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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ We can say, that the localization of energy and momentum in the aether in the old sense, in not in agreement with the relativity principle. However, we not at all need the energy and momentum theorems (in the considered form) for the formation of the equations of motion. Rather the definition given on p. 46 completely suffices. The energy theorem is then added to the three equations of motion as a statement depending on them, that the sum of the works of the resultants of the external and the interior field are always equal to zero. ”

“ 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) . ”

“ A bundle of such hyperbolas represents a motion, at which the rigid body comes from infinity and is approaching the origin, then it is turning back and is moving away into infinity again, where its velocity decreases from the speed of light to zero first, and after the turning back increases again to . This motion, to some extent analogous to the uniformly accelerated motion of old kinematics, we want to call hyperbolic motion shortly. ”

“ 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. ”

“ 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. ”

“ 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 . ”

“ 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. ”

“ At given dimensions of the body, the curvature of world lines thus cannot exceed a certain limit, and vice versa. From that it follows, that a rigid body is necessarily extended into all directions, and has to be the smaller, the bigger the accelerations are that it should experience. Here, we have the first hint at the fundamental importance of atomistics in the new dynamics. If the rigid body carries a substance of rest density , then it is independent of , and it is a function of , which we denote by ”

“ 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. ”

“ When are the initial values of , then matrix is equal to unit matrix 1 for , thus (12) reads:
It is now an elementary theorem of infinitesimal geometry [13] , that when these conditions are satisfied everywhere, it is about the motion of a rigid body.
This infinitesimal rigidity condition (13) can now easily be transfered to the kinematics of the relativity principle.
§ 2. The differential conditions of rigidity. In the following, only such quantities shall have a meaning, which are invariant with respect to the Lorentz transformations (7) , (8) . ”

“ 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 ”

“ On the other hand, it is not immediately valid for the magnetic deflection; though it can be easily seen, that (for quasi-stationary motion) also the magnetic deflection is reproduced by the theory.
(Received June 13, 1909.)
Correction to the paper: The theory of the rigid electron in the kinematics of the principle of relativity.
by Max Born.
In § 13, p. 54, of the paper published in Ann. d. Phys. 30. p. 1. 1909, is to be replaced by , in formula (161) as well as in some lines below. Consequently, formula (162) must read ”