CHAPTER XIV
RELATIVISTIC MECHANICS
WE saw in the preceding chapter that the time and space variables which entered into the Lorentz-Einstein transformations should be considered of absolutely general validity. They were assumed to represent the duration actually sensed and lived by the observer and the spatial measurements actually determined by his rigid rods in his Galilean frame. It was no longer a case, as in Lorentz’s theory, of supposing that these space and time variables applied only in certain electromagnetic cases and not in others. They were now of absolutely general application, and in particular should hold for all mechanical phenomena.
But this realisation brought us face to face with a peculiar difficulty. Einstein’s fundamental postulate was that Galilean motion was meaningless, or at least could never be detected by experiment—whether of a mechanical or electromagnetic nature. Now the complete relativity of Galilean motion is expressed mathematically by the invariance of form of the laws of nature for transformations corresponding to that type of motion. The classical laws of mechanics satisfied this requirement in classical science, since they remained invariant when submitted to the classical transformations. But if we accept Einstein’s views, these laws must remain invariant no longer to the classical transformations but to the Lorentz-Einstein ones.
The classical laws of mechanics did not fulfil this new condition of invariance. From this it would appear that the relativity of Galilean motion which Einstein had succeeded in establishing for electrodynamic and optical experiments could be maintained only at the price of abandoning this same relativity for mechanical ones. What had been gained on one side had been lost on the other. Inasmuch as the laws of mechanics would certainly have to remain invariant to the Lorentz-Einstein transformations if absolute velocity were to remain meaningless, and inasmuch as the classical laws did not satisfy this condition, there was only one way out of the difficulty. It was necessary to assume that the classical laws were incorrect.
Accordingly Einstein was compelled so to modify the classical laws of mechanics as to ensure their invariance to the Lorentz-Einstein transformations. This modification was then found to entail the relativity of mass. By this we mean that the mass of a given body could no longer be regarded as an invariant, independent of the body’s relative motion. On the contrary, it was now found to increase with the relative velocity of the body, becoming infinite when the velocity of light was reached. In the same way the relativity of force was established, and classical Newtonian mechanics, with its invariance of mass and force, was proved to be incorrect, and acceptable only as a first approximation for slow velocities.
It also followed that no material body could ever move with a speed greater than or even equal to the speed of light, since it would require a force of infinite magnitude to give it this critical velocity. A number of paradoxes associated with the theory are due to the critic’s initial assumption that the observer is moving with the speed of light. But all these paradoxes may be dismissed at the outset, since no observer could ever move with such a speed.
When the mathematical expression of the mass of a body in relative motion was considered, it was found to be decomposable into several parts. First of all we had the mass of the body as at rest (this was the mass which classical science assumed to be invariant, i.e., independent of the body’s motion). To this was added the vis viva, or energy of motion of the body with respect to the observer, as well as a number of other terms depending on the relative velocity. These last terms were, however, of very minor importance unless the relative motion of the body was considerable. Thus the mass of a body in relative motion was increased over its mass at rest by its energy of motion or vis viva, so that the distinction between mass and energy became obscured.
Since part of the mass of the body when in motion was due to its vis viva, it appeared probable that the entire mass of the body, whether at rest or in motion, was due to its energy of one kind or another. Now matter even at rest was known to contain energy, as exhibited by radioactive phenomena and by the thermal effects produced by chemical combinations. So it was natural to assume that the mass of the body at rest was due to its contained or bound energy, which would be released were the body to disintegrate into nothingness. The formula connecting this energy of constitution of the body with its mass was obtained from the mathematical expression of the mass, and it was found that in C.G.S. units the contained energy was equal to the mass multiplied by the square of the velocity of light. From this formula it is easy to realise the tremendous liberation of energy and the dire consequences that would result from the complete disintegration of even a small parcel of matter.
Another consequence of these discoveries was that any variation in the internal energy of a body maintained at rest, such as would result from compressing it, heating it or electrifying it, would be accompanied by an increase in its mass at rest.[50] This discovery enabled us to account for a curious discrepancy in the comparative masses of the helium and of the hydrogen atom, a mystery which had never been accounted for by classical science.
To illustrate: The nucleus of the hydrogen atom is composed of one proton; and the nucleus of the helium atom is composed of four protons, held together by two electrons. The protons alone having a mass of any importance, it seemed logical to assume that the mass of the helium atom would be four times that of the hydrogen atom. Accurate measurements proved, however, that there was a slight deficiency, and that the mass of the helium atom was less than four times that of the hydrogen atom. Classical science had recognised that the closer the electrons and protons were packed together in the nucleus, the smaller would be the electric energy of the atom, but it appeared impossible to account for the deficiency in mass. The theory of relativity by its identification of mass and energy gave an obvious solution to the problem. The deficiency in mass of the helium atom was to be ascribed to the loss of energy radiated under the form of heat and light when the electrons and protons were being packed together. At the same time it became possible to calculate with precision from the loss of mass the precise amount of energy that had been liberated. It followed that in regions of the cosmos where the helium atoms were in process of formation owing to the grouping together of the protons and electrons, enormous amounts of heat and light would be radiated continually. This consequence of the theory enabled astrophysicists to account for the enormous temperature of the sun, maintained through thousands of millions of years in spite of incessant radiation in a frigid space. Helium atoms are in constant process of formation in the sun, and in this fact resides the source of its apparently inexhaustible supply of energy.