Arnold Sommerfeld

“ By that also these well-known transformation formulas of the field appear to be the immediate consequence of our geometrical interpretation of the six-vector and its component formation derived from the projection of four-dimensional lines. About Maxwell's equation, from which these formulas were derived by Lorentz and Einstein, we didn't speak at all (except in so far, as we have denoted f as six-vector by referring to those equations) . Generally, according to Einstein and Minkowski the relativity principle is to be seen as a superior principle above electrodynamics. ”

“ The deviation of both paths in unit time (OC in the first, OD in the second case) is small of second order, if are small of first order. The reason for this is, that in the first case the velocity from the moving system (ruler) is differently estimated than it is estimated from the stationary system (drawing plane) , because of the motion dependence of time; and correspondingly in the second case this applies to velocity . Exactly the same fact we have expressed above, that by execution of one space-time rotation in space xyl, the plane of the other space-time rotation will be displaced. ”

“ In electrodynamics, which was founded by Einstein on the principle of relativity, superluminal velocities are impossible under any circumstances, both as convection velocity of a charge or as propagation velocity of an electrodynamic disturbance. ”

“ The theorem of Stokes. If means an closed one-dimensional convolution (arbitrarily located in the world) , a two-times extended surface limited by , a four-vector, then Stokes' theorem is given as a direct consequence of definition equation (21) in the location and order of directions in the form
One can remark, that one cannot speak (even with respect to the ordinary three-dimensional formulation of Stokes' theorem) , as it usually happens, of the normal component, but of the tangential component of rotation, since rotation is also at that place a vector of second kind. ”

“ The replaceability of the vector of third kind by its supplement is reaching even further, like the relation of the supplement with the vector of second kind in the case of three dimensions, in so far as in this case the supplement (due to the even dimension number) correctly reproduces the behavior of the vector of third kind, not only regarding coordinate transformations of determinant +1, but also regarding such of determinant -1 (mirroring, inversion) . As the summarizing denotation for the vector of first and third kind, the name "four-vector" is recommended. ”

“ The single reason for this short paper was to show, that the deep space-time interpretation of Minkowski not only simplifies the general construction of the theory of relativity, but it is also reliable as a guidance concerning specific questions. ”

“ 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 . ”

“ 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. ”

“ We want to speak about signal velocity - following Einstein's terminology, and to avoid the word group velocity that is not easily related to individual processes, but rather to natural processes that are statistically given - and understand it as the velocity at which a suddenly starting signal, i.e. a sequence of electrodynamic oscillations of a specific wavelength, is perceived by an observer inside the dispersive medium. According to Einstein, it would be an objection against the relativity theory, when this signal velocity may be under any circumstances > c. ”

“ Based on the world-line of a certain charge element (see Fig. 4) we denote the point of the world-line, which is cut by a cone constructed at point , with Minkowski as light-point of . Its coordinates are unequivocally determined when the charge element never moves at superluminal velocity, and the fourth coordinate can be determined, as previously shown, by equation . As it is known, it says that a light signal emanating from world-point , reaches world-point (i.e. it reaches the space-point at time ) . ”

“ 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. ”

“ 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 . ”

“ 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. ”

“ Now we shall see that it is exactly equal to c, if we think of the observer as equipped with an ideal, i.e. infinitely sensitive detector, no matter whether the medium is dispersive in a normal or anomalous way, whether it is isotropic or anisotropic, whether it contains conduction electrons or not. ”

“ 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. ”

“ The sum of the angles in a spherical triangle is greater then two right angles, the sum of the angle of the sides in a right angled spherical triangle is thus greater then a right angle, as it is shown in Fig. ”

“ 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 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. ”

“ 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 . ”

“ 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 . ”

“ 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 . ”

“ 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. ”

“ 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 . ”

“ 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. ”