When Einstein had discovered this influence of relative velocity over mass, it was suggested that the curious motion of the planet Mercury might be explained without our having to abandon Newton’s law of gravitation, but merely by taking into consideration the variations of the mass of this fast-moving planet at different times of the year. Calculation proved that under the circumstances a perihelial precession in Mercury’s orbit would indeed be in order, but that it would be considerably smaller than that actually observed. Only when the general theory compelled Einstein to abandon Newton’s law was the precise motion of Mercury accounted for, as has been explained in a previous chapter.
On the other hand, this variation of mass with velocity was soon to lead to the discovery of very important phenomena in the realm of intra-atomic motions. The atom, we must remember, is a miniature solar system, with electrons revolving round a central nucleus just as the planets revolve round the sun. Sommerfeld, by taking into consideration the relativistic variation of mass with velocity, proved that if Bohr’s conception of the atom was correct, the spectral lines emitted from an incandescent atom should be in the nature of bundles of very fine lines closely packed together. Very refined optical tests soon proved that these anticipations were correct; the thick spectral lines formerly observed turned out, upon more refined investigation, to be formed by a number of fine lines in the precise manner predicted by Sommerfeld. This discovery had therefore a twofold effect. It vindicated Bohr’s theory of the structure of the atom, while at the same time it supported Einstein’s special theory of relativity.
By affording a further empirical proof of the correctness of Einstein’s views, these experiments, in conjunction with Bucherer’s, proved conclusively that the mass of a body was a relative and not an invariant, so that the mass of a billiard ball in motion through our frame of reference would have to be considered greater than its mass when at rest. Nevertheless, as was shown by Einstein, it was still possible to adhere to the classical belief in the principle of the conservation of mass, provided that by mass we understood the relative mass as now formulated, and not the mass of the body at rest. Imagine, for instance, two billiard balls colliding and then rebounding in the space defined by our Galilean frame. After their rebound the masses of the two individual balls will have varied, since their velocities with respect to the observer will have changed; but the relativity theory proves that the sum total of both these masses will remain the same for all Galilean observers, after as before the impact. This is what is meant by the conservation of mass.
Were we to conceive of only one billiard ball, first at rest in our Galilean frame and then in motion, its mass, of course, would vary, according to relativity. But it must be realised that in order to set the body in motion we should have to submit it to the action of a force, so that we should be introducing a foreign influence which would have to be taken into consideration.
Now, this expression of the mass of a body, varying as it did with the relative motion of the body, was proved by Einstein to be equal to the mass of the body at rest, plus a certain mathematical expression which for slow velocities became identical with the vis viva, or classical energy of motion, of the body in our frame. Accordingly, the energy of motion of a body could be regarded as identical with the increase of its mass in motion over its mass at rest. It became, then, highly probable, for a number of reasons, that what we called the mass of a body at rest was itself due to the contained energy of the body. In a general way, therefore, a body, when heated or electrified or compressed, would increase in mass.
This identification of the mass of a body at rest with its contained energy, and of its mass when in motion with its contained energy plus its generalised vis viva, or energy of motion, allows us to assert that the principle of the conservation of mass is none other than the principle of the conservation of energy. Hence, Einstein’s special theory is also in harmony with the latter principle. In a similar way, the principle of the conservation of momentum, or Newton’s law of action and reaction, would be found to stand in full harmony with Einstein’s special theory, provided that by momentum we meant relative mass multiplied by velocity, instead of mass at rest multiplied by velocity.
In short, the principles of conservation endured only when we enlarged our understanding of mass, momentum and energy; and Newtonian mechanics had to be amended as a result of Einstein’s discoveries in the special theory of relativity.
Let us now study the problems of conservation in the general theory when we make use not merely of a Galilean frame of reference, but of any frame whatever, hence when our space is permeated with a field of inertial forces or, again, with a field of gravitation. We shall discover once more that conservation is required by the general theory; though, as we shall see, it will not be the type of conservation to which we were accustomed in classical science.
In order to make these points clearer we must revert to Einstein’s law of gravitation. We remember that assuming the correctness of Einstein’s views of gravitation as being due to space-time curvature, the principle of stationary action led us to the law of gravitation:
