Page:BatemanElectrodynamical.djvu/21

The general infinitesimal spherical wave transformation is

$x'=x+\epsilon \left[p\left(x^{2}-y^{2}-z^{2}+t^{2}\right)+2qxy+2rxz+2sxt+\mu x+hy+gz+lt+a\right],$ $y'=y+\epsilon \left[q\left(x^{2}-y^{2}-z^{2}+t^{2}\right)+2pxy+2ryz+2syt-hx+\mu y+fz+mt+b\right],$ $z'=z+\epsilon \left[r\left(z^{2}-x^{2}-y^{2}+t^{2}\right)+2pxz+2qyz+2szt-gx-fy+\mu z+nt+c\right],$ $t'=t+\epsilon \left[s\left(x^{2}+y^{2}+z^{2}+t^{2}\right)+2qxt+2qyt+2rzt+lx+my+nz+\mu t+d\right],$

where the coefficients are all constants, and ε is a quantity whose square may be neglected. Since there are fifteen arbitrary constants the group is a fifteen parameter group.

We have

${\frac {\partial x'}{\partial x}}={\frac {\partial y'}{\partial y}}={\frac {\partial z'}{\partial z}}={\frac {\partial t'}{\partial t}}=1+\epsilon [\mu +2px+2qy+2rx+2st]=\lambda ,$ $dt'^{2}-dx'^{2}-dy'^{2}-dz'^{2}=\lambda ^{2}\left[dt^{2}-dx^{2}-dy^{2}-dz^{2}\right].$

If (x, y, z) is kept constant as t varies, the corresponding point (x', y', z') moves along a parabola, but in one type of transformation it moves along a straight line with constant acceleration. This is the case, for example, when

${\begin{array}{lll}x'=x+\epsilon \left(y^{2}+z^{2}-x^{2}-t^{2}\right),&&y'=y(1-2\epsilon x),\\\\z'=z(1-2\epsilon x),&&t'=t(1-2\epsilon x),\end{array}}$

for, since quantities of order ε² may be neglected, the first equation may be written

$x'=x+\epsilon \left(y^{2}+z^{2}-x^{2}-t^{2}\right)$

Hence, if (x, y, z) are kept constant, the point (x', y', z') moves with constant acceleration γ given by

$\gamma =-2\epsilon .$

Substituting for γ, we get

${\begin{array}{lll}x'=x-{\frac {1}{2}}\gamma \left(y^{2}+z^{2}-x^{2}-t^{2}\right),&&y'=y(1+\gamma x),\\\\z'=z(1+\gamma x),&&t'=t(1+\gamma x).\end{array}}$

The last equation agrees with the one obtained by Einstein.^[1]

In the case of the general infinitesimal transformation, the expression

$(x'-x)^{2}+(y'-y)^{2}+(z'-z)^{2}-(t'-t)^{2},$

1. Jahrbuch der Radioaktivität, Band iv. (1907), p. 457.