Translation:On the Dynamics of the Electron (June)

by Henri Poincaré

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. This is not the case: the experiments in which we take into account only the first order of aberration were initially unsuccessful and an explanation was easily found; but Michelson, who imagined an experiment by which terms depending on the square of the aberration could be measured, had no luck either. It seems that this inability to demonstrate absolute motion is a general law of nature.

An explanation was proposed by Lorentz, who introduced the hypothesis of a contraction of all bodies in the direction of motion of the earth; this contraction would account for the Michelson-Morley experiment and all those that have been conducted to date, but leaves room for other experiments even more delicate and more easily conceived than executed, which might demonstrate the absolute motion of the Earth. 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).

The importance of this question made me determined to return to it; and the results I obtained are in agreement on all important points with those of Lorentz; I was only led to modify and complete them in a few points of detail.

The essential point, established by Lorentz, is that the electromagnetic field equations are not altered by a certain transformation (which I shall call after the name of Lorentz), which has the following form:

(1)	$x^{\prime }=kl(x+\epsilon t),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=kl(t+\epsilon x)$

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

$k={\frac {1}{\sqrt {1-\epsilon ^{2}}}}$,

and $l$ is an arbitrary function of $\epsilon$. One can see that in this transformation the $x$-axis plays a particular role, but one can obviously construct a transformation in which this role would be played by any straight line through the origin. The sum of all these transformations, together with the set of all rotations of space, must form a group; but for this to occur, we need $l=1$; so one is forced to suppose $l=1$ and this is a consequence that Lorentz has obtained by another way.

Let $\rho$ the electric density of the electron, $\xi ,\eta ,\zeta$ the velocity before the transformation; we obtain for the same quantities $\rho ',\xi ',\eta ',\zeta '$ after processing

(2)	$\rho ^{\prime }={\frac {k}{l^{3}}}\rho (1+\epsilon \xi ),\quad \rho ^{\prime }\xi ^{\prime }={\frac {k}{l^{3}}}\rho (\xi +\epsilon ),\quad \rho ^{\prime }\eta ^{\prime }={\frac {\rho \eta }{l^{3}}},\ \quad \rho ^{\prime }\zeta ^{\prime }={\frac {\rho \zeta }{l^{3}}}$

These formulas differ somewhat from those which had been found by Lorentz.

Let now $X,Y,Z$ and $X',Y',Z'$ the three components of force before and after transformation, and the force is expressed in unit volume; I found

(3)	$X^{\prime }={\frac {k}{l^{3}}}(X+\epsilon \Sigma X\xi ),\quad Y^{\prime }={\frac {Y}{l^{3}}},\quad Z^{\prime }={\frac {Z}{l^{3}}}$

These formulas are slightly different from those of Lorentz; the additional term in $\Sigma X\xi$ reminds us on a result previously obtained by Liénard.

If we now denote by $X_{1},\ Y_{1},\ Z_{1}$ and $X'_{1},\ Y'_{1},\ Z'_{1}$ the components of a force, not referred to unit volume, but to unit mass of the electron, we obtain

(4)	$X_{1}^{\prime }={\frac {k}{l^{3}}}{\frac {\rho }{\rho ^{\prime }}}(X_{1}+\epsilon \Sigma X_{1}\xi ),\quad Y_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Y_{1}}{l^{3}}},\quad Z_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Z_{1}}{l^{3}}}$

Lorentz was also led to assume that the electron in motion takes the form of an oblate spheroid; this is also the hypothesis made by Langevin, however, while Lorentz assumed that two axes of the ellipsoid remain constant, which is consistent with the hypothesis $l=1$, Langevin assumed that the volume remains constant. Both authors have shown that these two hypotheses are consistent with the experiments of Kaufmann, as well as the original hypothesis of Abraham (spherical electron). The hypothesis of Langevin would have the advantage that it is self-sufficient, because it suffices to regard the electron as deformable and incompressible, and to explain that it takes an ellipsoidal shape when it moves. But I show, in agreement with Lorentz, that it is incapable to accord with the impossibility of an experiment showing the absolute motion. As I have said, this is because $l=1$ is the only case for which all the Lorentz transformations form a group.

But with the hypothesis of Lorentz, the agreement between the formulas does not occur all alone; we obtain at the same time a possible explanation for the contraction of the electron, assuming that the electron, deformable and compressible, is subjected to a constant external pressure whose work is proportional to volume changes.

I show, by applying the principle of least action, that under those conditions the compensation is complete, assuming that inertia is an electromagnetic phenomenon exclusively, as generally admitted since Kaufmann's experiments, and apart from the constant pressure that I just mentioned and which acts on the electron, all the forces are of electromagnetic origin. 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).

It was important to examine this hypothesis more closely and in particular to examine what changes it would require us to make on the law of gravitation. That is what I sought to determine; I was first led to suppose that the propagation of gravitation is not instantaneous, but happens with the speed of light. This seems at odds with results obtained by Laplace, who announced that this propagation is, if not instantaneous, at least much faster than that of light. But in reality, the question posed by Laplace differs considerably from that which occupies us here. The introduction of a finite velocity of propagation was the only change Laplace introduced to Newton's law. Here, on the contrary, this change is accompanied by several others; it is possible, and that is indeed what happens, that a partial compensation occurs between them.

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.

So if $x,y,z$ 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 that of the attracting body $\xi _{1},\eta _{1},\zeta _{1}$, the three components of the attraction (which I can still call $X'_{1},\ Y'_{1},\ Z'_{1}$) are functions of $x,y,z,\xi ,\eta ,\zeta ,\xi _{1},\eta _{1},\zeta _{1}$. I asked myself whether it was possible to determine these functions in a way that they are affected by the Lorentz transformation according to equations (4) and found the ordinary law of gravitation, whenever the velocities $\xi ,\eta ,\zeta ,\xi _{1},\eta _{1},\zeta _{1}$ are small enough that one can neglect the squares in respect to the square of the speed of light.

The answer must be affirmative. It is found 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 to the ordinary law of gravitation, as I have said, is of order $\xi ^{2}$; if we only assume, as Laplace did, that the speed of propagation is that of light, this discrepancy is of order $\xi$, that is to say 10.000 times larger. It is therefore, at first sight, not absurd to assume that astronomical observations are not precise enough to detect a difference as small as the one which we imagine. But this is what only a thorough discussion will make possible to decide.