mc² dt/dτ = mc² /√(1 - v²/c²)
i.e., if we deduct from this the additive constant mc², we obtain the expression ½ mv² of Newtonian-mechanics up to magnitudes of the order of 1/c². Hence it appears that the energy depends upon the system of reference. But since the t-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting case c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.[[33]]
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 √-1 t = l, in the place of l, then the differential expression
dτ² = -(dx² + dy² + dz² + dl²),
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
3·10⁵ km = √-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 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/e is to be reckoned as the measure of PQ. Now at the world-point P₁, the vector-potential of the field excited by e is represented by the vector in direction PQ, having the magnitude e/cr, in its three space components along the x-, y-, z-axes; the scalar-potential is represented by the component along the t-axis. This is the elementary law found out by A. Lienard, and E. Wiechert.[[34]]
If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the field into electric and magnetic forces is a relative one, and depends upon the time-axis assumed; the two forces considered together bears some analogy to the force-screw in mechanics; the analogy is, however, imperfect.