Page:PrasadSpaceTime.djvu/3

$c^{2}t^{2}-x^{2}-y^{2}-z^{2}=1.$

Fig. 1

Like a hyperboloid of two sheets, it consists of two sheets separated by $t=0$. We consider the sheet in the region $t>0$, and we conceive now those homogeneous linear transformations of $x,y,z,t$ into four new variables $x'$, $y'$, $z'$, $t'$, by which the expression of these sheets in the new variables becomes similar. Evidently the rotations of space about the zero-point belong to these transformations. A complete understanding of the remainder of these transformations is acquired, if we fix our eyes upon such of them as leave $x$ and $z$ unaltered. Let us trace (Fig. 1) the section of these sheets with the plane of the $x$- and $t$-axes, viz., the upper branch of the hyperbole $c^{2}t^{2}-x^{2}=1$, together with its asymptotes. Further, let us mark an arbitrary radius vector $OA'$ of this hyperbolic branch from the zero-point $O$; lay down the tangent at $A'$ to the hyperbola up to $B'$, the point of intersection with the right-hand side asymptote; complete the parallelogram $OA'B'C'$; and, finally, produce $B'$ $C'$ to $D'$, its point of intersection with the $x$-axis. If we take now $OC'$ and $OA'$ as axes for the parallel co-ordinates $x'$, $t'$ with the scales $OC'=1$, $OA'={\tfrac {1}{c}}$, then the hyperbolic branch is again expressed by

$c^{2}t'^{2}-x'^{2}=1,\ t'>0$

and the transition from $x$, $y$, $z$, $t$ to $x'$, $y$, $z$, $t'$ is one of the transformations in question. We take up with these transformations the arbitrary displacements of the zero-points of space and time, and thus constitute a group of transformations which is evidently dependent on the parameter $c$, and which I denote by the symbol $G_{c}$.

Now let $c$ increase indefinitely, i.e., let ${\tfrac {1}{c}}$ converge to zero; then it is clear from the adjoined figure that the hyperbolic branch approaches closer and closer to the $x$-axis, the angle between the asymptotes becomes broader and broader, and the transformation $G_{c}$ changes in the limit in such a manner that the $t'$-axis can have an arbitrary direction upwards and $x'$ approaches closer and closer to $x$. Hence, it is clear that from $G_{c}$ in the limit when $c$ tends to $\infty$, i.e., as the group $G_{\infty }$, we have exactly