The evolution of scientific thought from Newton to Einstein

by A. D'Abro

It is to be noted that as space-time can be split up into a space and a time by any given observer (at least in his immediate vicinity), the space-time-structure will automatically be split up into a space-structure and a time-structure, or a spatial and a temporal metrical field. The behaviour of rods and bodies at rest with respect to the observer will be controlled by the space-structure, while the beatings of perfect clocks such as vibrating atoms will be controlled by the time-structure. When we consider space and time together, we may say that the difference between space-time and separate space and time can be expressed as follows: Whereas the three-dimensional Euclidean space of classical science could be considered as cleaved by parallel time-stratifications, defining the spatio-temporal situations of simultaneous events and separating neatly space from time, no such stratification exists in space-time. If we wish to adhere to the older conception, we may still speak of time-stratifications through space-time, but we must be careful to add that these stratifications manifest no fixed direction. This is due to the fact that those stratifications which would be time-stratifications defining simultaneous events for one observer would cease to be time-stratifications defining simultaneous events for another observer in relative motion. The relativity of simultaneity is but another way of expressing the same facts. But, of course, with this admission, the conception of time-stratifications loses all deep significance. It follows that, when discussing the space-time structure, it would be preferable to omit mentioning such relative concepts as those of space and time. Accordingly, we will say that the structure of space-time is defined by the light-cone, that is, by the locus of the world-lines of light rays issuing from any given point at any given time. It is this light-cone which will split up space-time into its permissible space and time directions.[161]

Let us revert to the metrical field, as defining the space-time structure. Although Riemann had attributed the existence of the structure, or metrical field, of space to the binding forces of matter, there is not the slightest indication in Einstein’s special theory that any such view is going to be developed later on; in fact, it does not appear that Einstein was influenced in the slightest degree by Riemann’s ideas. At any rate, in the special theory, the problem of determining whence the structure, or field, arises, what it is, what causes it, is not even discussed in a tentative manner. Space-time, with its flat structure, is assumed to be given or posited by the Creator.

But in the general theory the entire situation changes when Einstein accounts for gravitation, hence for a varying lay of the metrical field, in terms of a varying non-Euclidean structure of space-time around matter. We are then compelled to recognise not only that the metrical field regulates the behaviour of material bodies and clocks, as was also the case in the special theory, but, furthermore, that a reciprocal action takes place and that matter and energy in turn must affect the lay of the metrical field. But we are still a long way from Riemann’s view that the field is not alone affected but brought into existence by matter; and it is only when we consider the cosmological part of Einstein’s theory that this idea of Riemann’s appears to be vindicated.

And here we come to a parting of the ways with de Sitter and Eddington on one side, Einstein and Thirring on the other, and Weyl somewhere between the two extremes. The differences of opinion arise from the views to be entertained on the subject of the origin of the metrical field. It is impossible to accept Einstein’s general theory without admitting that the field regulates the behaviour of matter and all processes of change, and is in turn affected by matter yielding under its influence; but the point to be determined is whether this field or space-time structure is entirely created by matter, or whether some type of structure would subsist in the absence of all matter.

Now we saw when discussing the form of the universe, that if de Sitter’s spherical universe or the infinite quasi-Euclidean universe is adopted, inasmuch as they can both exist in the total absence of matter, we must infer that the metrical field is manifestly not brought into existence by matter. If, on the other hand, we accept Einstein’s cylindrical universe necessitated by the apparent stability of the star distribution, a direct relationship is found to exist between the size and geometry of the universe and the presence of matter. It is not necessary, however, to assume with Einstein that matter creates the field. We might also hold with Eddington that it is the field which creates matter.[162] If Eddington’s views are entertained, what our senses recognise as matter is constituted by certain curvatures of the metrical field. Were the universe to contract, a certain amount of matter would be annihilated; were it to expand, a corresponding amount of matter would be brought into existence. If we accept Einstein’s views it is the reverse. More matter would cause the universe to expand and an annihilation of matter would cause it to contract. We saw that Einstein’s universe would appear to be in perfect harmony with Mach’s belief in the radical relativity of rotation with respect to the stars, and in the relational nature of mass. Just as classical science recognised that weight was due to the proximity of other bodies, so do Mach and Einstein believe that similar considerations must be extended to mass. It is the same eternal conflict between those who, like Maxwell and Faraday, hold to the pure physics of the field, exalting the electromagnetic field over electricity and magnetism proper; and those who, like Lorentz, exalt the substantial electron over the field, or at least consider the field and the electron realities of equal importance, but of different categories.

