But when it is considered that Einstein’s theory is based on the relativity of velocity, then that it establishes, in a partial way at least, the relativity of acceleration, it cannot be denied that a logical extension of its triumphs would be for it to succeed in establishing the complete relativity of all motion and of inertia. Although Einstein never conceals his hope that this will be the ultimate outcome, there is no desire on his part to force the issue in defiance of facts.
It may, however, be of interest to ascertain what modifications in the theory would be required for the relativity of all motion and of inertia to be established. Inasmuch as it is the inherent structure of space-time persisting even in the absence of matter which is responsible for centrifugal force in an empty world, hence for absolute rotation and inertia, a vindication of Mach’s ideas would entail the following condition: It would be necessary to assume that in the absence of all matter, space-time would lose all trace of a structure and become amorphous. In other words, matter would have to createspace-time and its structure (i.e., the-distribution, or the metrical field), andnot merely modify locally a pre-existing structure. It is interesting to note that the relativity of rotation is thus leading us to a concept of space which could not have been anticipateda priori. Mach himself, when defending the relativity of motion, does not appear to have realised that the abandonment of Newton’s absolute space would entail a matter-created universe.
Now what Einstein has done has been to prove that it is mathematically possible to conceive of a universe in which the entire space-time structure (hence the entire-distribution, or metrical field) would be conditioned by matter. This, as we know, is the cylindrical universe. But, here again, we must remember that he has never asked us to accept this cylindrical universe merely because it entails the relativity of inertia and of rotation. However strongly the relativity of all motion may appeal to us, it reduces in fine to a mere philosophical fancy; and more imperative arguments must be presented before it can be entertained.
Such arguments were forthcoming when the gravitational equations were studied more closely. It was seen that they entailed those peculiar forces which Mach’s relativity of rotation compels us to attribute to the stars. Over and above these motives, Einstein was guided by a desire to avoid the conception of a gradually impoverished nuclear universe of stars. When, in addition, account was taken of the low star-velocities, he found himself driven once more to the cylindrical universe. Also, we may recall that the possibility of reducing matter to electricity and gravitation necessitates the same cylindrical universe.
When all these facts are taken into consideration, one can realise that the finite universe, entailing as it does the relativity of rotation, rests on solid ground. We do not say that Mach’s idea is vindicated, since the cylindrical universe follows from the stability of the star distribution, and this may be a mistaken premise. All that is maintained is that the complete relativity of motion is mathematically possible, a point which had never been established by Mach or by any one else. Indeed, as we have said, Einstein does not insist that we accept the cylindrical universe. He states expressly that the theory of relativity does not require it as an inevitable necessity; it merely suggests it as a possibility. If we assume that our stellar universe is concentrated into a nucleus surrounded by infinite space, the quasi-Euclidean infinite universe must take the place of the cylindrical one. With this alternative universe, matter still modifies space-time locally, but can never create itin toto. As Einstein tells us: “If the universe were quasi-Euclidean, then Mach was wholly in the wrong in his thought that inertia, as well as gravitation, depends upon a kind of mutual action between bodies.”
In short, we realise that the only means of coming to any decision on the form of the universe, hence incidentally on the problem of the relativity of rotation, must be through the medium of astronomical observation; in particular, by measuring the velocities of the stars in the Milky Way and globular clusters.
There exists also a third type of universe, de Sitter’s hyperbolical universe, which also is mathematically possible. In de Sitter’s universe, just as in the quasi-Euclidean one, matter can never create space-timein toto, and as a result Mach’s idea must be abandoned. But, here again, it is not a philosophical dislike for a matter-created universe which guides de Sitter to the hyperbolical universe; it is the restoration of absolute time in the cylindrical universe, and also the lack of invariance of the boundary conditions, which disturb him. This lack of invariance restricts the covariance of the natural laws, hence, from a purely mathematical standpoint, goes counter to the general principle of relativity. De Sitter criticises the cylindrical universe in the following words:
“It should be pointed out that this relativity of inertia is only realised by making the time practically absolute. It is true that the fundamental equations of the theory, the field equations, and the equations of motion ... remain invariant for all transformations. But only such transformations for which at infinitycan be carried out without altering the values of (I)”.[154]
These discussions on the form of the universe bring to light another important aspect of the methodology of science. We may notice that the accord which was practically unanimous in the earlier part of the theory now gives way to dissenting opinions. The reason for this is of course obvious. In the earlier part, experimental verifications were soon forthcoming and permitted us to decide matters one way or another. But when we are dealing with the form of the universe, verifications are extremely difficult; they depend on tedious astronomical observations which may require centuries to be completed. Hence, the only method of advance is to explore all mathematical possibilities in the hope that somehow or other we may be led to anexperimentum crucis. De Sitter places the emphasis on the mathematical principle of relativity. Einstein adopts the attitude of the physicist and prefers to stress the dynamical aspect of relativity. Weyl and Eddington adhere more closely to the pure physics of the field, following the procedure inaugurated by Maxwell and Faraday.
But in all cases the driving force behind these attempts has been a desire to co-ordinate mathematically a maximum number of experimental facts in the simplest way possible. These facts have been collected from the various realms of physical science, and, indeed, it is owing to their wide variety that the conclusions obtained are compelling. Furthermore, it should be noted that the facts appealed to are the results of highly refined experiments. This again is important, since we have seen that a difference one way or another of a tiny fraction of an angle or a length might be sufficient to overthrow an entire theory.
