The evolution of scientific thought from Newton to Einstein

by A. D'Abro

Instead of considering an accelerated enclosure pursuing a rectilinear path, we could of course consider a rotating enclosure, and the argument would be the same. Atoms situated at greater distances from the centre of rotation would appear to vibrate more and more slowly; and a parcel of radium, for instance, placed on the rim of a rotating disk, would be slowed down in its rate of disruption, so that the effect of the inertial field would be to decrease the radioactivity of the atom.

Now that we have discussed the effects due to the curvature of time and have seen that this type of curvature accounts for all the major effects of gravitation, let us consider the additional effects due to the curvature of space. The presence of this curvature of space, so far as the motions of bodies are concerned, will modify the effects due to the curvature of time. It will therefore be responsible for the small discrepancies that differentiate Einstein’s law from Newton’s. However, so long as the motions of bodies in the gravitational field are small compared with that of light, the curvature of space will not affect their motions to any appreciable extent.

Among the planets, Mercury alone is animated with a speed (with respect to the inertial frame of the sun) sufficient to render observable the influence due to the curvature of space; and it is this influence which is responsible for the slight precessional advance of Mercury’s perihelion. In the case of light waves, which move with a speed many times greater than Mercury, the influence of the curvature of space will be marked to a still greater degree. It will cause an increased bending of the ray of light equal to that already produced by the curvature of time alone, which, as we know, would correspond to the bending required by Newton’s law. In this way the double bending is accounted for.

In order to exhaust this discussion, we must mention that there exists still another case, in which the curvature of space would exert a perceptible influence, even in the case of slowly moving bodies. This would be in the event of a very intense field of gravitation, many times greater than that of the sun. In such a field, even slowly moving bodies would deviate perceptibly from the course required by Newton’s law of gravitation.

It may appear strange that this curvature of space should seem to be of such minor importance as contrasted with that of time, and produce observable effects only for bodies moving with great velocities with respect to the sun. The solution of this puzzle will be found, however, when we consider some of the peculiarities of the space-time continuum of relativity. In an ordinary continuum like space, where all the dimensions are of exactly the same nature, we have no difficulty in specifying some unit of length which will apply in exactly the same way to measurements computed along the dimensions of height, length and breadth. In space-time, however, the situation is somewhat modified, since the fourth dimension (time) represents something different from the other three. Thus, whereas our measurements along the spatial dimensions will be conducted with material rods, our time measurements will require time-rods, or clocks. The question is, then, how are we to co-ordinate our measurements with clocks and our measurements with rods; or, in other words, how shall we be able to establish any sort of comparison of magnitude between these measurements so different in nature?

The answer to this question is that the variable which represents measurements along the time direction is not t but ct. This means that one second in time corresponds in our formulæ to the distance which light covers during one second, namely, 186,000 miles. It follows, therefore, that if we measure time in seconds we must, in order to co-ordinate results, measure space in terms of unit rods 186,000 miles long.

Obviously there exists a vast discrepancy in actual practice between the magnitudes of our temporal and spatial units; for the great majority of bodies with which we come in contact cover far less than a unit of space (186,000 miles) in a unit of time (one second). Only in the case of bodies approaching the velocity of light in our frame are the two progressions, that in space and that in time, in any way comparable. Thus, consider our own earth. It moves in relation to the sun with a speed of some eighteen miles a second, which is equivalent to approximately one ten-thousandth part of a unit of space per unit of time. This emphasises the fact that to all intents and purposes the motions of bodies with which we are concerned reduce to a motion in time, the displacement in space being insignificant in comparison.

Now when we come to investigate the respective effects of the curvatures of time and space, taken separately, in the gravitational field of the sun, we must remember that for a curvature to manifest itself we must survey relatively large areas on the curved surface (large as compared with the intensity of the curvature). Thus, on the curved surface of the earth, for example, if we content ourselves with surveying restricted areas, no effects of its curvature will be apparent. It is only when we conduct measurements over large areas, say over a few hundred miles, that the direction of the vertical will change perceptibly and that geodesical surveys will render the curvature manifest. It is much the same with the curvatures of space and of time in a gravitational field. Only when we consider planets moving through the sun’s inertial frame with a speed approaching that of light (hence covering large distances in unit time) will the space-curvature produce effects comparable to those produced by the curvature of time. We understand, therefore, how it comes that the curvature of time alone is able to produce the major effect of Newtonian attraction, such as weight, whereas the effects due to the curvature of space remain imperceptible until we consider the motion of Mercury, which is moving fairly fast with respect to the inertial frame of the sun. When we consider the propagation of light rays, the effects due to the curvature of space reassert themselves fully, and as a result the double bending of a ray of light can be attributed in equal proportion to the curvature of time and to the curvature of space.

In short, we now understand why it is that Einstein’s law of gravitation deviates appreciably in its effects from Newton’s only when we consider bodies moving with velocities approximating to that of light. Once again we realise that if this invariant velocity of the universe had happened to be much smaller, the deviations between Einstein’s law of gravitation and that of Newton would have been far easier to detect; and, conversely, had the invariant velocity happened to be much greater than 186,000 miles a second, had it been infinite, for example, there would have been no discrepancies left between the two laws. Thus, once again we reach the same conclusions that we mentioned when discussing space-time. The fundamental difference between Einstein’s theory and classical science arises from the discovery that the invariant velocity of the universe is finite and not infinite.

We may summarise all these results by stating that the major effects of gravitation are produced by the curvature of time, i.e., of the time component of the metrical field of space-time; the curvature of space exerts perceptible effects only when the velocities of the moving bodies are very great or when the gravitational field is of very great intensity. Hence, for weak fields such as that of the sun, and for slow motions, Einstein’s law of space-time curvature leads us to results practically identical with those necessitated by Newton’s law of gravitation.