Translation:Concerning the Dynamics of the Electron in the Kinematics of the Principle of Relativity

by Max Born

Max Born (Göttingen)

Concerning the Dynamics of the Electron in the Kinematics of the Principle of Relativity.

Gentlemen! It's known that the determinations of electromagnetic mass as a function of velocity performed in recent times, were interpreted by the relevant experimenters as being in favor of Lorentz's formula, and against Abraham's formula related to this dependency (Bucherer,^[1] Hupka^[2]). It seems to be necessary to subject the theoretical foundations of Lorentz's formula – which is indeed very closely connected with the relativity principle – to a detailed test; because it is known (I cannot discuss this more detailed because of the limited time at my disposal) that objections have been raised against Lorentz's original derivation of the formula, because different values for the mass were given depending on whether one uses the energy theorem or the momentum theorem. On the other hand, Einstein's derivation of the formula given by him in his first paper^[3] on the relativity principle, can hardly be called an electrodynamic derivation of the inertia phenomenons of the electrons; because the validity of Newton's laws of motion for infinitely slow motions of a mass point are presupposed there, and then the laws of motion for arbitrary motions of the point were concluded by the aid of the relativity principle. Therefore I worked out a new theory of the dynamics of the electron^[4], which has the relativity principle as its foundation, namely by using the views formed and applied by Minkowski^[5].

The principal difficulty we are dealing with at that occasion, is the kinematic constitution of the electron at arbitrary motions. It's known that Lorentz's theory is based on the idea of the "deformable electron", which is only defined at uniform motion, namely in a way by which it suffers a velocity-dependent contraction in the direction of motion: the validity of the theory thus only encloses quasi-stationary motions, i.e. such ones that only slightly deviate from uniform ones. For a strict theory, Lorentz's deformation law is to be generalized from uniform motions to arbitrary motions in a suitable way. In which way this has to be done, can be explained at the following figure (Fig. 1). If a point is moving along the $x$-axis, then its motion can be represented by a curve whose abscissa axis denotes the coordinate $x$, and whose ordinate axis denotes the time $t$. Two rigidly connected points (in the sense of ordinary mechanics), which move at the $x$-axis, provide two curves in this representation, whose distance measured along the line $t$=const. is invariable. This condition of rigidity contradicts the relativity principle; because it is not invariant against those Lorentz-Einstein transformations, which express the equivalence of all such coordinate systems, whose axis-directions are conjugated to the form

$x^{2}+y^{2}+z^{2}-c^{2}t^{2}=\pm 1$

(where $c$ denotes the speed of light). Now one easily sees, that in general there is no analogue invariant relation at all between the coordinates of two moving points alone, as it is the case with the distance in ordinary mechanics. But there is such an invariant relation between any two infinitely close points, which expresses that its distance in a comoving coordinate system, i.e. measured along the direction conjugated to the tangent of the trajectory curve with respect to the form

$x^{2}+y^{2}+z^{2}-c^{2}t^{2}=\pm 1$,

is invariable. Thus one comes to introduce an infinitesimal concept of rigidity, which finds its expression in a system of 6 partial differential equations. In order to arrive at the finite laws of motion, one has to integrate them. I didn't succeed yet to conduct this in a general way, but I succeeded in the special case where the points of the body are moving upon parallel straight lines. It then suffices to consider the points upon one of these lines, for instance the $x$-axis, and then one can describe the kind of the connection of the points so that the similarity with ordinary rigidity (as explained by Fig. 1) is evident. Namely when the movement of one point, i.e. a curve of the $xt$-plane, is known, then one finds the movements of the points rigidly connected with it in the following way (Fig. 2):

One constructs the two invariant hyperbolas

$x^{2}-c^{2}t^{2}=\pm 1$,

which have the straight lines $x\pm ct=0$ as asymptotes. Then draw the tangent at one point $P$ of the given curve, and a ray with respect to the direction conjugated with respect to that hyperbola, whose length is made equal to the multiple of $r$ of the radius of the hyperbola $x^{2}-c^{2}t^{2}=1$ which lies in this direction, where $r$ is the distance between both points when they are at rest. If one conducts this construction in every point of the given curve, then one finds the sought curve of the rigidly connected point.^[6] In the case of uniform motions, this construction is identical to Lorentz's deformation law.

