Friedrich Kottler

“ The metric nature of the relativity principle suggests the consideration of the metrically preferred ones under the possible spacetime lines; these are the ones which allow a shift into themselves, and whose three curvatures are thus constant. Those as well as a second class of curves, the parallel curves representing the orthogonal trajectories of spaces, have already been studied by Herglotz in his work on the Born rigid body. ”

“ Infinitely thin spacetime-threads as an image of a pointlike charge. pointlike charges in the mathematical sense of this word do not exist; potentials and fields would indeed become infinitely large within them. The employment of pointlike sources in potential theory is only legitimate in the integral representation of the potential, or otherwise as approximation at distances which are not infinitely small. The same also holds in our case. Therefore we have to imagine the pointlike charge as physically very small. However, the individual points of this charge will exert forces upon each other. ”

“ Generalizations to curves, in which only is constant. One could say: “The passage from one point to a consecutive causes an orthogonal (infinitesimal) transformation of the comoving tetrad; for such one, however, the fundamental equations are covariant according to the relativity principle, thus invariant in a suitable reference system, namely the tetrad in ; now, if I define a reference-point fixed in the tetrad , thus
where the are constants satisfying the condition , do the fields in have to be constant in the general case of an arbitrary world line ?” ”

“ The nature of this reference system can only be determined in § 8. Here it is sufficient, that the three family parameter of or remain unchanged during this motion, while and experience the same increase . Thus during the transport of the potentials and fields to these four generalized coordinates, it has to be shown that these potentials and fields at the reference-point do not depend on .
In the following, we will only give the formulas for the potentials and the differentiation formulas stated in § 5, by which the (rather complicated) expressions for the field emerge from the potentials. ”

“ Regarding A it follows, that when for instance only contains spacelike directions, then as the completely perpendicular plane must also contain timelike directions (its infinitely distant line cuts the absolute measure-surface in a real way) . In our representation, must be an imaginary angle. Regarding B (1) it follows, that the rotation angle can be real or imaginary and correspondingly the displacement can be imaginary or real: cases (B) 1 and 2 of § 6. The types of curves of constant curvatures are thus obtained the same way as there. ”

“ The normal-plane , which definitely belongs to the framing- by virtue of the Euclidean metric, will be oriented correctly, if one predefines a direction within it in an arbitrary way, and in this way determines the necessary second direction : One intersects the normal plane with the curvilinear , by which an intersection curve emerges which forms with an even number of intersections ”

“ The previously mentioned two classes of curves will be shown to be the trajectory curves of a radius vector which is fixed in the comoving tetrad of the principal curve. The application to the world lines shows, that the mentioned constancy of the fields with respect to the first class takes place in the comoving tetrad of the light-point or of the reference-point – when reciprocity is present for both of them, so that the shape of the charge whose points possess such world lines can be seen as constant in this reference system. ”

“ Furthermore, a material point of plane world line is connected to a plane or straight trajectory depending on the choice of reference system, the latter being the case when the -axis is directed parallel to the plane of the world line ”

“ If a -fold integral extended over an arbitrary “open” shall only depend on their (closed) boundary-, then to that end it is necessary and sufficient, that
for all formations ; the related integral form of -th order is then called an exact differential and their coefficients allow the representation as , by which the transformation into a -fold integral extended over the closed boundary-, appears to be given. ”

“ Now, it cannot be done without making assumptions on the shape of the infinitely thin spacetime-thread. Since we do not want to make them, we have to confine ourselves with an approximation. So let the world line of the charge be given, for instance as a function of the arc. The pointlike charge cannot act on itself ”

“ Here, the line integral is orbiting around the normal in the negative sense, thus clock-wise if the coordinate system is a right-system; because by § 1 the directions , , are following each other like the coordinate axes, where is the normal (which is directed outwards of the area framed on ) of the framing-. Ordinarily, one prefers a positive sense of circulation and therefore the generalized rotation:
etc. ”

“ It also cannot act upon a later position of itself; because this would require superluminal speed. Outside of the world line there should by now charges though. If we therefore extend the integral
over a piece of which does not contain the world line , it will (being extended over a charge-free space) vanish. This is indeed a known property of the ponderomotive force of the electron theory of Lorentz. ”

“ The mutual influence is therefore constant in a system or ; the electron, whose points describe such world lines as and (thus curves of constant curvatures belonging to one and the same family) , retains its shape in such a continuously varying reference system . ”

“ Our world line is now
Consequently
thus
in which one recognizes Minkowski's proper time multiplied by , which here appears to be connected in the most simple way with the (imaginary) Euclidean arc of the world line. ”

