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On the dynamics of the electron
Henri Poincaré June 5, 1905
Reports of the Academy of Science (Comptes Rendus de l'Académie des Sciences),
T. 140, p. 1504-1508

It appears at first sight that aberration of light and related optical phenomena will provide us with a means to determine the absolute motion of the Earth, or rather, its motion, not compared to the other stars, but compared to the ether. It is not so; experiments where one accounts for only the first power of aberration have failed and one easily discovered the explanation; but Michelson, having imagined an experiment where one could detect the terms that depend on the square of the aberration, had not any more luck. It seems that this impossibility to demonstrate absolute motion is a general law of nature.

An explanation was proposed by Lorentz, who introduced the assumption of a contraction of all bodies in the direction of the terrestrial motion; this contraction would account for the experiment of Michelson and all those which were carried out up to here, but it would leave the place to other, still more delicate experiments, and easier to conceive than to carry out, [1505] which would be likely to highlight the absolute motion of the Earth. But, if one views the impossibility of such an observation to be highly probable, one may foresee that these experiments, if one ever manages to carry them out, will still give a negative result. Lorentz sought to supplement and modify his assumption in order to bring it in agreement with the postulate of the complete impossibility to determine absolute motion. This is what he succeeded to do in his article entitled 'Electromagnetic phenomena in a system moving with any velocity smaller than that of light' (Proceedings of the Academy of Amsterdam, May 27, 1904). http://en.wikisource.org/wiki/Electromagnetic_phenomena

The importance of this question made me determined to take it up again and the results that I obtained are on all important points in agreement with those of Lorentz; I was only led to modify and supplement them in some small points. The essential point, established1 by Lorentz, is that the equations of the electromagnetic field are not modified by a certain transformation (which I will call by the name of ''Lorentz'') and which is of the following form:

:(1)


x, y, z are the coordinates and t the time before the transformation, x' , y' , z' and t' after the transformation. Further, epsilon is a constant which defines the transformation:

,

and l is an unspecified function of epsilon. One sees that in this transformation the x-axis plays a particular part, but one can obviously build a transformation where this part would be played by an unspecified line passing by the origin. All these transformations together, joint together with all rotations in space, must form a group; but for this to be the case, it is necessary that l=1; one is thus led to suppose that l=1 and this is a consequence that Lorentz has obtained in another way.

Are rho the electric electron density, and xi, eta , zeta its velocity before the transformation; then one will have for the same quantities, rho', xi' , eta' , zeta' after the transformation :

:(2)


[1506] These formulas differ a little from those which had been found by Lorentz.

Are now X, Y, Z and X' , Y' , Z' three components of the force before and after the transformation, the force is reported to the unit of volume ; I find :

:(3)


These formulas also differ a little from those of Lorentz; the complementary term in Sigma X xi reminds of a result obtained formerly by Mr. Liénard.

If we indicate now by X1, Y1, Z1 and X'1, Y'1, Z'1 components of the force, not reported to the unit of volume, but to the unit of mass of the electron, we will have :

:(4)
.

Lorentz was also led to assume that the electron motion takes the form of a flattened ellipsoid; this is also the assumption made by Langevin, only, while Lorentz assumed that two axes of the ellipsoid remain constant, which is consistent with the hypothesis l = 1, Langevin assumes that it is the volume that remains constant. Both authors have shown that these two hypotheses agree with the experiments of Kaufmann, as well as the original hypothesis of Abraham (spherical electron ). The hypothesis of Langevin had the advantage of being self sufficient, since it suffices to look at the electron as deformable and incompressible to explain that it takes an ellipsoidal shape when it moves. But I show, in agreement with Lorentz, that such cannot be made to agree with the impossibility of an experiment showing absolute motion. This upholds, as I have said, that l = 1 is the only hypothesis for which the ensemble of the Lorentz transformations form a group.

But with the hypothesis of Lorentz, the agreement between the formulas is not the only thing: one obtains this, and at the same time a possible explanation for the contraction of the electron, assuming that the deformable and compressible electron, is subjected to a kind of constant external pressure whose work is proportional to changes in volume.

I show2, by applying the principle of least action, that under [1507] these conditions the compensation is complete, assuming that inertia is exclusively an electromagnetic phenomenon, as generally admitted since the Kaufmann experience, and apart from the constant pressure that I just mentioned and which acts on the electron, all the forces are of electromagnetic origin. One has thus the explanation for the inability to show the absolute motion and the contraction of all bodies in the direction of the Earth's motion.3

But that's not all: Lorentz, in the cited work, 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 on their components by the Lorentz transformation is still defined by equations (4).

It was important to examine this hypothesis more closely and in particular to pursue what changes it would require us to make to the laws of gravity. This is what I sought to determine; I was first led to suppose that the propagation of gravity is not instantaneous, but done with the speed of light. This seems inconsistent with results obtained by Laplace, who announced that this spread is, if not instantaneous, at least much faster than light. But in reality, the question that was posed by Laplace differs considerably from the one with which we are occupied here. For Laplace, the introduction of a finite speed of propagation was the only change that he brought to Newton's law. Here, however, this change is accompanied by several others; it is possible, and it so happens, that it produces a partial compensation between them.

When we speak therefore of the position or velocity of the attracting body, it will be the position or the velocity at the moment that the gravitational wave parted from the body; when we speak of the position or velocity of the attracted body, it will be the position or the velocity at the moment when this attracted body has been reached by the gravitational wave emanating from the other body; it is clear that the first time precedes the second.

If, then x, y, z are the projections on the three axes of the vector joining the two positions, if the velocity of the attracted body is xi, eta, zeta, and of the attracting body xi1, eta1, zeta1, the three components of attraction (which I can still call X'1, Y'1, Z'1) are functions of x, y, z, xi, eta, zeta, xi1, eta1, zeta1.
I wondered if it would be possible to determine these functions in such a [1508] way that they will be affected by the Lorentz transformation according to equations (4) and that one finds the ordinary law of gravitation, whenever speeds xi, eta, zeta , xi1, eta1, zeta1 are small enough that one can neglect the square in front of the square of the speed of light.

The answer must be affirmative. We find that the corrected attraction consists of two forces, one parallel to the vector x, y, z, the other to the velocity xi, eta, zeta.

The difference with the ordinary law of gravitation is, as I have said, of the order of xi²; if we only assume, as did Laplace, that the speed of propagation is that of light, this divergence is of the order of xi, that is to say 10 000 times greater. It is therefore not prima facie absurd to assume that astronomical observations are not precise enough to detect differences as small as the ones we imagine. But only a more thorough discussion will permit to decide.



Footnotes by this translation:

1. Not exactly; for details see for example http://www.bourbaphy.fr/darrigol2.pdf
2. Published one year later in his full paper (not yet translated):
http://fr.wikisource.org/wiki/Sur_la_dynamique_de_l%E2%80%99%C3%A9lectron_(juillet)
3. Earlier, Lorentz had also described the effect of motion on the time period of vibrations. See the last section of:
http://en.wikisource.org/wiki/Simplified_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Systems

*Note: apparently an English translation of both articles has been made available some years ago:
A. A. Logunov (1995) On the Articles by Henri Poincaré "On the Dynamics of the Electron" (English translation G. Pontecorvo),
Publishing Department of the Joint Institute for Nuclear Research, Dubna. (1st Russian edition 1984.)