The evolution of scientific thought from Newton to Einstein

by A. D'Abro

The existence of these potentials applies generally to all fields of force, regardless of whether these fields are of the uniform or non-uniform variety.[78] It is, as we have said, the variations in value of these potentials from point to point that are associated with the existence of forces. When this variation is uniform in one direction and no variation exists in directions at right angles, we are in the presence of a uniform field of force. When the variation is irregular, we have a non-uniform field of force. When the potential has the same constant value throughout space, so that it does not change in value from place to place, no forces can be present; there is then no field of force.

From this we see that a certain degree of indeterminateness surrounds the precise numerical value of a potential, since it is only its variation from place to place, and not its absolute value, that defines the force. This indeterminateness can be removed in the case of a non-uniform field of force by stipulating that in those regions of the field where the force vanishes (such as would be the case with the gravitational force at an infinite distance from matter), the potential itself vanishes. A definite value having been attributed to the potential in one part of the field, we can determine its precise value from place to place in all other parts of the field.

Let us now give a few concrete illustrations of fields of force and of potentials in classical science. We have already mentioned the field of force surrounding matter. This was the gravitational or Newtonian field, and the potential at every point derived therefrom was called the Newtonian Potential at the point. The distribution of the field of force was known in a perfectly definite manner when the potential distribution was known. In fact Newton’s law, which tells us how the field of force is distributed around matter, can also be expressed in an equivalent form by Laplace’s Equation, in which it is the potential distribution, and no longer the force distribution, that is described.

Let us now consider another type of field of force known as the inertial field. In perfectly free space, far from matter, there exists no field of force so long as we refer our observations to a Galilean frame. But if now we step into an accelerated or a rotating frame, we shall experience the effects of a field of force which we will ascribe to forces of inertia. Not only our own human bodies, but, in addition, all free bodies, will be subjected to the actions of these forces. Thus, consider a disk that is rotating, and a ball rolling without friction on the disk’s surface. If the ball is sent from the centre of the disk to its periphery it will of course follow a straight line at constant speed with respect to the earth or to any other Galilean frame. But then, with respect to the rotating disk itself, its course can no longer be rectilinear and uniform. Instead of following one of the radii of the rotating disk, it will follow a curve as though it had been pulled sideways by some force.

This result is general. If we ourselves tried to advance along one of the radii of the rotating disk we should experience a real physical force pulling us sideways. To all intents and purposes, when we referred events to the rotating disk and not to the non-rotating earth (the rotation of the earth being so slow that we may neglect it as a first approximation), the physical existence of a field of force would have to be taken into consideration. The precise type of force we have mentioned is called the Coriolis force; in addition, there also exists another type of force, the better-known centrifugal force. Both these types of force are called forces of inertia. It is their ensemble which constitutes the field of inertial force existing in a rotating frame.

If, in place of a rotating frame, we had considered an accelerated railway compartment, the field of inertial force would have been disposed in a different manner. In the case of a train moving with constant acceleration along a straight line, the field of inertial force generated would have been of the uniform variety, disposed longitudinally through the train.

All these different illustrations show us that whereas, in a Galilean frame, no field of force exists, yet a field springs into existence automatically as soon as we place ourselves in any accelerated frame. It is for this reason that accelerated frames can be distinguished physically from Galilean frames; and it is owing to this generation of physical fields of force that accelerated motion must be regarded as absolute, whereas velocity, giving rise to no such fields, yields us no means of distinguishing one velocity from another, hence is relative.

Now we have seen that fields of force are accompanied by potential distributions. Hence it follows that, whereas, with respect to a Galilean frame, the inertial potential is zero at every point, or at least maintains some constant value, in the case of accelerated frames this inertial potential must vary from place to place in the frame. The magnitude of the force at any point of an accelerated frame is given, therefore, by a mathematical expression involving the variations in value of the potential in the neighbourhood of the point considered.

All we have explained so far pertains to classical science, but it appeared necessary to mention these results briefly before proceeding to a more systematic study of Einstein’s theory. We shall now see how all these results find a natural place in the space-time theory.

Consider a Galilean frame and an object moving freely in this frame. Its path will of course be rectilinear and its speed uniform, hence it will possess no acceleration, in accordance with Newton’s law of inertia. If we interpret these results in terms of space-time, we see that the world-line of the body is a straight line in space-time as referred to our mesh-system of equal four-dimensional cubes. If we now examine the motion of this same free body from some definite accelerated frame, all we have to do is to change our four-dimensional space-time mesh-system in an appropriate way. This change of mesh-system does not affect the world-line of the body, since this line remains a straight line or geodesic through flat space-time; but, on the other hand, it will certainly alter the appearance of the straight line when we refer our measurements to our curvilinear mesh-system. The erstwhile straight world-line will now appear curved in a definite way, and its precise mathematical equation as referred to our new curved mesh-system can be obtained without difficulty.