Page:DeSitterGravitation.djvu/6

Mar. 1911. Thus, if we wish the two events to be simultaneous with respect to the time $$t',$$ or $$t'_{1}=t'_{2}$$, we have

Denoting the coordinates of simultaneous events by non-italic letters, we find thus—

or, remarking that $$q_{1}-q^{2}=-q_{1}\sqrt{1-q^{2}},$$

and similarly for the other coordinates.

We find easily, denoting the distance between simultaneous positions by $$\boldsymbol{r}$$:—

Therefore (6) can be written—

5. If we consider the action on $$m_1$$ at time $$t_1$$ of a force emanating from $$m_2$$ at time $$t_2$$, we will suppose—

The expression

is of the general form (4), and is thus an invariant of the transformation. We have thus also $$c\left(t'_{1}-t'_{2}\right)=\Delta'$$. The equation (8) states that the force is propagated through space with the velocity of light. This is, of course, an arbitrary assumption, which is not a necessary consequence of the principle of relativity. The velocity of propagation might be defined by any invariant of the transformation containing $$c\left(t_{1}-t_{2}\right)$$, put equal to zero. But it is a natural assumption, and the most simple which can be made.

Denoting now the simultaneous relative coordinates by letters of another type, $$\boldsymbol{x, y, z}$$, we have, for time $$t_{1}$$,

where