Translation:On the spacetime lines of a Minkowski world/Introduction

by Friedrich Kottler

The theory of the pointlike charge is dealt with in the present work on the basis of Minkowski's views.

The introductory paragraphs treat the integral forms in accordance with Goursat and the theory of invariants of a quadratic differential form in accordance with Ricci and Levi-Civita. It will be shown with their aid, how to construct a vector analysis in a space of arbitrary many dimensions and arbitrary metric in arbitrary generalized coordinates. They will be applied to $S_{3}$ and $S_{4}$, from which the generalized form of Maxwell's equations follow.

Consequently we will go over to the pointlike charge. On that occasion, the Schwarzschild formulas for the fields will be derived by differentiation from the Wiechert formulas for the potential following the method of Abraham, however, the respective computation will be essentially simplified by stating new formulas for the displacement of the light-point along with the reference-point, of which the Doppler effect will be proven to be a special case. Consequently the Wiechert-Schwarzschild formulas will be stated in generalized coordinates in accordance with the method given in the introduction.

The next paragraph brings dynamical considerations; since there can be no strictly pointlike charge and since the computation of forces which a non-pointlike charge exerts on itself is not entirely doable without assumptions about their shape, we will only determine the part which is computational without these assumptions, which is denoted by Abraham as the reaction force of radiation, by evaluation of a triple integral and the following boundary transition to infinite large distances. In agreement with this, the motions of the charge which is to treated as pointlike at a sufficient distance, will be simply seen as given.

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 $\infty ^{1}$ spaces, have already been studied by Herglotz in his work on the Born rigid body. Both classes will be considered here from another viewpoint: a principal curve, the world line of the charge, will be given here as well, but the correlation of the $\infty ^{3}$ curves of the group is not given by spacelike vectors as with Born's body - with which, by the way, our considerations have nothing to do -, but by minimal vectors. In this way, the motion of the reference-point is predefined, and for the first class (i.e. the curves of constant curvatures), the constancy of the fields in a generalized axis-cross follows, a property that was hitherto only known for a species of this class (i.e. Born's hyperbolic motion).

The last paragraphs then bring the representation of differential geometry of $S_{4}$ in accordance with the works of Brunel and Landsberg as well as in accordance with Scheffers, in which the orthogonal infinitesimal transformation on which the Frenet formulas are based is particularly referred to.

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. The second class results in the disappearance of the magnetic components of the fields in a system comoving with the reference-point.

Also the consequent use of the Euclidean metric is formally to be mentioned, which is made possible by the mapping $x^{(4)}=ict$. The disadvantages of such a representation in terms of questions of reality appear to be outweighed by their vividness and the possibility of following the known results of $S_{3}$.

Denotations

§ 1, 2

$x^{(1)}x^{(2)}x^{(3)}x^{(4)}$ coordinates in $S_{4}$

$A_{\alpha _{1}\alpha _{2}\dots \alpha _{p}}$ covariant system of $p$-th order.

$A^{\left(\alpha _{1}\alpha _{2}\dots \alpha _{p}\right)}$ contravariant system of $p$-th order.

$dS_{p}$ element of $S_{p}\ \left(dS_{1}=ds\right)$, $c_{\alpha \beta }$ coefficients of $ds^{2}$.

$N_{\alpha }$, or $N_{\alpha \beta }$ etc. Directions of the normal, or normal plane ($\sum N_{\alpha }^{2}=1$, $\sum N_{\alpha }^{2}=1,\sum _{(\alpha \beta )}N_{\alpha \beta }^{2}=1$ etc. in Cartesian orthogonal coordinates).

§ 3.

