that the anomaly might be due to the attraction of an unknown body lying between Mercury and the sun. He hoped that he would thus discover, by calculation, an intra-Mercurial planet, just as he had discovered the trans-Uranian planet Neptune.