Summing up these discoveries, we see that a matter-created universe entailing the complete relativity of all motion to the universal masses appears to be mathematically possible. If this solution is the correct one, the cornerstone of Newton’s absolute space is finally destroyed. If, on the other hand, the cylindrical universe shouldprove untenable, we could scarcely claim that an absolute background to motion had been disposed of. In this measure, at least, Newton’s ideas would stand.
In short, we see that this problem of the relativity of rotation, of the relationships between matter and extension, of the finiteness of the universe, is not one for philosophy anda prioriarguments to solve. A mathematical co-ordination of the facts developed by ultra-precise experiment can alone yield us an answer. As Weyl tells us:
“The historical development of the problem of space teaches how difficult it is for us human beings entangled in external reality to reach a definite conclusion. A prolonged phase of mathematical development, the great expansion of geometry dating from Euclid to Riemann, the discovery of the physical facts of nature and their underlying laws from the time of Galileo together with the incessant impulses imparted by new empirical data, finally, the genius of individual great minds, Newton, Gauss, Riemann, Einstein, all these factors were necessary to set us free from the external, accidental, non-essential characteristics which would otherwise have held us captive.”
To be sure, as our knowledge accumulates and our mathematical ability increases, we may be led to revise former conclusions. This is why the conclusions of Einstein differ from those of Newton, just as those of our descendants may differ from ours of to-day. But the main point to grasp is that these variations in our philosophical outlook are brought about by our increase of knowledge both mathematical and physical; and this increase requires centuries: it cannot be obtained overnight. Indeed, had Einstein lived in Newton’s time and ignored the mathematical and physical discoveries of the present day, he could never have improved upon Newton’s solution. Inversely, had Newton lived in our time and been acquainted with the facts known to modern science, it is very possible that he would have created the theory of relativity.
In our analysis of the methodology of scientific theories we have attempted to show that the deductions of the great scientists were always based on experiment and not on wild guesses. But this does not mean that other factors have not also come into play. For instance, we mentioned that a proper choice of facts was of supreme importance. Whereas a certain choice may lead to nothing of interest, some other choice may issue in a great discovery. A case in point was afforded by Einstein’s attitude towards the well-known identity of the two masses.
But even this is not all. In mathematics, as in architecture, certain co-ordinations are beautiful, others top-heavy or unsymmetrical. When, therefore, physical phenomena are translated into mathematical formulæ, the theoretical physicist will always endeavour to obtain beautiful equations rather than awkward ones. Of course, since he must restrict himself to a slavish translation of physical results, his initiative in this respect may be extremely limited. Nevertheless, in certain cases it may be possible to add an unobtrusive term to the equations, though this term may not actually be demanded by experiment. If thisadditional term renders the equations more beautiful, there will be every incentive to retain it. It was an urge of this sort that guided Maxwell in his discovery of the equations of electromagnetics. The experimental data known in his day could not establish whether a certain additional term was necessary or not. Nevertheless, Maxwell introduced it because it beautified his equations, and was thus led to anticipate the existence of electromagnetic waves. As we know, subsequent experiment has justified Maxwell’s deep vision. It is much the same with the theory of relativity. One of the reasons for its great appeal to mathematicians is the extreme harmony, beauty and simplicity which it permits us to bestow on many of the equations of physics.
However, if we leave these aesthetic urges aside, we may say that the methodology that has yielded us the theory of relativity is the same methodology that has yielded all the great scientific discoveries. It is the methodology of Galileo, Newton and Maxwell: First, ascertain experimental facts; then, as needs be, frame tentative hypotheses or scaffoldings for the sole purpose of co-ordinating these facts into a consistent whole with a maximum of simplicity. Only when the theory has succeeded in accumulating a sufficient number of facts can its philosophical implications be studied. It follows that any attempt to reverse the normal order and posit the philosophy before the science will result in hampering future discovery by subordinating accuracy of treatment to loose guesswork. The history of human thought is full of discarded philosophic prejudices swept away by the onward march of science. Every new scientific discovery reveals aspects of nature, or even of the mind, as in pure mathematics, which we never suspected before. To-day, both the discoveries of relativity and the quantum hypothesis are case in point. It is precisely because the philosophy of a theory of mathematical physics can only be attaineda posteriori, coming as a crowning achievement, that the philosophy which is beginning to disentangle itself from Einstein’s discoveries is still in an embryonic stage. It is not that a scientific philosopher of sufficient scope could not be found; it is because there are many scientific aspects of the subject which are still in doubt. The following passage from Einstein will explain what we mean:
“From the present state of theory it looks as if the electromagnetic field as opposed to the gravitational field rests upon an entirely new formalmotif, as though nature might just as well have endowed the gravitational ether [metrical field[155]] with fields of quite another type, for example with fields of a scalar potential instead of fields of the electromagnetic type.
“Since according to our present conceptions the elementary particles of matter are also in their essence nothing else than condensations of the electromagnetic field, our present view of the universe presents two realities which are completely separated from each other conceptually, although connected causally, namely, gravitational ether [metrical field] and electromagnetic field, or—as they might also becalled—space and matter.
“Of course it would be a great advance if we could succeed in comprehending the gravitational field and the electromagnetic field together as one unified conformation. Then, for the first time, the epoch of theoretical physics founded by Faraday and Maxwell would reach a satisfactory conclusion. The contrast between ether [metrical field] and matter would fade away and through the general theory of relativity the whole of physics would become a complete system of thought like geometry and the theory of gravitation.”
We have quoted this passage at length, because it appears to us that by reading between the lines it is possible to realise the deeper philosophical problems pertaining to our knowledge of nature, hence to knowledge in general, which still obscure the significance of relativity.