Page:The principle of relativity (1920).djvu/159

 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 [sqrt]-1 t = l, in the place of t, then the differential expression

dτ^2 = -(dx^2 + dy^2 + dz^2 + dl^2),

becomes symmetrical in (x, y, r, 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

3·10^5 km = [sqrt]-1 Sec.

V

The advantages arising from the formulation of the world-postulate 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 P_{1} let us construct the fore-cone belonging to P_{1} (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 P_{1} the normal to this tangent. Let r be the measure of P_{1}Q. According to the definition of a fore-cone, r/e is to be reckoned as the measure of PQ. Now at the world-point P_{1},