Arnold Sommerfeld

“ This vector thus agrees for the primed system in all four components with the specific force . Under the single assumption, that the force behaves like a four-vector, must be given for any reference system by the four-vector (Pf) . In relativity theory, Lorentz's force-definition directly follows from the definition of the electric field-strength by the single assumption, that the specific force behaves as a four-vector perpendicular to the worldline of the contact-point. ”

“ The content of the relativity principle can be formulated according to Minkowski: In the physical equations only space-time vectors are allowed to occur, i.e. quantities that have vector-character in the four-times manifold of space and time, the Minkowskian "world". Therefore when passing from one to a new reference system, its components can be transformed according to the scheme of the coordinate transformation for that manifold ("Lorentz transformation") . Instead of absolute space of the older theory, the absolute world takes its place, i.e. ”

“ If we especially choose the primed system as "co-moving", so that the point of contact of force is at rest in this reference frame, then and also , since the force on the worldline of its point of contact is perpendicular. ”

“ It's well known that in space of three dimensions we have to distinguish between two kinds of vectors, vectors of first kind or polar vectors and vectors of second kind or axial vectors, also denoted as rotors or plan-quantities. ”

“ The six components of T can be most conveniently, geometrically interpreted as coefficients of the equation of a surface of second degree.
In the four-dimensional case we accordingly start with a six-vector f, and define a tensor (ff) by all possible scalar products of the special four-vectors (derived by equation (6a) from f) for h, j = x, y, z, l. From that, at first 16 tensor components follow
which, however, are reduced to 10 because of . ”

“ The components of the first ones are thus related to one axis for each component, those of the second ones are related to two axes for each component, thus the first has to be written with one and the latter with two indices. In mechanics, force and velocity belong to the first, torque and angular velocity to the second. E. g. the components of an angular velocity were to be written as ; only after passing to Grassmann's "supplement", i.e. the perpendicular upon the rotating plane, the ordinary notation is justified. ”

“ The scalar products (considered in A) are pure invariants of the Lorentz transformation. The vector products (considered in B) were transformed covariantly as a six- or a four-vector; their components in the new reference frame are thus only connected to those of the old reference frame by the formulas of §2.
For the vector product (Pf) we can calculate this directly from equation (9) . Namely, if we scalar-multiply the four-vectors located at the left and right sides with the four-vector P, then
i.e. ”