Henri Poincaré

“ 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. ”

“ But there are forces which we can not assign an electromagnetic origin, as for example gravitation. It could happen, indeed, that two systems of bodies produce equivalent electromagnetic fields, that is to say, exerting the same action on the electrified bodies and on the currents, and yet these two systems do not exercise the same gravitational action on the Newtonian mass. ”

“ If the electron continues to move in a rectilinear and uniform way with the velocity it had at the instant t1, that is to say, with the velocity -ε, 0, 0, the point (5) would be the one occupied at the instant t.
Taking the acceleration wave, we can, through the Lorentz transformation, reduce its determination to the case of zero velocity. This is the case if we imagine an electron whose oscillation amplitude is very small, but very fast, so that the displacements and velocities are much smaller, but the accelerations are finished. ”

“ In this theory, two equal lengths are, by definition, two lengths for which light takes the same time to traverse.
Perhaps it would be enough to give up this definition, so that the theory of Lorentz is as completely rejected as it was the system of Ptolemy by the intervention of Copernicus. If that happens one day, it will not prove that the effort made by Lorentz was useless; because Ptolemy, no matter what we think about him, was not useless for Copernicus. ”

“ It seems at first sight that the indetermination remains, since we have made no hypothesis about the value of t, that is to say about the speed of transmission; and that also x is a function of t, but it is easy to see that x - ξt, y, z (which appear in our formulas) do not depend on t.
We see that if two bodies are simply in motion by a common translation, the force acting on the body is drawn normal to an ellipsoid with its center at the attracting body.
To go further we must look for the invariants of the Lorentz group. ”

“ The potentials ψ, F, G, H depend on the position of the electron and also its velocity, because not only ξ1, η1, ζ1 show up in the numerator of F, G, H, but the radial component ω shows up in the denominator. It is of course its position and its velocity at the instant t1.
The partial derivatives of φ, F, G, H with respect to t, x, y, z (and therefore the electric and magnetic fields) will also depend on its acceleration. Moreover, they depend linearly, since the acceleration in these derivatives is introduced as a result of a single differentiation. ”

“ It seems at first sight that the aberration of light and the associated optical phenomena will provide a means of determining the absolute motion of the Earth, or rather its motion, not in relation to other stars, but in relation to the ether. ”

“ But Copernicus, by simply changing the axes of coordinates regarded as fixed, destroyed this appearance; each planet does not describe any more than only one circle and the durations of the revolutions become independent (until Kepler restores between them the binding which was believed to be destroyed) .
Here it is possible that there is something analogue; if we admit the postulate of relativity, we would find in the law of gravitation and the electromagnetic laws a common number which would be the speed of light ”

“ The conclusion is that if the electron is subject to a binding between its three axes, and if no other force intervenes except the binding forces, the shape of that electron, when it is given a uniform velocity, may be such that the ideal electron corresponds to a sphere, except the case where the binding is such that the volume is constant, in conformity with the hypothesis of Langevin. ”

“ When we therefore speak of the position or velocity of the attracting body, it will be the position or the velocity at the time when the gravitational wave leaves the body; when we talk about the position or velocity of the attracted body, it will be the position or the velocity at the moment when this body was reached and attracted by the gravitational wave emanating from the other body; it is clear that the first instant precedes the second. ”

“ Thus Lorentz's hypothesis is the only one consistent with the inability to demonstrate absolute motion; if we accept this impossibility, we must admit that the moving electrons contract and become ellipsoids of revolution where two axes remain constant; it must be admitted, as we have shown in the previous §, the existence of an additional potential which is proportional to the volume of the electron. ”

“ We have thus the explanation of the impossibility of demonstrating absolute motion and of the contraction of all the bodies in the direction of the terrestrial motion.
But that's not all: Lorentz, in the work quoted, found it necessary to complete his hypothesis by assuming that all forces, whatever their origin, are affected by translation in the same way as electromagnetic forces and, consequently, the effect produced on their components by the Lorentz transformation is still defined by equations (4) . ”

“ The propagation velocity is much faster than that of light, but in some cases t may be negative, which, as we have said, seems hardly acceptable. We will add this to hypothesis (A) .
2° The four invariants (7) must be functions of the invariants (5) .
3° When the two bodies are in absolute rest, X1, Y1, Z1 must have the value deduced from the law of Newton, and when they are in relative rest, the value deduced from the equations (4) . ”

