CHAPTER XXXIITHE PRINCIPLES OF CONSERVATION
CLASSICAL science had assumed that the mass of a body was an invariant, which would not change in value when the relative motion of the body was changed. It is true that highly refined experiments on electrons moving at enormous speeds (Bucherer’s experiment) had shown that the mass of an electron increased with its velocity; but it was assumed that this increase in mass was due to other causes, either to a modification in the electromagnetic field of the electron, or possibly to the mass of the ether which the electron in its rapid motion would drag along with it.
One of the first triumphs of the special theory of relativity was to prove that mass, just like time and space, must be a relative; and that a body in motion with respect to the observer would suffer an increase in mass in all ways identical with that disclosed by Bucherer’s experiment. This increase could no longer be attributed to the drag of the ether, since in the special theory of relativity exactly the same increase would have to be expected, regardless of whether it was the observer or the electron that was in motion through the ether. As in all previous examples, the essential factor was the relative motion between observer and observed. The ether played no part.
Calculation then showed that the mass of a body would become infinite when the velocity of the body reached that of light; for this reason no material body could ever move faster than light, since it would require an infinite force to increase its velocity any further.
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, provedthat 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 thevis 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 generalisedvis 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. Ina 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:
Now the tensor on the left possesses the remarkable mathematical property of expressing a form of curvature which is preserved regardless of our mesh-system. It follows that in virtue of the equations of gravitation written above, the same conservation must hold for,hence for mass, energy and momentum.
The mathematical identity that expresses this property, let it be noted, does not depend on Einstein’s theory; it is a purely mathematical property between the’s of a continuum, which had been discovered by the mathematician Ricci before the theory of relativity was born. But in view of the physical relationships which Einstein had established between the curvatures of the space-time continuum and the presence of matter, it now followed as a necessary consequence of the theory that the right-hand side of the gravitational equations must present the same characters of conservation as the left-hand side. If we interpret this result physically, we see that it implies the conservation of mass, momentum and energy.
We thus reach the remarkable conclusion that the principles of conservation, which in classical science were of an empirical nature and were totally unconnected with the law of gravitation, now appear as immediate and necessary consequences of Einstein’s law of gravitation. In other words, granting the correctness of the law of gravitation, conservation cannot help but exist in this universe. To many thinkers this is one of the most beautiful aspects of Einstein’s theory, showing as it does the wonderful unity and relatedness of nature.
Nevertheless we must be careful to understand the deeper significance of conservation in relativity. We shall see that the type of conservation we are now discussing is somewhat different from the classical conception of conservation, under which matter, for instance, disappeared from here only to reappear there. In the present case conservation is more ethereal and abstract. We may explain these points better by examining conservation in classical science. Here, when fields of force are present, the principles of the conservation of energy and momentum would appear at first sight to be contradicted by experiment. Thus, a body falling freely towards the earth can certainly not be said to possess a constant vis viva, or energy, since, originally starting from rest, it acquires an increasing speed as it falls towards the ground. Rather than recognise the breakdown of the principle of the conservation of energy, classical science preferred to assume that the body possessed two types of energy: a kinetic energy given by,and a potential energy depending on the position of the body in the field of gravitation. It was the sum total of these two types of energy that remained constant and was therefore conserved during the fall of the body.
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 vivaand 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 theenergy 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.