H. A. LORENTZ
As to the second-order effect, the situation was much more difficult. The experimental results could be accounted for by transforming the co-ordinates in a certain manner from one system of coordinates to another. A transformation of the time was also necessary. So I introduced the conception of a local time which is different for different systems of reference which are in motion relative to each other. But I never thought that this had anything to do with the real time. This real time for me was still represented by the old classical notion of an absolute time, which is independent of any reference to special frames of co-ordinates. There existed for me only this one true time. I considered my time transformation only as a heuristic working hypothesis. So the theory of relativity is really solely Einstein’s work. And there can be no doubt that he would have conceived it even if the work of all his predecessors in the theory of this field had not been done at all. His work is in this respect independent of the previous theories.
I shall have little to say about the theory of the Michelson-Morley experiment, which was the first ever made of those in which we are concerned with effects of the second order. That here again the result must be negative is immediately clear if we follow the theory of relativity. If, instead of that, we apply to the experiment our old stationary ether, we must carefully consider the paths of the interfering rays of light and the time in which the light is propagated along each of them from the source of the point where the interference takes place.
For this purpose we can again use the fundamental equation (i). Confining ourselves to the propagation in ether, we may put u = c, k = i so that the equation becomes
Taking into account terms of the second order w2/c2, we deduce from it
C2=V2 + W2 — 2VW COS &
T T ( W W2
- = - i i— cos$H—- (cos2 #+|sin2#) V C ( c c
CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 351
Now let there be two paths, 1 and 2, along which light can go from the point P to the point Pr (Fig. 5). For each of them the time required for propagation will be represented by expressions of the form
fë-ljds — —2^w cos & w2 (cos2 sin2 ê) ds , (3)
and we shall be able to calculate the two times, if we know the lines along which the integrals are to be taken. Let the lines h and l2 (Fig. 5) represent the paths of the two rays as they would be if the ether did not move through the diagram.
As has been shown, these lines are not altered by the motion so long as we confine ourselves to terms of the order w/c. They may, however, be somewhat changed when, as is now proposed, quantities of the second order are taken into account.
We shall then have, for instance, the dotted p lines l[ and l2 whose distances from h Fig
and Z2, reckoned along the normals to
these lines, are of the second order. We must now calculate the times of propagation for the paths l[ and lf2, say Tl[ and Tl2. Since, however, T is a minimum for lx, as compared with neighboring lines, and since the displacements from lx to l'x are of the second order, the difference between Th and Tl[ will be of the fourth order. This may be neglected when we confine ourselves to quantities of the second order. Similarly, we may replace Tl'2 by Tl2. This means that, in the determination of the phase differences, we may use the values of (3) for the rays, such as they would be according to the ordinary laws of optics in the absence of the earth’s motion.
We are thus led to the ordinary theory of the experiment, which would make us expect a displacement of the fringes, the absence of which is accounted for by the well-known contraction hypothesis (Lorentz contraction).
Asked if I consider this contraction as a real one, I should answer “yes.” It is as real as anything that we can observe.