The practical working physicist, confronted with two rods which were of equal lengths when they left London, but of dif ferent lengths when they reached Paris, would also conclude that they had travelled by different routes, or at least had experienced different physical conditions on their journey. If the measured rods had travelled side by side packed in the same box he would find the difference in length inexplicable ; for why, he would say, if one has shrunk, did not the other shrink by an equal amount. But he would consider the difference in length quite natural if they had travelled by different routes. One might have passed through a succession of beatings and coolings so that fatigue phenomena had permanently changed its molecular structure, or might have experienced a series of magnetic changes so that hys teresis had affected its length.
The foregoing elementary considerations will show why Weyl's geometry, which attempts to take account of physical forces, and so of varying physical conditions, must necessarily break away from Einstein's geometry, which took no cognizance of physical changes. On Einstein's scheme a measuring rod, or any other piece of matter, had a standard physical state which it eternally carried about with it ; it could not expand with heat or contract with cold, and its physical condition did not enter into the dis cussion at all.
Weyl begins, in effect, by supposing, quite generally, that mov ing a rod through a small distance will produce a small change in its length, the amount of this change depending on the physical conditions in the small region through which the rod is moved. Any motion of a measuring rod is represented by a world line in the four-dimensional space time continuum. Taking x, y, z as co-ordinates of space and t as time co-ordinate, a small journey, or a small piece of the world line of the measuring rod, will be represented by small changes dx, dy, dz, dt in the values, x, y, z, t. Weyl supposes that this small journey produces a change dl in the length of the rod, 1, given by The factor 1 occurs outside because the change dl must neces sarily be proportional to the length 1 of the rod; F, G, H and K then depend only on the conditions prevailing at the point xyzt of the continuum, i.e., on the physical conditions in the region occupied by the rod at the instant when the journey is performed.
If two rods move over the same journey, say from London to Paris, and one is found to be slightly longer than the other by a factor (1— 0) when the two are compared at the end of the journey, it is easily shown that The world lines of the two rods intersect twice, first in London at the beginning of the journey, and again in Paris at the end of the journey. The above integral must be taken round a complete
circuit in the four-dimensional continuum, starting from Paris, retracing the journey of the first rod to London, and then start ing out again from London and following the route of the second rod to Paris.
The condition that the two rods shall be of the same length when they reach Paris is that 0= o, or that the integrand Fdx+Gdy+Hdz+Kdt shall be a perfect differential. The con ditions for this are expressed by the six equations : Compare these with Maxwell's equations for the components (a, 3, -y) of magnetic force and the components (x, y, z) of elec tric 'force, namely where F, G, H are the components of the magnetic vector potential, T. is the electrostatic potential, and C is the velocity of light. We at once see that if we identify the F, G, H and K which specify the change of length of the measuring rod with Maxwell's F, G, H and —CT, then the condition that 0, the relative change of length, shall be zero, is identical with the con dition that all the electric and magnetic forces are zero, so that there is no electro-magnetic field. When this condition is satisfied Weyl's geometry becomes identical with that of Einstein. Since 0 is now always zero, the length of a given measuring rod depends only on its position in space and time, and not on the route by which it attained that position, this being a fundamental conse quence of Einstein's scheme.
Weyl's extension of Einstein's scheme consists in introducing the four quantities we have denoted by F, G, H and K, and identifying them with Maxwell's F, G, H and —CT. This is equivalent, as we have just seen, to introducing the magnetic forces a, j3, -y and the electric forces X, Y, Z into his geometry. It is now possible to explain all electro-magnetic phenomena in free space in terms of geometry, for all such phenomena can be deduced from Maxwell's equations.
Einstein's theory of gravitation differed slightly from that of Newton, and so predicted slightly different phenomena. These slight differences made direct experimental tests between the two theories possible and these were found to decide in favour of Einstein. Weyl's theory, on the other hand, is formally identical with that of Maxwell, and so predicts precisely the same phenom ena in free space. For this reason no experimental test between the two theories is possible.