Max Born

“ Here, the formation of the resultants by integration over the space at invariable , is justified. Because, when we confine ourselves to the case of rectilinear translation, then all points of the electron have the same velocity at a time ; the single points of the electron don't perform work relative to each other, the forces acting between them don't appear, and therefore one can see the integrals of the force components and the work over the volume of the electron, as resulting force components and total work. There, however, no relativity principle of any kind is satisfied. ”

“ From that it follows that the motion of a "rigid" body is in general completely determined by one worldline, i.e. by the motion of one point. This result was derived in a very complete and mathematically elegant way by G. Herglotz in his paper "On Bodies that are to be Designated as 'Rigid' from the Standpoint of the Relativity Principle" [3] , and he discussed the exceptional cases at which a larger freedom of motion is present.
Since the new concept of rigidity consequently doesn't contain the necessary 6 degrees of freedom, it certainly cannot be applied to ordinary material rigid bodies. ”

“ 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, one has to view as desirable the introduction of a rigidity definition, which represents the completest possible analogy to the rigid body of ordinary kinematics; because on one side the rigid body is necessary for the formation of mechanics, and in addition it seems to me that an electron theory without this concept is an essential step backwards. ”

“ My rigidity definition proves to be appropriate for the system of Maxwell's electrodynamics quite in the same way, as the old definition of rigidity for the system of Galilei-Newtonian mechanics. The rigid electron in this sense, represents the dynamically most simple motion of electricity. One can even go so far to assert, that the theory provides clear hints to an atomistic structure of electricity, which is not at all the case in Abraham's theory. ”

“ Definition: The motion of a point system is called rigid, when between the location and the velocity of any point the relation persists that the relative velocity towards the center is equal to the rest rotation:
or extensively:
This definition is covariant with respect to Lorentz transformations. Furthermore it is fully sufficient in connection with conditions (17) and (18) , in order to determine the motion as soon as the motion of center and the path of vector are given for all times in agreement with conditions (16) and (19) . ”

“ Yet, since every accelerated motion can be approximated by such hyperbolic motions as long as their acceleration doesn't vary too suddenly, one achieves in this way an electrodynamic foundation of the fundamental equations of mechanics. This theory fails only for very rapidly changing accelerations; then also radiation resistances arise besides the inertial resistances. It is remarkable, that an electron causes no actual radiation, as great as its acceleration may be, but it drags its field along with it, which was up to now only known for uniformly moving electrons. ”

“ 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. ”

“ The acceleration is thus constant for every world line of hyperbolic motion in terms of their magnitude; here lies the analogy with the uniformly accelerated motion of old mechanics represented by parabolic world lines. Thus it is the simplest accelerated motion, and every motion can approximated by hyperbolic motions. Based on that, we want to find out more precisely the dynamics of hyperbolic motions, above all we try to determine the force exerted by an electrically charged rigid body upon itself. ”

“ We call and the rest vectors belonging to the spacetime vector with respect to velocity , because in that coordinate system in which a point moving with velocity appears to be at rest, the first three components of and are identical with the space vectors and , which compose the transformed vector of second kind . ”

“ Yet, my theory strictly provides the dependency of the rest mass on acceleration for a class of motions, which corresponds – being the principally simplest accelerated motions – to the uniformly accelerated ones of the old mechanics, and which I call "hyperbolic motions", namely the rest mass proves to be constant up to enormous accelerations. Equations of motion in the form of the mechanical fundamental equations [9] which are adapted to the relativity principle, apply to this motion. ”

“ General properties of rigid bodies. Now we can derive those properties from the definition of rigid bodies, which are analogous to the ones of the ordinary rigid body discussed in § 1.
First it is clear, that the definition (28') in equation (1) goes over to ordinary kinematics, as soon as terms of order are neglected.
In order to prove the existence of an instantaneous rotation axis, we ask for the spacetime points , whose velocity is equal to that of the center. From
it follows by multiplication with because of (16)
,
thus
.
Thus it must be for the sought point ”

“ Since the simplicity of the dynamics is therefore not inferior to the simplicity of kinematics of the new rigid body, then one will ascribe to this concept of rigidity the same fundamental importance in the system of the electromagnetic world-picture, as the ordinary rigid body in the system of the mechanical world-picture. ”

“ Definition of the rigid body. The now developed concepts of rest vectors of first and second kind will now be used in the definition of rigidity.
We denote a certain point of the moving body as the center. This shall be represented in a moment by the spacetime point , and shall have the velocity defined by the spacetime vector ”