$F_{23}={\mathfrak {H}}_{x},\ F_{31}={\mathfrak {H}}_{y},\ F_{12}={\mathfrak {H}}_{z},\ F_{14}=-i{\mathfrak {E}}_{x},\ F_{24}=-i{\mathfrak {E}}_{y},\ F_{34}=-i{\mathfrak {E}}_{z}$;

$\Phi _{1}={\mathfrak {A}}_{x},\ \Phi _{2}={\mathfrak {A}}_{y},\ \Phi _{3}={\mathfrak {A}}_{z},\ \Phi _{4}=i\varphi$;

${\begin{aligned}P^{(1)}&=\rho {\frac {{\mathfrak {v}}_{x}}{c}}=i\rho _{0}V^{(1)},&P^{(2)}&=\rho {\frac {{\mathfrak {v}}_{y}}{c}}=i\rho _{0}V^{(2)},\\P^{(3)}&=\rho {\frac {{\mathfrak {v}}_{z}}{c}}=i\rho _{0}V^{(3)},&P^{(4)}&=i\rho =i\rho _{0}V^{(4)},\end{aligned}}$

where

${\begin{aligned}V^{(1)}&={\frac {{\mathfrak {v}}_{z}}{ic}}{\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},&V^{(2)}&={\frac {{\mathfrak {v}}_{y}}{ic}}{\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},\\V^{(3)}&={\frac {{\mathfrak {v}}_{x}}{ic}}{\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},&V^{(4)}&={\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},\end{aligned}}$

with

$\sum _{\alpha =1}^{4}\left(V^{(\alpha )}\right)^{2}=1$

and $\rho _{0}=\rho {\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}$ is the rest density of the charge.

§ 4.

In particular, $x$ means the light-point $L$, $y$ the reference-point $A$, $R$ the radius vector ${\overrightarrow {AL}}\ x-y$, $V$ the previous vector for $x$, $W$ the analogue one for $y$ (velocity of the reference-point: ${\mathfrak {w}}$). It is

$R^{2}=\Sigma (x-y)^{2}=0$,

if $L$ is the light-point for $A$.

$e_{x},\ e_{y}$ charges in $x$ or $y$;

$s_{x}$ or $s_{y}$ is the arc of the world lines in $x$ or $y$. It is

${\frac {dx^{(\alpha )}}{ds_{x}}}=V^{(\alpha )}$ or ${\frac {dy^{(\alpha )}}{ds_{y}}}=W^{(\alpha )}$.

Proper time $\tau _{x}$ or $\tau _{y}$; it is $s_{x}=ic\tau _{x}$, $s_{y}=ic\tau _{y}$. $(AB)$ scalar product, $[AB]^{(\alpha \beta )}=A^{(\alpha )}B^{(\beta )}-A^{(\beta )}B^{(\alpha )}$ vector of 2. kind from $A$ and $B$.

§ 5.

$F_{1}={\mathfrak {F}}_{x},\ F_{2}={\mathfrak {F}}_{y},\ F_{3}={\mathfrak {F}}_{z},\ F_{4}={\frac {i}{c}}({\mathfrak {wF}})$ the ponderomotive force in terms of unit volume, its mechanical power $F_{4}$ respectively.

$S_{\alpha \beta }$ tensions:

${\begin{array}{rl}S_{11}&={\frac {1}{2}}\left({\mathfrak {E}}_{x}^{2}-{\mathfrak {E}}_{y}^{2}-{\mathfrak {E}}_{z}^{2}\right)+{\frac {1}{2}}\left({\mathfrak {H}}_{x}^{2}-{\mathfrak {H}}_{y}^{2}-{\mathfrak {H}}_{z}^{2}\right),\\S_{12}&={\mathfrak {E}}_{x}{\mathfrak {E}}_{y}+{\mathfrak {H}}_{x}{\mathfrak {H}}_{y}\ {\text{etc.,}}\\S_{14}&=-i\left({\mathfrak {E}}_{y}{\mathfrak {H}}_{z}-{\mathfrak {E}}_{z}{\mathfrak {H}}_{y}\right)=-{\frac {i}{c}}{\mathfrak {S}}_{x}\ {\text{etc.,}}\\S_{44}&={\frac {1}{2}}\left({\mathfrak {E}}^{2}+{\mathfrak {H}}^{2}\right)=w,\end{array}}$

$icV={\frac {dx}{d\tau }}=({\mathfrak {v}},\ ic){\frac {1}{\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}},$

