ELECTROMAGNETIC FORCES The restricted physical theory of relativity introduced a revo lution into the foundations of scientific thought by destroying the objectivity of time and space. The gravitational theory has effected a hardly less important revolution by destroying our belief in the reality of gravitation as a "force." The physicist has, however, to deal with other "forces" besides those of gravitation, namely, the electromagnetic forces which are concerned in all electrical, magnetic and chemical phenomena, and the question inevitably arises as to whether these also must be regarded as illusions, arising only from our faulty interpretation of the special metrical properties of the continuum. Analysis makes it clear that the continuum as imagined by Einstein has no capacity for simu lating any forces beyond those of gravitation. If electromagnetic forces are evidence of special geometrical properties of the con tinuum, then the continuum is more complicated even than Ein stein imagined.
In a brilliant paper published in 1918 (Berlin Sitzungsber., 1918, P. 465) H. Weyl proved that Einstein's geometry is far from being the most general geometry which is consistent with the general principle of relativity, and showed that certain at least of the forces of electromagnetism could be explained very naturally by adding an extra degree of complexity to the geometry of the continuum. In the continuum as imagined by Einstein, the length of a measuring rod might change as it was moved about in a gravitational field, but its length at any instant would depend solely on its position in space. Weyl imagined a continuum in which it is meaningless to speak of comparing two lengths at different points; we can compare two lengths at the same point, such as two measuring rods pointing in different directions at the same place, but we can only measure a length by comparison with a standard length already in existence at the point at which the measurement is made. The standard length was peculiar to the point at which it existed, so that Weyl had to imagine the whole of the continuum, every point of space at every instant of time, filled with an infinite series of gauges against which a moving object might be measured. Whether the length of a measuring
rod changed as it was moved about would now of course depend on whether the arbitrary selection of gauges had been arranged so as to make it change or so as to keep it constant.
We might, for instance, select the yard as unit length in Lon don, and the metre in Paris. In this case a standard yard trans ferred to London from Paris might be said to shrink by about a tenth of its length. Suppose now that two standard yards are taken from London to Paris. When we compare them with the standard length at Paris they may both be said to have shrunk, but if we compare them with one another in Paris—and this is possible, even in Weyl's geometry—we may expect to find them both of the same length. If they appear to have shrunk when measured against the Paris unit of length, they have at least shrunk equally.
Imagine first that the two standard yards travel from London to Paris lying side by side in the same box, so that they pass through the same experiences on their journey. If physical changes are governed by a law of causation, so that similar causes always produce the same effects, then the two rods will neces sarily have the same length when they arrive in Paris.
Suppose, however, that one standard travelled via Calais and one via Havre. The two rods have now been subjected to differ ent influences on their journey, and there is no a priori reason why their lengths should not be different when they arrive in Paris. Weyl's geometry expects, as a matter of course, that their lengths will be different, at least until the contrary is proved. And if the length of one proves to be 0.9143 metres, while that of the other proves to be 0.9144 metres, Weyl's geometry would merely interpret this as evidence that the two rods had travelled by different routes.