# Page:SahaSpaceTime.djvu/16

APPENDIX

85

From the very beginning, we can establish the ratio between the units of time and space in such a manner, that the velocity of light becomes unity. If we now write ${\displaystyle {\sqrt {-1}}t=l}$, in the place of t, then the differential expression

${\displaystyle d\tau ^{2}=-\left(dx^{2}+dy^{2}+dz^{2}+dl^{2}\right),}$

becomes symmetrical in (x, y, z, l); this symmetry then enters into each law, which does not contradict the world-postulate. We can clothe the essential nature of this postulate in the mystical, but mathematically significant formula

3·105 km = ${\displaystyle {\sqrt {-1}}}$ Sec.

V

The advantages arising from the formulation of the world-potulate are illustrated by nothing so strikingly as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner according to the Maxwell-Lorentz theory.

Let us conceive of the world-line of such an electron with the charge (e), and let us introduce upon it the "Proper-time" τ reckoned from any possible initial point. In order to obtain the field caused by the electron at any world-point P1 let us construct the fore-cone belonging to P1 (vide fig. 4). Clearly this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from P1 the normal to this tangent. Let r be the measure of P1Q. According to the definition of a fore-cone, r/c is to be reckoned as the measure of PQ. Now at the world-point P1,