Page:PrasadSpaceTime.djvu/3

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



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