Max Abraham

“ We imagine a framework rigidly connected with the electron. In all points of the field, at which the density of the electromagnetic momentum increases with time, a corresponding force acts at the framework. ”

“ However, he thought it necessary to add an additional force to the ponderomotive electromagnetic force, which is incompatible [4] with my system of electrodynamics. In § 4 I will write the equations of motion so that they satisfy the principle of relativity, without introducing the additional force of Minkowski. However, it must be admitted that the "rest density" of mass is not constant, yet it increases every time, when an electric current generates heat (in Joule) in matter; this hypothesis was already before stated by A. Einstein and M. Planck in relation to the principle of relativity. ”

“ H. A. Lorentz announces it with all restraint; he doesn't go so far as to state it as probable. Indeed, most serious objections can be raised against this hypothesis.
If such an electron is accelerated, its oblateness becomes increased; thus work must be performed against the electric forces. While (for the undeformable electron) the increase of energy is equal to the work performed by the external electric forces, this is not taken place here any more; the energy increase when the velocity increases, is greater as the work of the external forces. ”

“ The system of "relative stresses"
is so defined, that (6) and (8) lead to the equal values of the ponderomotive force.
From (6a) and (10) , the relations follow
which can be written vectorially
is the unit volume related to the torque of relative stresses; it vanishes in ordinary mechanics, since the momentum vector is in agreement with the velocity vector in terms of direction. In electromagnetic mechanics it is not to be neglected in general, but it will be compensated (when related to a fixed point of moment) by that torque which stems from the entrained momentum. ”

“ Only for a special class of motions, for "preferred motions", it was possible to deduce the Lagrangian equations in the form known from analytical mechanics; to such motions, also that formulation of Lagrangian mechanics applies which is called "Hamiltonian principle". If the applicability of analytical mechanics is restricted in some way, it again experiences an essential extension in other ways. Because ordinary mechanics of material bodies is related to very small velocities, while the dynamics of the electron is valid close up to the speed of light. ”

“ Now we imagine the motion as altered by an external force , for instance by a homogeneous electric or magnetic field. Angular forces and rotation are excluded by us; we satisfy the equations of motion, when we accordingly alter the momentum vector of force , and let the motion be parallel to the respective direction of the momentum vector. ”

“ The relations are quite similar at this place; at very rapid accelerations of the electron it becomes inadmissible to calculate with electromagnetic mass, especially then, when the speed of light is reached or even exceeded. Though we still may apply the theory of quasi-stationary motion when the velocity remains behind the speed of light only by a few kilometers per second, and when the the acceleration assumes the values attainable in the strongest fields. ”

“ In view of the perplexity, which is caused by the increasing number of rival theories, it appeared to me as desirable, to possess a system of electrodynamics of moving bodies on a Maxwellian basis, which at first is free from special assumptions of the individual theories. ”

“ The negative result of experiments, whose sensitivity is sufficient to discover an influence of second order, provides considerable difficulties for the electron theory. In two papers [4] , Lorentz sought to overcome these difficulties. In the second of the cited papers, he states a system of hypotheses, which are sufficient to give account for all negative experimental results:
H. Due to Earth's motion, the bodies experience a certain contraction parallel to the direction of motion. ”

“ In the electrodynamics of moving bodies, the equations of the momentum and energy apply: [8]
In order to give to these four equations a more symmetrical from, we put:
Then they become: Now, in Minkowski's theory, the system of four variables
,
where the first three are the components of which determines the ponderomotive force per unity of space, must form a . The 16 variables , from which that system is derived using equations (16) , must be transformed in such a way that this condition is satisfied. ”

“ That the electron can move force-free and stationary in any direction, is caused by the symmetry that we attributed to it. For instance, if the electron would be an ellipsoid uniformly charged with electricity, then force-free stationary motion would only be thinkable parallel to one of the 3 major axes, since only here, the momentum vector is directed parallel ”

“ However, for our system of electrodynamics of moving bodies, the vectors and are of lower importance than the vectors . When we, under elimination of and , directly connect those four vectors with each other, then the relatedness of the theories of Minkowski and Lorentz will become clear. ”

“ At pure rotation, nothing remarkable occurs. At slow translatory motion, one only commits an error of order , when one sets . Angular forces occur at the passage of cathode rays through inhomogeneous fields. Though the angular force is very small, since the electron's radius is so small; thus according to my estimation, the energy in the angular motion emerging in the fields of the greatest attainable inhomogeneity, only amounts the part of the translatory energy. At Becquerel rays, the consideration of rotation is far more difficult ”

“ The necessity to introduce non-electromagnetic forces shows, that the hypothesis of the deformable Heaviside ellipsoid, although it is mathematically simpler in a certain sense, is physically far more complicated though, as the hypothesis of the rigid spherical electron. ”

“ The first two terms are contributions of the aether in the sense of Lorentz's theory, the two latter ones are to be seen as contributions of polarized matter to the electromagnetic momentum density. We begin to calculate the energy current. ”

“ Authors, who doesn't provide a clear representation of their fundamental hypotheses and a thorough development of the consequences derived from them, cannot demand to be noticed by serious consideration. ”

“ This partially stems from the fact, that E. Cohn's approach for the relative stresses is not entirely identical with (Va) ; he namely sets at this place instead of , probably with the intention to remove the torque of the relative stresses. The difference in the value of the force contribution stemming from the relative stresses, which is caused by that, can immediately be found
Only then we have seen the vanishing of as necessary, when – as in the theory of Hertz – no electromagnetic momentum comes into play. ”

“ There, denotes the boundary of the field, which can be determined by other bodies or by an (only imagined) surface; is the inner (scalar) product of vectors and , and the first member of the left-hand side is the "power of the inner forces". The second member contains the normal component of "Poyntings radiation vector".
and gives the radiation sent through the boundary by the electron. Power of the inner forces and radiation appear at the cost of the scalar quantity , which we will call "electromagnetic energy", and which occupies the field with the density ”

“ A statement is supplemented to them concerning the force (at a given field) acting upon a volume element filled by electricity:
D. The electromagnetic force is additively composed of forces, acting in the electric field upon resting electrons, and in the magnetic field upon moving electrons. ”

“ So the following relation holds:
a formula already present in the first paper. [10]
Equation (24b) therefore can be written
namely
Evidently from (26b) and (18b) , the energy current postulated by the Poynting theorem follows in the case of rest. ”

“ It is the only one that takes into account (and by that it is preferred over the other proposed extensions) those physical properties of space, that find their mathematical expression in the Maxwell-Hertz differential equation. ”

“ If gives the momentum exerted by the electromagnetic field, then the energy converted into non-electromagnetic forms, is given by the sum of Joule-heat and work of the ponderomotive force. For the Joule-heat, according to main equation (III) and (59d) , it is given
while the performance of work of force , is given from (60)
If one now considers, that the calculation of the ponderomotive force is based on the presuppositions ”

“ The comparison with (20) gives the important relation
If one inserts Minkowski's connecting equations between the electromagnetic vectors into our system, then the momentum density in the moving body becomes equal to the energy current divided by . ”

“ One can – because according to the things said, the component of coinciding with the direction of vector , is equal to zero – also write
By that, condition (18a) is shown to be valid, and at the same time the gap in the previous derivation of the value of is closed. ”