${\begin{array}{rl}-c^{2}{\frac {dV}{ds}}&={\frac {d^{2}x}{d\tau ^{2}}}=({\dot {\mathfrak {v}}},0){\frac {1}{1-{\mathfrak {v}}^{2}/c^{2}}}+({\mathfrak {v}},ic){\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}/c^{2}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}=\\&=({\dot {\mathfrak {v}}}_{\bot },0){\frac {1}{1-{\mathfrak {v}}^{2}/c^{2}}}+({\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}},0){\frac {1}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}+\left(0,{\frac {i}{c}}{\frac {{\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}}{\mathfrak {v}}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}\right),\end{array}}$

where ${\dot {\mathfrak {v}}}={\mathfrak {{\dot {\mathfrak {v}}}_{\Vert }}}+{\dot {\mathfrak {v}}}_{\bot }$ was split into two components parallel and normal to ${\mathfrak {v}}$,

${\frac {c^{4}}{R_{1}^{2}}}=\left({\frac {dV}{ds}}\right)^{2}c^{4}=\sum \left({\frac {d^{2}x}{d\tau ^{2}}}\right)^{2}={\frac {{\dot {\mathfrak {v}}}_{\bot }^{2}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{2}}}+{\frac {{\dot {\mathfrak {v}}}_{\Vert }^{2}}{\left(1-{\mathfrak {v}}^{2}/c^{2}\right)^{3}}}$,

${\begin{array}{rl}-ic^{3}{\frac {d^{2}V}{ds^{2}}}&={\frac {d^{3}x}{d\tau ^{3}}}=({\ddot {\mathfrak {v}}},0){\frac {1}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{3}}}+\\&+({\ddot {\mathfrak {v}}},0){\frac {3{\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}}{c^{2}}}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+({\mathfrak {v}},ic)\left\{{\frac {{\mathfrak {\dot {v}}}^{2}/c^{2}+{\frac {{\mathfrak {v}}{\mathfrak {\ddot {v}}}}{c^{2}}}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+{\frac {4\left({\frac {{\mathfrak {v}}{\mathfrak {\dot {v}}}}{c^{2}}}\right)^{2}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{7}}}\right\}.\end{array}}$

${\begin{array}{rl}-ic^{3}\left({\frac {d^{2}V}{ds^{2}}}+{\frac {1}{R_{1}^{2}}}V\right)&={\frac {d^{3}x}{d\tau ^{3}}}-{\frac {c^{4}}{R_{1}^{2}}}\cdot {\frac {i}{c}}V=\\&={\frac {d^{3}x}{d\tau ^{3}}}-{\frac {c^{4}}{R_{1}^{2}}}\cdot {\frac {1}{c^{2}}}\cdot {\frac {dx}{d\tau }}=\\&=\left({\ddot {\mathfrak {v}}}_{\bot },0\right){\frac {1}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{3}}}+\left({\ddot {\mathfrak {v}}}_{\Vert },0\right){\frac {1}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+\left(0,\ {\frac {i}{c}}{\frac {\left({\mathfrak {\ddot {v}}}_{\Vert }v\right)}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}\right)\\&+\left({\dot {\mathfrak {v}}}_{\bot },0\right){\frac {3{\mathfrak {v}}{\mathfrak {\dot {v}}}/c^{2}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{5}}}+\left({\dot {\mathfrak {v}}}_{\Vert },0\right){\frac {3{\mathfrak {v}}{\mathfrak {\dot {v}}}/c^{2}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{7}}}+\left(0,\ {\frac {i}{c}}{\frac {\left({\mathfrak {\dot {v}}}_{\Vert }v\right)3{\mathfrak {v}}{\mathfrak {\dot {v}}}/c^{2}}{\left({\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right)^{7}}}\right)\end{array}}$

where $({\mathfrak {A}},\ B)$ denotes a vector of first kind, whose three spatial components are summarized to ${\mathfrak {A}}$; $B$ is its timelike component.

§ 7, 8

$c_{k}^{(\alpha )}\ \alpha =1,2,3,4$ direction cosines of the $k$-th axis of the comoving tetrad ($c_{1}$ tangent, $c_{2}$ principal normal, $c_{3}$ binormal; $c_{4}$ trinormal).

$c_{1,x}^{(\alpha )}=V^{(\alpha )},\quad {\frac {c_{2,x}^{(\alpha )}}{\mathrm {R} _{1,x}}}={\frac {dV^{(\alpha )}}{ds}}={\frac {d^{2}x}{ds^{2}}}$

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