Arnold Sommerfeld

“ The scalar divergence. Let be an arbitrarily formed, four-dimensional, and infinitely small section of space [2] in the surrounding of the considered spacetime-point , the element of the (three-dimensional) boundary of , the outer normal to . Let be an arbitrary four-vector, its normal component formed in the sense of equation (7) . From the four-vector a scalar magnitude Div emerges, which we define as follows [3] :
where the integration with respect to is to be extended over the entire boundary of . ”

“ For a far distant point, one can view and as approximately constant for all charge elements and execute the three-times integration, where the total charge of the system emerges. Thus one has for the limiting case:
In consequence of Fig. 6 one can easily convince himself, that direction and magnitude of vector are only expressed by the state of motion at light-point . ”

“ We want to presuppose the motion of as uniform (acceleration evidently comes into play only secondarily, since an influence of has never been demonstrated even on gravitation) . Then, Minkowski's proposal is simplified to the following expression of Newton's law , when we also include the inessential factor in
On the other hand, Poincaré [16] raises himself the task, to form (in most general manner) that Lorentz invariant force law, which reduces itself for small velocities (neglecting and ) to Newton's law in the ordinary sense. ”

“ At first, we want to circumscribe the formulas (34b) , so that only four-vectors related to the light-point occur
is the vector of to , the velocity vector in (see equation (33a) ) and the acceleration vector in . In the previous definition (equation (29b) ) of by the world-line element , since in our cyclic coordinates applies, it evidently becomes , thus according to equations (31)
consequently is follows:
thus has in the light-point the direction of radius and the magnitude . ”

“ To that, however, a remark has to be made concerning the reality relations. As always, we have implicitly presupposed as real the coordinates as well as , and assumed for example, that is only vanishing at one point . ”

“ Fig. 5a represents the relations in the -plane with and imagined as real: the world-lines are circles , the rest-form is projected into the variable radius . Fig. 5b shows, as to how the things are with respect to the imaginary constitution of . If one draws and as real coordinates, and puts , where is a real angel, then the world-lines become equally sided hyperbolas and the rest-form is given by . The asymptotes under 45° are corresponding to a motion with speed of light , which is approximated by hyperbolic motion for . ”

“ Here, Poincaré decidedly operates with four-vectors (the introduction of six-vector, however, was reserved to Minkowski) ; they were represented by its components in an arbitrary xyzl-system. Namely, the following combination between our vectors (left) and Poincaré's components (right) applies:
with the abbreviations
Furthermore, it is to be noted that Poincaré has put , and also has taken for the attracting and attracted mass; to write his formulas in agreement with dimensions, one has to replace his velocity components by and his force components by . ”

“ For that, we would have (in the direction normal to ) to separate an infinitely small, linear area , and the difference as replacement for the degenerated integration over the boundary points of , and eventually to form as component of the emerging vector of third kind:
Instead, we will dually reverse our process by starting with a vector of zero kind , i.e. ”