Henri Bergson: Half-Relativity – Lorentz Equations (12)

So would Paul speak. And at the same time he would have established the famous “transformation equations” of Lorenz, equations which, moreover, if we place ourselves at the more general point of view of Einstein, do not imply that the system S is definitively fixed. We will indeed show in due course how, according to Einstein, we can make of S any system, provisionally immobilized by the force, and how we will then have to attribute to S’, considered from the point of view of S, the same temporal and spatial deformations that Peter attributed to Paul’s system. In the hypothesis, always admitted until now, of a unique Time and a Space independent of Time; it is evident that if S’ moves relative to S with the constant speed v, if x’, y’, z’ are the distances from a point M’ of the system S’ to the three planes determined by the three rectangular axes, taken two by two, O’X’, O’Y’, O’Z’, and if finally x, y, z are the distances from this same point to the three fixed rectangular planes with which the three moving planes first coincided, we have:

x = x’ + vt’

y = y’

z = z’

As moreover the same time unfolds invariably for all systems, we have:

t = t’.

But if the motion determines contractions of length, a slowing down of time, and the fact that, in the system of dilated time, the clocks no longer mark more than a local curve, it results from the explanations exchanged between Peter and Paul that we will have:

(1)	x = (1/√(1 – v²/c²))(x’ + vt’) y = y’ z = z’ t = (1/√(1 – v²/c²))(t’ + vx’/c²)

From there a new formula for the composition of speeds. Let us suppose in fact that the point M’ moves with a uniform motion, inside S’, parallel to O’X’, with a speed v’, measured naturally by x’/t’. What will its speed be for the spectator seated at S and who relates the successive positions of the mobile to its axes OX, OY, OZ? To obtain this speed v”, measured by x/t, we must divide member by member the first and fourth of the equations above,, and we will have:

v” = (v + v’)/(1 + vv’/c²)

whereas until now mechanics posed:

v” = v + v’

Source: Henri Bergson, Durée et simultanéité : à propos de la théorie d’Einstein, Deuxième édition, qugmentée, Paris, 1923. Translation and interpretation Nicolae Sfetcu. © 2024 MultiMedia Publishing