It must be noted that the necessity for introducing potential energy, as an additional requirement in order to ensure conservation, exists only when we are considering a phenomenon occurring in a gravitational field or in an inertial field of force, such as would be present were we to take our stand in an accelerated frame. In the absence of such a field, potential energy would become completely meaningless, and conservation of energy would endure without it.
If now we take into consideration Einstein’s discoveries, in the special theory where mass is identified with energy, we see that in the general case of a gravitational or inertial field being present, conservation of both mass and energy can be upheld, but only provided we introduce the concept of potential energy. It remains to be seen whether potential energy possesses any intrinsic significance in the world, or whether it represents a mere mathematical artifice introduced for the purpose of saving the principles of conservation. The general theory proves that this last alternative is the correct one and that conservation in the classical sense must be abandoned. The reason is that when we express this space-time equation of conservation in a semi-Cartesian mesh-system of space and time (a strictly Cartesian one being impossible, owing to the intrinsic curvature of space-time round matter), we find that it expresses true conservation in space and time, in the sense of the special theory, without the adjunction of potential energy; but when we select an arbitrary curvilinear mesh-system of space and time, the mathematical expression we are studying splits up into two terms, and the conservation of this expression implies, therefore, the conservation of the sum total of these two terms. The first term refers to the mass, vis viva and momentum of the matter, while the second term refers to the energy and momentum of the gravitational or inertial field. It would thus appear that conservation could be maintained provided we extended it so as to include the energy and momentum of the gravitational field as well as that of the matter. This, indeed, was Einstein’s original stand. Later, he seems to have abandoned it for the following reasons:
The second term, which represents gravitational or potential energy in our equations of conservation, is given by a mathematical expression which is not a tensor. It expresses, therefore, merely a relationship between the motion of the observer and the structure of space-time; it does not describe the structure of absolute space-time itself. We must realise, then, that potential energy represents nothing inherent in the absolute world, and we must exclude it as a fundamental entity in the formulation of our laws.
As soon, however, as we refuse to take gravitational or potential energy into consideration, conservation breaks down in a field of force. Hence, it follows that conservation of mass, energy and momentum holds only for Galilean observers. Conservation is not, therefore, a true law of nature, valid for all observers. In the light of these discoveries, the permanent nature of mass and matter can no longer be upheld. Lavoisier’s principle of the conservation of mass must be abandoned, and the conception of matter as a substance is shown to be fundamentally incorrect.
But then if the ordinary laws of conservation are discarded, by what shall we replace them? Obviously, they will now give way to the more general principle of relativistic conservation, a sort of space-time principle. This type of conservation cannot in the general case be expressed in terms of the conservation of our habitual physical quantities in space and time. Its significance will be deeper, referring as it will to space-time uniformities existing in the absolute world of space-time.
Only when the observer adopts a Galilean or a semi-Galilean system of reference will physical conservation endure; only then will it be possible for a man to conceive of a universe of permanent spatial entities. As Eddington remarks, it is only when these Cartesian or semi-Cartesian mesh-systems are used that the separation of space-time into four directions yields the habitual space and time of men. Thus there appears to be a relationship, between the natural separation of time and space by our consciousness, and the insistent demands of the mind to seek out the permanent rather than the transient in order to construct an edifice of knowledge.
CHAPTER XXXIII
OTHER ASPECTS OF THE GRAVITATIONAL EQUATIONS
WE remember that in classical science the Newtonian law of gravitation was expressed by Poisson’s equation,
where