Page:DeSitterGravitation.djvu/7

The values of $\xi _{2}$ and the higher differential coefficients must be taken for the time $t_{2}$, defined by (8). We consider $1/c$ as a small quantity of the first order, and we wish our formulæ to be exact to the second order inclusive.

We find easily

$\Delta ^{2}\left(1-\phi _{2}^{2}\right)={\boldsymbol {r}}^{2}+2{\boldsymbol {r}}\epsilon _{2}\Delta +2{\boldsymbol {r}}\chi _{2},$

where

${\begin{aligned}{\boldsymbol {r}}\epsilon _{1}&={\boldsymbol {x}}\xi _{1}+{\boldsymbol {y}}\eta _{1}+{\boldsymbol {z}}\zeta _{1},\\{\boldsymbol {r}}\epsilon _{2}&={\boldsymbol {x}}\xi _{2}+{\boldsymbol {y}}\eta _{2}+{\boldsymbol {z}}\zeta _{2},\\{\boldsymbol {r}}\chi _{2}&={\boldsymbol {x}}\delta x_{2}+{\boldsymbol {y}}\delta y_{2}+{\boldsymbol {z}}\delta z_{2}.\end{aligned}}$

$\epsilon _{1}$ and $\epsilon _{2}$ are of the first order, $\chi _{2}$ is of the second order.

In the equations of motion there appear the invariants—

${\begin{aligned}\mathrm {A} &=c\left(t_{2}-t_{1}\right)(\kappa )_{1}-\left(x_{2}-x_{1}\right)(\xi )_{1}-\left(y_{2}-y_{1}\right)(\eta )_{1}-\left(z_{2}-z_{1}\right)(\zeta )_{1},\\\mathrm {B} &=c\left(t_{1}-t_{2}\right)(\kappa )_{2}-\left(x_{1}-x_{2}\right)(\xi )_{2}-\left(y_{1}-y_{2}\right)(\eta )_{2}-\left(z_{1}-z_{2}\right)(\zeta )_{2},\\\mathrm {C} &=(\kappa )_{1}(\kappa )_{2}-(\xi )_{1}(\xi )_{2}-(\eta )_{1}(\eta )_{2}-(\zeta )_{1}(\zeta )_{2}.\end{aligned}}$

To express these in simultaneous relative coordinates we have—

${\begin{aligned}\mathrm {A} {\sqrt {1-\phi _{1}^{2}}}&=-\Delta \left(1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}\right)+{\boldsymbol {r}}\epsilon _{1},\\\mathrm {B} ^{2}\left(1-\phi _{2}^{2}\right)&={\boldsymbol {r}}^{2}\left(1-\phi _{2}^{2}+\epsilon _{2}^{2}\right)+2{\boldsymbol {r}}\chi _{2},\end{aligned}}$

(10)

$\mathrm {C} ={\frac {1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}}{\sqrt {\left(1-\phi _{1}^{2}\right)\left(1-\phi _{2}^{2}\right)}}}.$

6. The equations of motion can be given in three forms:—

$mc{\frac {d(\xi )}{d\tau }}=m{\frac {d^{2}x}{d\tau ^{2}}}=(\mathrm {X} )={\frac {\mathrm {X} }{\sqrt {1-\phi ^{2}}}},$

(11)

$mc{\frac {d(\xi )}{d\tau }}=m{\frac {d}{dt}}\left({\frac {dx}{d\tau }}\right)=\mathrm {X} =(\mathrm {X} ){\sqrt {1-\phi ^{2}}},$

(12)

$m{\frac {d^{2}x}{dt^{2}}}=\left\{\mathrm {X} -\xi (\mathrm {X} \xi +\mathrm {Y} \eta +\mathrm {Z} \zeta )\right\}{\sqrt {1-\phi ^{2}}},$

(13)

and similarly for the other coordinates.

$\mathrm {X}$ is the force according to the ordinary definition, or "Newtonian" force; $(\mathrm {X} )$ is called the "Minkowskian" force.^[1] The mass $m$ is a constant.

Differentiating the formula $(\kappa )^{2}=1+(\phi )^{2}$, we derive a fourth equation analogous to (11) or (12), viz.:

$mc{\frac {d(\kappa )}{d\tau }}=m{\frac {d^{2}(ct)}{d\tau ^{2}}}=(\mathrm {T} )={\frac {\mathrm {T} }{\sqrt {1-\phi ^{2}}}},$

(11′)

1. See Lorentz, Göttingen Lectures, October 1910 (Physikalische Zeitschrift, vol. xi. p. 1234). The names quoted above occur on page 1237.