It is easy to see that these physical conditions are precisely those which we have found to exist in a gravitational field produced by matter. The straightest type of mesh-system around a point (namely, a semi-Cartesian mesh-system) would correspond to that of the observer in the falling elevator or in the hollow projectile, since for an observer in a frame of this type the force of gravitation disappears in the immediate interior of his frame, but reappears at more distant points. Further, space-time being curved, its geodesics would no longer be Euclidean straight lines. They would still appear approximately straight when referred to a semi-Cartesian mesh-system in the immediate neighbourhood of the point for which the mesh-system was approximately one of equal four-dimensional cubes; but their curvature would become more and more apparent when we surveyed them from this same mesh-system in more distant regions. The physical interpretation of this geometrical fact would be afforded by the seemingly uniform and rectilinear motion of free bodies and rays of light in the immediate neighbourhood of the falling elevator when viewed by an observer situated in this falling frame, and by the apparent acceleration of these bodies in regions more distant from the falling elevator.
It is therefore obvious that the existence of a non-Euclideanism of space-time around matter would betray itself physically in exactly the same way as a field of gravitation produced by matter.[85] Assuming this generalisation to be correct, we see that no essential difference can exist between a field of gravitation and one of inertia. Both types of field arise from a relationship between the mesh-system adopted by the observer, and the fundamental structure of the space-time continuum. The only difference that exists is that in interstellar space far from matter, space-time being flat, it is always possible to select a Cartesian mesh-system corresponding to a Galilean system in which no field of force is present, whereas in the neighbourhood of matter, owing to the inherent curvature of space-time, the nearest approach to Galilean frames is given by semi-Cartesian or semi-Galilean frames (falling elevator). In such frames the field of force cannot vanish throughout the frame at more or less distant points.
It follows that these same
’s whose values in a given mesh-system represented the potentials of the field of inertia can now with equal justification be considered to yield the potentials of the field of gravitation. The mathematical expression comprising the variations of the
’s from point to point, which was formerly proved to represent the force of inertia, can now with equal justification be deemed to represent the force of gravitation. As was the case with the force of inertia, this mathematical expression of the force, depending as it does on the variations in the values of the
’s from point to point, varies with our choice of a mesh-system; in exactly the same way, the magnitude of a gravitational force in our frame of reference varies with the acceleration of our frame (as was established when we discussed the postulate of equivalence). Furthermore, the value of the mathematical expression of this force will vanish completely throughout a mesh-system when this mesh-system is Cartesian, whereas it will vanish only around a definite point when, owing to the curvature of space-time, a Cartesian mesh-system is impossible, so that we have to be content with a semi-Cartesian one. We see that the analogy is complete in all respects.
Here it must be noticed that, identifying as we are the