The evolution of scientific thought from Newton to Einstein

by A. D'Abro

along successive plane areas ([Fig. VII]). We may say, therefore, that from the standpoint of space and time, the red space-time volume appears as a splash of red colour squirming and varying in shape; in spite of the fact that from the space-time standpoint the volume is fixed and motionless. But the varying shape of the red patch as described above corresponds to the patch as it exists in our instantaneous present. It is not this that we perceive with our eyes, for whatever we perceive is already situated in our past. Hence, what we perceive at a given instant will be given, not by the intersection of the red volume with the plane of our instantaneous present, but by its intersection with the surface of our instantaneous cone of the passive past ([Fig. VIII]). This is the light-cone which passes through those space-time events which are in our present range of vision. All other events are from a visual standpoint either in our visual past or future. (More detailed explanations will be given in [Appendix I].)

Looking back on what has been accomplished, we see that the motion and change with which we are familiar in our daily experience are now interpreted as arising from a passage of our consciousness through space-time. The cause of this passage remains utterly mysterious; but, as we said before, it would not solve our difficulties to attribute it to dynamic properties of the continuum itself—dynamic properties which would urge our consciousness on. Even were this solution to be adopted, the cause of the dynamic properties and their mode of action on our consciousness would remain as mysterious as ever.

Fig. VIII

Now, in this interpretation of space-time, there is no room for free will, since everything is already predetermined and pre-exists in the future. To be sure, this is a most unsatisfactory aspect of the presentation. But how are we to avoid it? In classical science we could always reserve a part for free will by assuming that the present and the past alone were fully determined while the future might be modified at will. But in relativity this neat separation is no longer possible. For if we consider two observers passing each other, the present of the one comprises both the past and the future of the other. Hence, we cannot banish the future without annihilating a portion of the past together with the present, or at least that portion of the present which stands elsewhere than in our immediate vicinity. Neither may we assume boldly that the seat of free will resides in some suprasensible world transcending that of space-time, for the position of an object on the table is, after all, an effect of the will that placed it there, and the body as a material object occupies a position in space-time.

At any rate, there appears to be very little use in speculating on puzzles of this sort. The fact is that science is necessarily deterministic, not through an act of faith, not because it has convinced itself that free will is non-existent, but because determinism is for science a dire necessity if phenomena are to be co-ordinated and linked to one another. Regardless of what the future may hold in store, the physicist is therefore compelled to act as though a rigidly deterministic scheme were existent in nature, and to consider space-time and its content as given, even though he may doubt whether such is really the case.

Interesting sidelights on causality are offered by the space-time theory. Since no effects can be propagated with a speed greater than that of light, the passage of time is an essential if causal connections are to operate. This was not necessarily true when instantaneous effects (or action at a distance), were contemplated. The theory of relativity thus imposes restrictions on causal connections; it allows us to assert that between certain space-time events no causal connections can ever exist. It is well to note, however, that the category of causality has no place in space-time as such. Only when we consider the perceptions of the observer can causality be considered, since only then is the passage of time introduced. When discussing space-time from an impersonal point of view as a four-dimensional continuum which is static, it would therefore seem preferable to refer to functional relationships.

The next most important aspect of the theory of relativity which we have to consider relates to our conception of real space. We remember that the geometrical space of mathematics was amorphous. It possessed no intrinsic metrics, no special geometry, no size, no shape; and the geometry which the mathematician credited to space was purely a matter of choice. On the other hand, the real space of physics appeared to possess a definite metrics. Furthermore, the amorphous nature of space was belied by the existence of centrifugal forces and the like, proving that not all motions through space were equivalent.

When it came to determining the precise geometry of real space, various methods could be considered. We might proceed by appealing to rigid bodies, that is, to bodies which visually and tactually appeared to be the same wherever we might carry them. More generally, rigid bodies were assumed to be exemplified by material bodies maintained as far as possible under permanent conditions of temperature and pressure. We could also investigate space by means of light rays, assuming that their courses in vacuo would define geodesics. Over a century ago Gauss attempted experiments of this sort between the summits of two mountains. A still more general method consisted in appealing to the laws of physics in general, and determining the type of geometry which would have to be credited to space in order to permit us to co-ordinate natural phenomena with the maximum of simplicity. This procedure was suggested by Riemann in 1854 in his inaugural address at Göttingen.

It will be noted that all these various methods of determining the geometry of space are essentially physical. Hence, the accusation could always be made, as it was indeed made by Poincaré, that all we could determine in this way would reduce to the geometrical properties of material bodies or of light rays, and that space itself was amorphous and would therefore escape us completely. On the other hand, it was an undeniable fact of experience that measurements in space, whether conducted with material rods or with light rays, always yielded the same Euclidean results. Furthermore, the same Euclidean geometry seemed to be imposed upon space when we sought to interpret the laws of physics in the simplest way possible. It was scarcely conceivable that this general concordance of results obtained by these various methods of approach should be attributed to chance. Hence, the problem was reversed and the belief arose that space itself possessed a Euclidean structure, and that it was this structure which controlled the behaviour of material solids, the paths of light rays, and the course of free bodies moving far from matter according to the law of inertia. When this attitude was adopted the significance of measurements with material rods was altered. The geometry of space was not merely the Euclidean geometry of material bodies; it was rather the reverse. The geometry of material bodies was Euclidean by reason of this pre-existing Euclidean structure of space, which moulded them into shape and thus gave them their geometrical properties. Under the circumstances, measurements in space with rigid bodies and light rays would resolve themselves into an exploration of the pre-existing geometry of space, and the rôle of the rigid body and the light ray became similar to that of the thermometer in detecting pre-existing variations of temperature.