Page:DeSitterGravitation.djvu/4

Further, we put

$r^{2}=x^{2}+y^{2}+z^{2},\quad \phi ^{2}=\xi ^{2}+\eta ^{2}+\zeta ^{2}$

The modulus of the transformation is $q$. We also introduce $q_{1}=1-{\sqrt {1-q^{2}}}$. If $q$ is a small quantity of the first order, then $q_{1}$ is of the second order. The cosines of the angles which the axis of the transformation makes with the axes of $x,y,z$ are denoted by $\alpha ,\beta ,\gamma ,$ so that $\alpha ^{2}+\beta ^{2}+\gamma ^{2}=1$.

The transformation-formulæ are then—

$x'=x+{\frac {\alpha \left[q_{1}r_{q}-qct\right]}{\sqrt {1-q^{2}}}},$ $y'=y+{\frac {\beta \left[q_{1}r_{q}-qct\right]}{\sqrt {1-q^{2}}}},$ $z'=z+{\frac {\gamma \left[q_{1}r_{q}-qct\right]}{\sqrt {1-q^{2}}}},$ $ct'={\frac {ct-qr_{q}}{\sqrt {1-q^{2}}}}.$

(1)

We find easily

$\xi '={\frac {dx'}{cdt'}}={\frac {\xi {\sqrt {1-q^{2}}}+\alpha \left[q_{1}\phi _{q}-q\right]}{1-q\phi _{q}}},$

(2)

and similarly for $\eta '$ and $\zeta '$.

Further,

$1-\phi '^{2}={\frac {\left(1-\phi ^{2}\right)\left(1-q^{2}\right)}{\left(1-q\phi _{q}\right)^{2}}}.$

In these formulæ $r_{q}$ and $\phi _{q}$ are the projections of $\rho$ and $\phi$ on the axis of transformation:—

$r_{q}=\alpha x+\beta y+\gamma z,$ $\phi _{q}=\alpha \xi +\beta \eta +\gamma \zeta .$

If we put

$(\kappa )={\frac {1}{\sqrt {1-\phi ^{2}}}},$ $(\xi )={\frac {\xi }{(\kappa )}},\ (\eta )={\frac {\eta }{(\kappa )}},\ (\zeta )={\frac {\zeta }{(\kappa )}},$ $(\phi )^{2}=(\xi )^{2}+(\eta )^{2}+(\zeta )^{2},$

we can easily verify that the transformation-formulas for $(\xi ),(\eta ),(\zeta ),(\kappa )$ are the same as those for $x,y,z,t$, viz.—

$(\xi )'=(\xi )+{\frac {\alpha \left[q_{1}\left(\phi \right)_{q}-q(\kappa )\right]}{\sqrt {1-q^{2}}}},$ $(\kappa )'={\frac {(\kappa )-q(\phi )_{q}}{\sqrt {1-q^{2}}}}.$

(3)