numbers which accompanies all curvilinear mesh-systems (accelerated frames).[79] When this occurs, the mathematical expression of the force of inertia assumes a definite numerical value at each point, whereas, when the
’s are constants, as in a Cartesian mesh-system, this expression of the force maintains a zero value.
From this we see that forces of inertia arise from an attempt on our part to cut up space-time with curvilinear mesh-systems instead of Cartesian ones, just as they arise from our substitution of accelerated frames for Galilean ones. We cannot help but feel, however, that having proceeded thus far, a further generalisation is required. To be more explicit, it should be understood that the scheme of physics we have developed has compelled us to attribute a fundamental rôle to space-time. But Newton’s great law of universal attraction is expressed in terms of the separate space and time of classical science. If space-time is indeed as fundamental as Einstein has led us to believe, it appears incredible that Newton’s law should remain outside its scope. Yet this it certainly does, for if Newton’s law were a space-time law, it would preserve the same form in spite of any change in our space-time mesh-system. And this it fails to do.
Einstein’s next attempt was therefore to weld Newton’s law into the general fabric, a result he achieved about 1914. The mathematical generalisation which would allow this result to be obtained seems almost obvious to-day, since (as will be explained in [Chapter XXV]) it reduces to assuming that in the neighbourhood of matter, four-dimensional space-time loses its flatness and becomes non-Euclidean or curved. However, it appears to have been through the medium of physical observation that Einstein was led to his superb generalisation, so we shall proceed to follow the historical order by explaining the significance of his postulate of equivalence.
CHAPTER XXIV
THE POSTULATE OF EQUIVALENCE
EINSTEIN’S Postulate of Equivalence consists essentially in an identification of forces of gravitation and forces of inertia. This identification he considered permissible because of certain well-known empirical facts which we shall discuss presently.
Let us first recall that a field of force is exemplified by a region of space, at each and every point of which a definite force would be found to be acting on a test-body with which the field might be explored. Several different types of fields were known to classical science. Electric fields acted on electrified bodies, magnetic fields acted on magnets, and both inertial and gravitational fields acted on material bodies in general, whether electrified or not. For the present we shall be concerned solely with the inertial and gravitational fields.
These two species of fields of force were regarded by classical science as of a totally different nature. There appeared to be very good reasons for this distinction. Suppose, for instance, that a train is slowing down. If we are standing in the train, we shall feel a force pulling us towards the engine and it may require a certain effort on our part to resist its pull. As such, the force is obviously real, in that it is experienced. But suppose now that an observer on the embankment views these same happenings. He will argue as follows: No force is pulling the passenger; but as the train is slowing down and as the passenger’s body tends to maintain a constant velocity along a straight line, in conformity with the laws of motion (law of inertia), the net result is that he will overtake the engine unless he holds on to the seat. Thus we see that according to whether we judge these same happenings from the standpoint of the train or of the embankment, the force exists or becomes a fiction.