The evolution of scientific thought from Newton to Einstein

by A. D'Abro

We may say, then, that in the absence of forces—for instance, in the interior of a falling elevator—the ball, if started with some initial speed, would describe that geodesic of the surface which lay tangentially to its initial velocity. In addition, the ball’s speed along the geodesic would remain constant (in the absence of friction).

And so, were we to discover one of the geodesics of the surface by stretching a string, and were we to mark the lay of this geodesic with a line of ink, we should find that the ball, when constrained to move without friction over the surface, would follow this line provided its initial velocity were directed tangentially to it. To all intents and purposes, geodesics act, therefore, as grooves guiding the progress of free bodies; and the free bodies follow these grooves with whatever initial speed we may have given them. All these considerations apply solely when no external forces are acting on the moving body. The presence of a force tears the body away from the geodesic it would normally follow, and also accelerates its speed.

We now come to Newton’s law of gravitation. It was a remarkable fact that although, according to the law of inertia, the motions of free bodies should be directed along the straight geodesics of three-dimensional Euclidean space, yet the courses and motions of the planets round the sun and of projectiles near the earth’s surface deviated widely from these geodesic paths. In the case of projectiles, a cause for this deviation was easy to discover. It was attributed to the disturbing action of the force of gravitation exerting itself in the earth’s neighbourhood. But in the case of the planets the discrepancy was harder to explain. Newton solved the problem by extending the scope of gravitation; namely, by assuming that a gravitational force varying inversely as the square of the distance, and directed towards the sun, was acting on the planets. He was thus able to account for the elliptical orbits of the planets, and also for their motions along these orbits, as given by Kepler’s laws.

Thus, in classical science, we were faced with straight geodesics and with forces emanating from matter and compelling bodies to depart from their normal geodesic courses when moving in the proximity of matter.

But we, who, in contradistinction to Newton, know of non-Euclidean geometry, might be tempted to offer another solution of gravitation. Viewed from the standpoint of non-Euclidean geometry, the motion of the point over the surface (discussed previously) can be interpreted as the free motion of a point in a two-dimensional non-Euclidean space (the space of the surface). Hence we may say that whether space be Euclidean or non-Euclidean, least action demands that free bodies follow its geodesics with constant speed. This holds regardless of the dimensionality of the space, whence we may conclude that were our three-dimensional space non-Euclidean instead of Euclidean, Newton’s law of inertia would have to be revised. It would no longer be correct to state that free bodies described Euclidean straight lines with constant speeds. Instead we should say: “Free bodies pursue the geodesics of space with constant speed.”

Thanks to this generalisation of the law of inertia, we are enabled to extend classical mechanics to spaces of all kinds. Only in the particular case where space is Euclidean does our generalised law of inertia coincide with that given by Newton.

From an observational standpoint, what would be the result of space manifesting itself as non-Euclidean? The geodesics would be curved, as contrasted with Euclidean ones, and free bodies unsolicited by forces would appear to follow curves with constant speeds. Under the circumstances, it would seem to be possible to account for the curved paths of planets and projectiles without having to introduce a disturbing force of gravitation. All we should do would be to assume that space was suitably curved around large masses, such as the sun and earth; and as a result the geodesics, hence the paths of free bodies, would be curved in turn. In particular, planets would now circle round the sun, not because the attraction of the sun compelled them to do so, tearing them away from their straight geodesics, but because the space around the sun being now curved, its geodesics would automatically become curved lines.

Ideas of this sort were advanced tentatively by Lobatchewski, Riemann and Clifford; but, as we shall proceed to explain, a solution along these lines was physically and mathematically impossible so long as we restricted our investigations to the separate space and time of classical science.

A first difficulty would be encountered when we contrasted an exploration of the metrics of space by the method of rod measurements, with that of free bodies following geodesics. We remember that when discussing the free motion of a ball constrained to move on a curved surface, we said that the ball would describe the shortest course between two points on the surface, in other words, a geodesic; that had we stretched a string over the surface between these two same points, the string in turn would have placed itself along the same geodesic (it was in this sense that the geodesic could be considered as defining the shortest line between the two points over the surface). In short, measurements with rods or motions of bodies yielded the same geometry, the same metrics, for the two-dimensional surface. Similar considerations would apply to a three-dimensional space.

But with our tentative theory of gravitation this accord would disappear and we should be faced with an insufferable dualism. Consider, for instance, the case of a stone thrown into the air; it describes a parabola. According to our present views, we should have to assume that this parabola represented a geodesic of three-dimensional space, and therefore the shortest distance between two points on the trajectory. But obviously, were a string to be stretched between the starting point and the point of fall of the stone, it would never coincide with the parabolic trajectory. In the same way, the distance between two points on the earth’s orbit—say, at six months’ interval—is not given by the line of the orbit. Thus the geometry of space established by freely moving bodies would be incompatible with that established by rod measurements. In the first case, intense non-Euclideanism would manifest itself, whereas, in the second case, space would appear to be Euclidean.