At the image arising this way, one recognizes a remarkable property of those new "rigid" bodies. Those rays which connects the first curve to the second curve, are generally not parallel as in Fig. 1, and therefore have an envelope over which they don't pass by (Fig. 3). Therefore such body can only have a finite extension, and since the envelope lies the closer at the starting curve the greater its curve is, it follows that the body must be the smaller the greater its acceleration is. One easily sees, that when the body fits in a sphere of radius $R$, it must always be

$b<{\frac {c^{2}}{R}}$,

where $b$, the "amount of acceleration",^[7] follows from the ordinary acceleration by division through the factor $\left(1-{\tfrac {w^{2}}{c^{2}}}\right)^{2}$. The actual occurrence of immense accelerations of electrons thus shows, that the electrons, in so far as they are rigid structures in the new sense, must be extraordinarily small. But this restriction provides values for the magnitude of the electrons, which are far above the ones that are ordinarily assumed; for instance, if one imagines an electron of ⅓ the speed of light to be brought to rest upon a path length equal to the diameter of a hydrogen atom ($10^{-8}$ cm), then it is given $b=5\cdot 10^{27}$ cm sec^-2, thus $R<1.8\cdot 10^{-7}$ cm, while the ordinarily assumed value amounts to $R<1.5\cdot 10^{-13}$; to this corresponds $b<6\cdot 10^{33}$ cm sec^-2 as the uppermost limit of acceleration. Thus it is not to be expected that experience will decide for or against this consequence of the theory, from which the necessity of the assumption of an atomistic structure of electricity arises.

Now there is a preferred accelerated motion, which corresponds to the uniform-accelerated motion of old mechanics and which I call "hyperbolic motion". Namely, while in the old mechanics the motion of a rigid body with constant acceleration (vertical throw) is represented in a $xt$-coordinate system by a bundle of parallel parabolas, the image of this hyperbolic motion is a bundle of hyperbolas, whose asymptotes are the lines $x\pm ct=0$. Also this motion is in a certain sense to be denoted as uniformly accelerated, since the previously introduced "amount of acceleration" $b$ is constant for every point. The parabolas of old mechanics are evidently the limiting case $c=\infty$ of such hyperbolas. Since furthermore every motion can be approximated by hyperbolic motions, I have used them for the following electrodynamic calculations.

In order to test the dynamic behavior of such a rigid body, one imagines that it is equipped with electric charge and determines the force exerted upon itself. First, the field of the electron must be calculated for this purpose. Without being forced to make any assumption concerning the form and charge distribution, the following is given:

At hyperbolic motion whose acceleration can indeed have arbitrary large values, the electron carries its field along with itself; radiation is not present. The field is of course different from the ordinary static one; however, no magnetic field is present as seen from the electron itself, exactly as in the case of uniform motion, and the electric one is derived by differentiation (even though in somewhat modified form) form a potential $\Phi$, which satisfies a differential equation

$\Delta \Phi +{\frac {1}{x}}{\frac {\partial \Phi }{\partial x}}-{\frac {\Phi }{x^{2}}}=\varrho$,

very analogue to $\Delta \Phi =\varrho$.

I interpreted the vector denoted by Minkowski as "electric rest force", as the ponderomotive forces upon the unit charge, in agreement with the approach of Lorentz ${\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {wH}}]$; i.e. at every point the electric field strength as it is measured in a coordinate system whose motion agrees with the instantaneous motion of the point. On the other hand one can easily see, that the way by which the resulting forces are ordinarily formed from these individual forces by integration over the volume of the electron at one instant, contradicts the relativity principle.

