虚のローレンツ変換

$x'=\gamma(x+iivit),$
$y'=y,$
$z'=z,$
$it'=\gamma(it-i\frac{iv}{(ic)^{2}}x).$
ここで、$\gamma\equiv\frac{1}{\sqrt{1-(\frac{iv}{ic})^{2}}}$、$x,y,z,it,iv,ic\in\mathbb{R}i$である。

$x'=\gamma(x+iivit),\ y'=y,\ z'=z,\ it'=\gamma(it-i\frac{iv}{(ic)^{2}}x),\ \gamma\equiv\frac{1}{\sqrt{1-(\frac{iv}{ic})^{2}}}.$
$(x')^{2}+(y')^{2}+(z')^{2}=-(ic)^{2}(it')^{2}$に、上式を代入すると、
$(\gamma(x+iivit))^{2}+y^{2}+z^{2}=-(ic)^{2}(\gamma(it-i\frac{iv}{(ic)^{2}}x))^{2}$となる。以下、計算。
$(\gamma(x-ivt))^{2}+y^{2}+z^{2}=c^{2}(\gamma(it-vx/c^{2}))^{2}$
$\gamma^{2}(x^{2}-2ixvt-v^{2}t^{2})+y^{2}+z^{2}=c^{2}\gamma^{2}(-t^{2}-2itvx/c^{2}+v^{2}x^{2}/c^{4})$
$\gamma^{2}(x^{2}-2ixvt-v^{2}t^{2})+y^{2}+z^{2}=\gamma^{2}(-c^{2}t^{2}-2itvx+v^{2}x^{2}/c^{2})$
$\gamma^{2}(x^{2}-v^{2}t^{2})+y^{2}+z^{2}=\gamma^{2}(-c^{2}t^{2}+v^{2}x^{2}/c^{2})$
$x^{2}-v^{2}t^{2}+\frac{1}{\gamma^{2}}y^{2}+\frac{1}{\gamma^{2}}z^{2}=-c^{2}t^{2}+v^{2}x^{2}/c^{2}$
$x^{2}-v^{2}t^{2}+(1-\frac{v^{2}}{c^{2}})y^{2}+(1-\frac{v^{2}}{c^{2}})z^{2}=-c^{2}t^{2}+v^{2}x^{2}/c^{2}$
$x^{2}-v^{2}t^{2}+y^{2}-v^{2}y^{2}/c^{2}+z^{2}-v^{2}z^{2}/c^{2}=-c^{2}t^{2}+v^{2}x^{2}/c^{2}$
$x^{2}+y^{2}+z^{2}-v^{2}x^{2}/c^{2}-v^{2}y^{2}/c^{2}-v^{2}z^{2}/c^{2}=-c^{2}t^{2}+v^{2}t^{2}$
$x^{2}+y^{2}+z^{2}-\frac{v^{2}}{c^{2}}(x^{2}+y^{2}+z^{2})=-(ic)^{2}(it)^{2}+\frac{v^{2}}{c^{2}}c^{2}t^{2}$
$x^{2}+y^{2}+z^{2}-\frac{(iv)^{2}}{(ic)^{2}}(x^{2}+y^{2}+z^{2})=-(ic)^{2}(it)^{2}-\frac{(iv)^{2}}{(ic)^{2}}(-(ic)^{2}(it)^{2})$