Page:DeSitterGravitation.djvu/6

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

$c\left(t_{1}-t_{2}\right)=q\Delta _{q}.$

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

$\mathrm {x} '_{1}-\mathrm {x} '_{2}=x_{1}-x_{2}+{\frac {\alpha \left[q_{1}\Delta _{q}-qc\left(t_{1}-t_{2}\right)\right]}{\sqrt {1-q^{2}}}},$

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

$\mathrm {x} '_{1}-\mathrm {x} '_{2}=x_{1}-x_{2}-\alpha q_{1}\Delta _{q},$

(6)

and similarly for the other coordinates.

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

${\boldsymbol {r}}'_{q}=\Delta '_{q}=\Delta _{q}-q_{1}\Delta _{q}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)=\left(1-q_{1}\right)\Delta _{q}=\Delta _{q}{\sqrt {1-q^{2}}}.$

Therefore (6) can be written—

$x_{1}-x_{2}=\mathrm {x} '_{1}-\mathrm {x} '_{2}+\alpha {\frac {q_{1}{\boldsymbol {r}}'_{q}}{\sqrt {1-q^{2}}}}.$

(7)

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—

$c\left(t_{1}-t_{2}\right)=\Delta ={\sqrt {\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}}}.$

(8)

The expression

$c^{2}\left(t_{1}-t_{2}\right)^{2}-\Delta ^{2}$

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.^[1]

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

${\boldsymbol {x}}=\mathrm {x} _{1}-\mathrm {x} _{2}=x_{1}-x_{2}+\Delta \xi _{2}-\delta x_{2},$

(9)

where

$\delta x_{2}={\tfrac {1}{2}}\Delta ^{2}{\frac {d^{2}x_{2}}{c^{2}dt^{2}}}+{\tfrac {1}{6}}\Delta ^{3}{\frac {d^{3}x_{2}}{c^{3}dt^{3}}}+\dots$

1. It is, of course, not allowed to put $c\left(t_{1}-t_{2}\right)=0$, which would mean instantaneous action at a distance, since the expression $c\left(t_{1}-t_{2}\right)$ by itself is not invariant.