The contradictions arising in Lorentz's theory stem from this circumstance. By a near-at-hand modification of that approach, one achieves such resultants which are invariant against Lorentz transformations; however, those are not identical with the space integral of the individual forces, and therefore also the concepts total energy, total momentum, etc. lose their meaning for the electron mechanics. The result of the calculation is: the components of the resulting force perpendicular to the direction of motion are not zero in general, even at quasi-stationary motion. However, since experience tells us that no external forces are required to sustain them, it follows that the charge must be distributed in a specific way, namely in homogeneous, spherical, concentric layers, in so far as one makes the assumption that the direction of motion is an arbitrary direction in the electron. If we finally consider the force component in the direction of motion, then it can be brought into the form

$-\mu b_{x}$,

where $b_{x}$ is the $x$-component of the acceleration vector in the center, $\mu$ a certain function of $b$, which is otherwise only dependent on the electric density; $\mu$ is to be denoted as rest mass.

It is only to be shown that an external force field exists, which is capable of sustaining a hyperbolic motion. Indeed this is performed by an external, constant, electric field $E_{x}$ in the direction of motion.

Then one obtains for such motions, which only slightly deviate from hyperbolic motions, the mechanical fundamental equation in Minkowskian form:

$\mu {\frac {d^{2}x}{d\tau ^{2}}}=eE_{x}{\frac {dt}{d\tau }}$,

where $\tau$ denotes the proper time.

Concerning the function $\mu (b)$, it is given that it is constant up to terms of second order in $b$, whose coefficient simultaneously vanishes with the radius of the electron, and namely one finds the known value

$\mu ={\frac {4}{3c^{2}}}U$,

where

$U={\frac {1}{8\pi }}\iint {\frac {\varrho {\bar {\varrho }}}{r}}dw\ d{\bar {w}}$

denotes the electrostatic energy of the electron. If one additionally introduces the "ordinary" mass by Lorentz's relation

$m={\frac {\mu }{\sqrt {1-{\frac {w^{2}}{c^{2}}}}}}$,

then one obtains the equation of motion in the Einstein-Planck form:

${\frac {dm\ w_{x}}{dt}}=eE_{x}$,

and the energy theorem assumes the form:

${\frac {dm}{dt}}={\frac {1}{c^{2}}}eE_{x}w_{x}$.

The theory can easily be generalized by superposition of Lorentz transformations, so that it involves all practical existent cases.

(Received 14 October 1909.)

Discussion

Sommerfeld: The great simplification and beautification of the theory, which is given by the definition of rigidity and the fortunate thoughts of Born, can nobody cherish better than me, because earlier I also considered the uniformly accelerated motion, and because of the old definition of rigidity I arrived at extraordinarily unclear expressions. Nevertheless, I would like to emphasize one thing: the relativity principle can only say something about uniform motion; in cases of accelerated motion it can surely suggest approaches, but it can never provide definite conclusions, e.g. with respect to the form of ponderomotive forces; for instance, in your approach one could add terms to the forces which not contain the velocity anymore. I am very much in favor of neglecting such terms (and the approaches of Born in any case seem to be the simplest among many other possible ones). But a conclusive reason cannot be provided by the relativity principle.

Lecturer: But not every assumption that one makes about accelerations is in agreement with the relativity principle (Sommerfeld: Of course not!). I have taken the simplest assumption and everywhere I principally followed Minkowski. The only thing new is maybe the composition of forces at rigid bodies into resultants.

---

1. A. H. Bucherer, this journal, 9, 755-762, 1908; Ann. d. Phys. 28, 513-536, 1909.
2. E. Hupka, Verh. d. Deutsch. Ph. Ges. 11, 249-258, 1909.
3. A. Einstein, Ann. d. Phys. 17, 891, 1905.
4. M. Born, |Ann. d. Phys. 30, 1, 1909.
5. H. Minkowski, Nachr. d. k. Ges. d. Wissensch. zu Göttingen, math. Phys. Kla., S. 54, 1908; this journal 10, 104, 1909 and Jahresber. d. deutsch. Mathematiker-Vereinigung, 18, 1909. (Also published as separate print.)
6. This construction applies only to the case of translation.
7. Minkowski, this journal 10, 108, 1909; Jahresb. d. deutsch. Math.-Ver. 18, 84.