From the standpoint of methodology there is an interesting difference between the special and the general theories. In the special theory the space-time structure is explored by physical means. It is the Euclidean behaviour of our rods, the behaviour of our clocks, and finally the rectilinear paths and constant invariant velocity of light rays, that indicate that the structure must be flat or semi-Euclidean, at least so far as the accuracy of our observations permits us to assert. Theoretically, the same physical method of exploration might have been continued in the neighbourhood of large masses of matter. But it must be remembered that by these methods we could never have explored regions of space-time in the interior of matter. Furthermore, as can be gathered from the minuteness of such effects as the bending of a ray of light passing near the sun, and from the elusiveness of the Einstein effect, we now know that variations from flatness are most certainly exceedingly minute. Under the circumstances, it would have been extremely hazardous to attribute to a non-Euclideanism of space-time effects which might also have been ascribed to contingent physical influences.

Furthermore, even in the case of falling bodies, which describe ellipses or parabolas deviating widely from straight courses covered with constant speeds, so long as the ordinary Newtonian force of attraction was deemed responsible for these deviations, no particular reason could be given for attributing them to a non-Euclideanism of space-time. It was only after Einstein had become convinced that a modification in the structure of space-time could not help but exist around matter that a new mathematical method suggested itself for determining the structure prior to all physical exploration. We have seen that this method was furnished mathematically by the general principle of relativity, according to which all laws of nature, and in particular the law of gravitation, should be capable of being expressed by tensor equations.

Now it is most important to understand that although, in the case of the space-time structure, or metrical field, the method of discovery differs in the special and general theories, yet the significance of the structure, once it has been determined, remains exactly the same. In either event it is the structure which conditions the behaviour of rods and clocks and directs the courses of light rays and of bodies falling along geodesics; so that although the general structure was not discovered by means of direct physical measurements, theoretically it should have been, had our measurements been susceptible of greater accuracy. Tests of the relativity theory consist, therefore, in discovering whether or not ultra-refined measurements with rods, clocks, light rays and free bodies will reveal the structure as predetermined by Einstein. We know that the motion of Mercury and the double bending of a ray of light have confirmed Einstein’s previsions. But in order to be doubly sure we should also require measurements with rods and with clocks. With rods, unfortunately, the verification is quite impossible, owing to the minuteness of the anticipated deviations from Euclideanism around matter; but with clocks the verification should be possible by appealing to the vibrations of incandescent atoms. This was the reason why the detection of the Einstein shift-effect was held to be of such vital importance. Had this shift been proved non-existent, the theory would have collapsed. It might have been possible to save it by assuming that the atom did not adjust the frequency of its vibrations to the time-curvature of the structure, or, in other words, did not behave like a perfect clock. But as every one will recognise, had this avenue of escape been taken, the entire beauty of the theory would have vanished along with its ability to enable us to foretell and to foresee. Of what significance would be this knowledge of the structure around matter, if among other things it did not result in moulding the time-rate of all processes of change to its curvature? At any rate, Einstein stated long before the shift-effect was finally observed on the companion of Sirius that he would abandon the theory if the shift should prove non-existent. Fortunately for this beautiful theory, such a course is now obviated.

Further verifications of the structure would relate to explorations with rods and clocks in the interior of matter. But, as may well be realised, explorations of this sort are exceedingly difficult and we have to be satisfied for the present with Einstein’s theoretical anticipations in this connection.

When we come to the form of the universe, hence to the space-time structure of the universe as a whole, theoretically again physical explorations should yield us the structure. For instance, a ray of light should circle round the universe, and the motions of the stars in the Milky Way and of the planets in the solar system should be affected by the universal curvature. But once more, in view of the difficulty of direct physical exploration, mathematical methods were resorted to; and, as we have seen, Einstein appealed to the structural tensor law of curvature, which would be compatible with the stable appearance of the star constellations. He thus obtained the cylindrical universe. If this law is adopted, space-time is never perfectly flat; the flat continuum of the special theory becomes an ideal limiting case.