Page:Eddington A. Space Time and Gravitation. 1920.djvu/219

Rh Note 2 (p. 47). Suppose a particle moves from $$(x_1, y_1, z_1, t_1)$$ to $$(x_2, y_2, z_2, t_2)$$, its velocity $$u$$ is given by Hence from the formula for $$s^2$$ (We omit a $$\sqrt{-1}$$, as the sign of $$s^2$$ is changed later in the chapter.)

If we take $$t_1$$ and $$t_2$$ to be the start and finish of the aviator's cigar (Chapter ), then as judged by a terrestrial observer, As judged by the aviator, Thus for both observers $$s$$ = 30 minutes, verifying that it is an absolute quantity independent of the observer.

Note 3 (p. 48). The formulae of transformation to axes with a different orientation are where $$\theta$$ is the angle turned through in the plane $$x \tau$$.

Let $$u = i \tan \theta$$, so that $$\cos \theta = (1 - u^2)^{-\tfrac{1}{2}} = \beta$$, say. The formulae become or, reverting to real time by setting $$i\tau = t$$,  which gives the relation between the estimates of space and time by two different observers.

The factor $$\beta$$ gives in the first equation the FitzGerald contraction, and in the fourth equation the retardation of time. The terms $$ut^\prime$$ and $$ux^\prime$$ correspond to the changed conventions as to rest and simultaneity.

A point at rest, $$x$$ = const., for the first observer corresponds to a point moving with velocity $$u$$, $$x^\prime - ut^\prime $$ = const., for the second observer. Hence their relative velocity is $$u$$.