“ Exactly these foundations of kinematics are the ones to be abandoned, when the electrodynamic relativity principle – as stated by Lorentz, Einstein, Minkowski and others – comes into play. Because here, the connection of space and time into the "world" is different: the independence of the natural laws from the uniform translation of the spatial reference system only then takes place, when also the time parameter experiences a change, which not only tantamounts to a displacement of the origin and the choice of another unit. ”

“ From this behavior of our fundamental solution, it follows exactly as in potential theory, that (when is a function defined for ) the expression
satisfies the inhomogeneous equation (137) (of course, only as long as is different from zero) . If one herein replaces by its value , which corresponded to the co-moving charges according to (134) , then one is led back to (119) . The different signs in different areas of the reference point are resulting from the considerations, that assembled by in the stationary coordinate system, are retarded potentials, not advanced ones. ”

“ The first of these equations, is the equation of motion in an analogous form of one of the Newtonian equations of old mechanics, the second one is the energy equation. The quantity thus corresponds to the kinetic energy of old mechanics. The dependency of mass from velocity is given by Lorentz's formula (157) ; more essential than this one (which is also valid for the ordinary (non-electromagnetic) mass in the new kinematics) , is the dependency of rest mass from the magnitude of acceleration according to formula (153) . ”

“ Phys. 30, 1, 1909↑ For instance, P. Ehrenfest (this journal, 918, 1909) shows in a very simple manner, that a resting body can never be brought into uniform rotation; I already discussed this fact with A. Einstein at the meeting of natural scientists in Salzburg.↑ G. Herglotz, Ann. ”

“ Furthermore we want to state the equations of motion of a rigid electron as follows: The rigid electron is so moving, that the resultants of its own field is oppositely equal to the resultant of the external field. ”

“ The verifications of the assertions stated here can be easily given analytically or geometrically. However, they are also implicitly contained in the considerations of Minkowski [9] concerning the transformation of the electric and magnetic field strength. ”

“ Functions thus are connected with the auxiliary functions by the same Lorentz transformation, which (9. 23, (74) ) transforms the rigid body to rest. Therefore we will call the "rest potentials"; functions of alone don't depend any more from .
From relations (118) we see, that the electron is carrying its field; the rest potentials observed by an observer co-moving with the electron, only depend on the rest coordinates . ”

“ 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.) ↑ This construction applies only to the case of translation.↑ Minkowski, this journal 10, 108, 1909; Jahresb. d. deutsch. Math.-Ver. ”

“ It is most closely connected to this, that measuring rods that maintain their length at uniform translation in the co-moving coordinate system, suffer a contraction in the direction of their velocity when viewed from a stationary system. By that, the concept of the rigid body fails, at least in its form adapted to Newtonian kinematics. ”

“ This force will be capable of sustaining the hyperbolic motion with acceleration , when one chooses:
If the external force field is only slightly variable, so that one can see it as constant in the interior of the electron, then it will generate a motion which only slightly deviates from hyperbolic motion. When we see equations (154) as valid also in this case, we are neglecting radiation.
Thus we obtain for slightly variable, yet arbitrary great accelerations, the following equations of motion and energy:
where and are related to the center of the electron. ”

“ One recognizes, that also the - and -components of the field strengths, do not become infinite at . We won't discuss the field-path in a fixed coordinate system in more detail. Only that much can immediately be seen, that and are vanishing at , thus that the force lines are parallel to the -axis there, and that and are vanishing at , i.e., in the moment where the electron turns back and therefore is momentarily at rest. ”

“ Thus we are faced with a bundle of Lorentz transformations depending on the parameter . The motion, or rather the corresponding bundle of world lines, can now be described as follows.
If one gives a certain value to , then and are given by equations (73) as specific functions of , which represent the world line of point . The components of the velocity world-vector with respect to axes and , are and . By that single curve , all curves of the bundle are co-determined. ”

“ The insight, that one can conclude the existence of a center and of the charge distribution in concentric layers from the behavior of electrons at quasi-stationary translation, is an additional contribution to fortify the atomistic view of electricity. ”

“ The distribution of these reference points can be derived from the figure.
Fig. 6. We want to presuppose mostly in the following, that ; only such points can be interior points of the electron at hyperbolic motion. Thus the positive square root is to be taken for them. When we sometimes also consider points for which , then we have to replace by everywhere.
Thus we have:
Now we calculate the denominator of integral (83) for these values of .
Due to , it becomes:
This is inserted into the integrals ”