“ When the previous formulas are applied to the special case of the passage from Cartesian orthogonal coordinates of an Euclidean to orthogonal curvilinear coordinates, the known Jacobian integral transformation follows. ”

“ The world lines of constant curvatures correspond to a constant magnitude of the first acceleration in Newtonian mechanics, and furthermore to a common helix or its variations: a circular line or straight line as hodograph of the velocity. ”

“ For the vectors arising in the Wiechert formulas
[2]
one finds:
as well as
and eventually
( has the parameter line !)
Thus we have by § 4 (15) :
I. Potentials in :
[3]
Note, that only appears in the relation , from which one can conclude that the increase of and by each, leaves the potentials unchanged. Furthermore by § 4 (18) :
II. Differentiation formulas:
By the differentiations, the relation is therefore not dissolved, and within that relation will therefore appear in the fields as well. But if is an arbitrary function in this relation, it follows
which was to be proven. ”

“ The supplement in an Euclidean is of course nothing other than the duality of polar correlation in the infinitely distant with respect to the absolute of Euclidean metric, therefore the name dual system is used.
As the reciprocal system [8] , on the other hand, we denote the following covariant or contravariant system, which emerges from the contravariant or covariant system of same order be means of the coefficients of :
with the solution
,
where the are not forming a special system this time. ”

“ These formulas can be derived in the shortest way as follows: The static case, i.e. a straight world line parallel to the -axis, is only a specialization of the coordinate system for the potentials of a pointlike charge in , as one can see e.g. from the solution of Herglotz, because
etc.
thus
where means the “rest”-density, and the assumption of the static case is equivalent with a rotation of the coordinate system in the plane (Lorentz transformation) , so that
.
However, since it is indeed
,
the solution is
. ”

“ This is at first true for every curve of three-fold curvature. If now , , , then the elements of the rotations remain the same from point to point, then the individual infinitesimal orthogonal transformations are added together to a finite orthogonal transformation having the same rotation planes; we write
and we obtain for curve A (simultaneously the trajectory of the most general orthogonal transformation of determinant +1 and fixed origin)
The related family. ”

“ Now, since is vanishing everywhere except along the infinitely thin spacetime-thread of the charge, one can write the integral on the right also as follows:
by introducing a vector of total force , thus
It is known that this result has already been derived by Abraham [3] in a different way. Now, guided by dimensional considerations, there is still a supplement; [4] to the previous expression we can add
,
if
,
i.e. if the motion in the location or is not accelerated.
We remark, that the first solution
gives the appearance of a timelike vector ”

“ On the polar nature of and the axial nature of . The most general direction-changing transformations can namely always be traced back to a transformation multiplied by a change of direction of the coordinate axes. ”

“ Again, the reciprocity of the representation between and only holds for the special case of hyperbolic motion. It can be furthermore seen, that these curves will sustain when is steadily approximated by hyperbolic motions (i.e. the curvature circle) , and if one combines the infinitesimal pieces of the corresponding circles (of the respective family belonging to ) that are going through the locations of , as well as the earlier curves are sustained when the general osculating “hyper helix” (A) is used. ”

“ We are now going to investigate the nature of these curves of constant curvatures more closely, for which we require results from the differential geometry of curves of three-fold curvature in , which should be developed in advance. ”

“ The polar vectors of are contravariants of first order, the axial vectors are special contravariants of second order, from which the so-called supplement (a covariant system of first order) has to be derived.
Supplement. We define the following covariant system -th order as the supplement of a contravariant system of -th order:
Let be a positive permutation of , then for the supplement of the system (which is assumed to be special) , we have
with the remark that in the case of motions changing the direction, i.e. ”

“ Vectors rigidly connected with the tangent. For those, we of course have
and , while the satisfy the differential equations (22) . Thus by (19) and (22)
Characterization of the world lines. 1. Using the real parameter :
(tangent is timelike) ; furthermore
therefore the angle is purely imaginary, and are real; regarding the radii of the three curvatures: is real, and are imaginary.
2. Using the imaginary arc gives:
3. ”

“ Only the integral over 2) can be evaluated by us, if we let go towards after finished evaluation. [2]
Evaluation of the integral over the mantle . With the aid of the Frenet formulas (see § 7, (19) ) one finds the value of
. ”

“ Wright, Invariants of quadratic differential forms. Cambridge Tracts in Mathematics etc. No 9, chapter II.↑ Covariant or contravariant systems of -th order will be marked by lowered or raised indices. The themselves are not contravariants.↑ Christoffel, Crelle's Journal, 70, p. 46 (1869) ”