“ So when we apply the Lorentz transformation when the real electron is spherical, the electron becomes a perfect ellipsoid. The equation of this ellipsoid is based on § 1:
But here we have:
so that the equation of the ellipsoid becomes:
If the radius of the real electron is r, the axes of the ideal electron would therefore be:
In Lorentz's hypothesis, however, the moving electrons are deformed, so that the real electron would become a ellipsoid, while the ideal electron is still always a perfect sphere of radius r ”

“ But if the impossibility of such a finding is considered highly probable, one can predict that these experiments, if they can ever be conducted, will give a negative result. Lorentz has sought to supplement and amend his hypothesis so as to bring it into accord with the postulate of the complete impossibility of determining absolute motion. This he managed to do in his article entitled Electromagnetic phenomena in a system moving with any velocity smaller than that of Light (Proceedings de l’Académie d’Amsterdam, May 27, 1904) . ”

“ Now a is a numerical constant which shows that: the pressure that creates our additional potential is proportional to the 4th power of the experimental mass of the electron.
As Newton's law is proportional to the experimental mass, we are tempted to conclude that there is some relation between the cause that generates gravitation and the one that generates the additional potential.
§ 9. — Hypotheses on gravitation Thus Lorentz's theory would completely explain the impossibility to demonstrate absolute motion, if all forces are of electromagnetic origin. ”

“ Langevin waves Langevin has put the formulas that define the electromagnetic field produced by the motion of a single electron in a particularly elegant form.
Let us remember the equations
We know we can integrate by the retarded potentials and we have: In these formulas we have:
whereas ρ1 and ξ1 are the values of ρ and ξ at the point x1, y1, z1 and the instant
x0, y0, z0 being coordinates of a molecule of the electron at the instant t ”

“ In order to obtain the second determinant, I look at the coordinates of different molecules of the electron at instant t', which is the same for all molecules, but in such a way that for the molecule considered we have . The coordinates of a molecule will then be:
U', V', W' is what become of U, V, W, when we replace t1 by t'1 ”

“ It's about the transformation of δJ. To begin the search for the continuity equation, we express how the charge of an electron is preserved by the variation.
Let x0, y0, z0 be the initial position of an electron. Its current position is:
We also introduce an auxiliary variable ε, which produces changes in our various functions, so that for any function A we have:
It is indeed convenient to switch from the notation of variation calculus to that of ordinary calculus, or vice versa. ”

“ Our functions should be regarded: 1° as dependent on five variables x, y, z, t, ε, so that we can remain at the same place when ε and t vary alone: we then indicate their derivatives by the ordinary d; 2° as dependent on five variables x0, y0, z0, t, ε so that we may always follow a single electron when t and ε vary alone, then we denote their derivatives by ∂. We will have then:
Denote now by Δ the functional determinant of x, y, z with respect to x0, y0, z0:
If ε, x0, y0, z0 remain constant, we give to t an increasement ∂t ”

“ Let on the other hand:
we see that the Lorentz transformation will cause to make δx, δy, δz and δ1x, δ1y, δ1z, δ1t undergo the same linear substitutions as with x, y, z, t.
We regard
as the coordinates of three points P, P', P" in a 4-dimensional space. We see that the Lorentz transformation is a rotation of that space around the origin, regarded as fixed. ”

“ Thus we find in addition to the four invariants (5) , four new distinct invariants, which are:
The last invariant is always zero, according to the definition of T1.
This granted, what are the requirements?
1° The left-hand side of relation (1) , which defines the velocity of propagation must be a function of the four invariants (5)
One can obviously make a lot of hypotheses, we only look at two:
A) It may be
where t = ±r, and since t must be negative, t = -r. ”

“ We denote by
the longitudinal electric energy; by
the transverse electric energy; by
the transverse magnetic energy. There is no longitudinal magnetic energy, since α = α' = 0. We denote by A', B', C' the corresponding quantities in the ideal system. ”

“ At first sight, the solution (8) seems the most straightforward, it may nevertheless be adopted and in effect – since M, N, P are functions of X1, Y1, Z1 and – we can draw from these three equations (8) the values of X1, Y1, Z1, but in some cases these values become imaginary.
To avoid this, we will operate in another way. Let:
This is justified by the analogy with the notation
which appears in the substitution of Lorentz. ”

“ We pose then:
it follows that C had disappeared from the denominator of (11a) :
and there will also:
Then λ, μ, ν or is a kind of electric field, while λ', μ', ν' or rather is a kind of magnetic field.
3° The postulate of relativity would require us to adopt solution (11) or solution (14) or any solution that would inferred by using the first remark; but the first question that arises is whether they are compatible with astronomical observations ”