Page:The principle of relativity (1920).djvu/150

 quantity c. At the first impression, the axiom seems to be rather unsatisfactory. It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in geometry.

The impulse and real cause of inducement ''for the assumption of the group-transformation G_{c}'' is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation G_{c}. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group G_{infinity}. Now if we have an optics with G_{c}, and on the other hand if there are rigid bodies, it is easy to see that a t-direction can be defined by the two hyperboloidal shells common to the groups G_{infinity}, and G_{c}, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations, with regard to the direction of progressive motion of the earth. But all efforts directed towards this object, and even the celebrated interference-experiment of Michelson have given negative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis which practically amounts to an invariance of optics for the group G_{c}. According to Lorentz every substance shall suffer a contraction

1:([sqrt](1 - v^2/c^2)) in length, in the direction of its motion

l/l´ = 1/[sqrt](1 - v^2/c^2) l´ = l(1 - v^2/c^2).