The Lorentz Transformation

Lorentz had adopted a particular system of units, so as to eliminate the factors 4π in the formulas.

I do the same. I choose the units of length and time so that the speed of light is equal to 1.

Under these conditions, the fundamental formulas become (by calling f, g, h the electric displacement, α, β, γ the magnetic force, F, G and H the vector potential, φ the scalar potential, ρ the electric density, ξ, η, ζ the electron velocity, u, v, w the current):

(1)

A material element of volume dx dy dz suffers a mechanical force whose components X dx dy dz, Y dz dx dy, Z dx dy dz are deduced from the formula:

(2)

These equations are capable of a remarkable transformation discovered by Lorentz and which owes its interest from the fact, that it explains why no experience is suited to show us the absolute motion of the universe. Let:

(3)

l and ε are two arbitrary constants, andIf we now set:

it follows:

Consider a sphere entrained with the electron in a uniform translational motion, and is the equation of that moving sphere whose volume is .

The transformation will change it into an ellipsoid, and it is easy to find the equation. It is easily deduced because of equations (3):

(3bis)

The equation of the ellipsoid becomes:

This ellipsoid moves in uniform motion; for t’ = 0, it reduces to and has the volume:

If we want that the charge of an electron is not altered by the transformation, and when we call ρ’ the new electrical density, it follows:

(4)

Those are the new velocities ξ’, η’, ζ ‘; we must have:where: 4bis

Here I should mention for the first time a discrepancy with Lorentz.

Lorentz poses (with different notations) (loco citato, page 813, formulas 7 and 8):

We thus find the formulas:

but the value of ρ’ differs.

It is important to note that formulas (4) and (4bis) satisfy the continuity condition

Let λ be an undetermined quantity and D the functional determinant

(5)

with respect to t, x, y, z. We will have:

with

Let 5bis

, we see that the four functionsare related to the functions (5) by the same linear relations as the old variables to the new variables. Then, if we denote by D’ the functional determinant of the functions (5bis) in relation to the new variables, we have: where:

C. Q. F. D

With the hypothesis of Lorentz, this condition is not satisfied, since ρ’ has not the same value.

We will define the new potentials, vector and scalar, in order to satisfy the conditions (6)

Then we obtain from this: (7)

These formulas differ significantly from those of Lorentz, but the difference is ultimately due to the definitions.

We will choose the new electric and magnetic fields so as to satisfy the equations: (8)

It is easy to see that:

and we conclude: (9)

These formulas are identical to those of Lorentz.Our transformation does not alter the equations (I). Indeed, the continuity condition, and the

equations (6) and (8), already provided us with some of the equations (I) (except the accentuation of letters).

Equations (6) close to the continuity condition give: (10)

It remains to establish that:

and it is easy to see that these are necessary consequences of equations (6), (8) and (10).