All we have to do is to consider a transparent plane and a man executing geometrical constructions with Euclidean rods on its surface. If we consider a point-source of light casting the shadows of these rods on some other surface, whether plane or curved, these shadows, considered as measuring rods in turn, will still yield exactly the same Euclidean results, their laws of disposition will remain unchanged, and hence they will be rigid Euclidean rods exactly to the same extent as the original ones. Yet as contrasted with these original rods the shadows would squirm when displaced varying in shape and size, and the Euclidean straight lines would appear curved.
In short we see that absolute shape, straightness, size and rigidity in conceptual mathematical space escape us completely, and the significance of rigidity as portraying the maintenance of an unchanging volume of space, even when considering Euclidean bodies, is indeterminate. In its most extended sense, this is what mathematicians mean by the relativity of space. The sole justification for the general acceptance of the concept of rigidity in the popular sense is due to the presence of material bodies in our universe which we agree to accept as standards. These, incidentally, yield Euclidean results.
Summarising, we see that mathematical space is amorphous; it has no particular metrics, no particular geometry. According to our methods of measurement, we may obtain one geometry or another in the same space. It is often convenient to express all these results by saying that space is Euclidean, Riemannian or Lobatchewskian; but we must be careful to note that space itself has very little to do with the matter.
Let us now examine another aspect of the problem. Until such time as we have fixed our choice on a system of measurements, the amorphous nature of space forbids us to attach any determinate significance to a distance between two points. This, in turn, prevents us from attaching any significance to what may be considered the shortest distance between two points. However, when we have adopted one of the three measuring conventions, the significance of a shortest distance becomes determinate and a definite line joining the two points will be found to embody this shortest distance. Lines of this type are known under the name of geodesics; and in any of the geometries with which we shall be dealing, they play the part taken by the straight line in Euclidean geometry.[9]
Corresponding to every one of the three congruence definitions we have mentioned, there exists a definite type of straight line or geodesic between any two points. With Euclidean congruence we obtain the Euclidean straight line which satisfies Euclid’s parallel postulate, and vice versa. Again, if we adopt one of the two non-Euclidean types of congruence we are led to the non-Euclidean straight lines or geodesics, those which satisfy the non-Euclidean postulates, and vice versa. In short, we see that the two methods of presenting non-Euclidean geometry, either through the metrical or congruence method or through the parallel-postulate method, are in the main equivalent.
In a more thorough treatment of non-Euclidean geometry other non-metrical methods of presentation are sometimes adhered to. One of these is based on the theory of groups and projective geometry; another is due to the discoveries of Levi-Civita and Weyl and depends on the fundamental concept of an infinitesimal parallel displacement; it opens the way to Weyl’s still more general geometry. Still another mathematical method of exploring space is that of continuous tracks which was investigated by Eisenhart and Veblen. In this book we shall refrain from discussing these more difficult methods, for they present too technical an aspect;[10] but it would be a great mistake to assume that these imaginative flights of the pure mathematicians were of no utility to physical science. Apart from the deep philosophical light which they throw on the entire problem of space, we know from past experience that what was at its inception a pure mathematical dream has more than once become at some later date the image of physical reality. Non-Euclidean geometry, and possibly Weyl’s still stranger geometry, are cases in point.
We have now to consider a number of questions pertaining to location and motion in space. Mathematical space, as we have seen, is
-dimensional and amorphous. But in order to bring it into closer contact with the physical space of experience we may assume the necessary postulates of order and contiguity to have been specified. We thus obtain three-dimensional mathematical space. Now mathematical space, in view of its amorphous nature, is essentially relative. The very homogeneity of space, its sameness “here” as “there,” precludes our being able to differentiate one absolute position from another. It will follow that absolute motion, i.e., a variation of absolute position in empty amorphous mathematical space, can have no significance.
Our sole means of giving significance to position and motion will be by selecting some three-dimensional frame of reference as standard. For instance, we might consider three mutually perpendicular axes meeting at a point. A more concrete illustration would be afforded by considering the two adjoining walls and the floor of our room. Then, after having selected the standard rods which we intend to employ, and also a clock, we should proceed to measure the position and change of position of an object with respect to our room. This would yield us the relative position and motion of the object (relative to our room).