Translation:On the spacetime lines of a Minkowski world/Paragraph 3

by Friedrich Kottler

The two Maxwell quadruples as coefficients of two mutually dual integral forms

The considerations shall now be based on Minkowski's $S_{4}$, the “world”, $S_{4}(xyz\,ict)$:

$x^{(1)}\equiv x,\quad x^{(2)}\equiv y,\quad x^{(3)}\equiv z,\quad x^{(4)}\equiv ict,\quad (i={\sqrt {-1}})$.

With

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

the Maxwell quadrupels become in orthogonal Cartesian coordinates

$\sum _{h=1}^{4}{\frac {\partial F_{gh}}{\partial x^{(h)}}}=\mathrm {P} ^{(g)},\quad g=1,2,3,4,$

$\sum _{h=1}^{4}{\frac {\partial F_{gh}^{\ast }}{\partial x^{(h)}}}=0,\quad g=1,2,3,4,$

where

$\mathrm {P} ^{(1)}=\rho {\frac {{\mathfrak {v}}_{x}}{c}},\quad \mathrm {P} ^{(2)}=\rho {\frac {{\mathfrak {v}}_{y}}{c}},\quad \mathrm {P} ^{(3)}=\rho {\frac {{\mathfrak {v}}_{z}}{c}},\quad \mathrm {P} ^{(4)}=i\rho$

and

${\begin{aligned}F_{23}^{*}&=-i{\mathfrak {E}}_{x},&F_{31}^{*}&=-i{\mathfrak {E}}_{y},&F_{12}^{*}&=-i{\mathfrak {E}}_{z},\\F_{14}^{*}&={\mathfrak {H}}_{x},&F_{24}^{*}&={\mathfrak {H}}_{y},&F_{34}^{*}&={\mathfrak {H}}_{z}.\end{aligned}}$

If we treat $F$ as covariant, then this is evidently somewhat arbitrary in terms of Cartesian orthogonal coordinates, since covariants and contravariantes coincide here. If we would require (following Abraham) that the $F$ transform like the products of the corresponding components of the radius vector, then we would have to postulate the $F$ as contravariants. By passage to the reciprocal system we can, however, represent them as convariants. For the sake of being easier related to the integral forms, we stick to the covariant description.

Now, these two quadruple are the coefficients of two mutually dual integral forms.^[1] Thus we write, still in Cartesian coordinates,

${\begin{aligned}\mathrm {F} _{234}^{*}&={\mathfrak {Div}}^{(1)}\left(F_{12}\dots F_{34}\right)=\mathrm {P} ^{(1)},&\mathrm {F} _{234}&={\mathfrak {Div}}^{(1)}\left(F_{12}^{\ast }\dots F_{34}^{\ast }\right)=0,\\-\mathrm {F} _{134}^{*}&={\mathfrak {Div}}^{(2)}\left(F_{12}\dots F_{34}\right)=\mathrm {P} ^{(2)},&\mathrm {F} _{134}&={\mathfrak {Div}}^{(2)}\left(F_{12}^{\ast }\dots F_{34}^{\ast }\right)=0,\\&{\text{etc.}}&&{\text{etc.}}\end{aligned}}$

Of course, $\mathrm {P}$ satisfies the continuity equations, because (see § 1):

$-{\frac {\partial }{\partial x^{(1)}}}\mathrm {F} _{234}^{*}+{\frac {\partial }{\partial x^{(2)}}}\mathrm {F} _{134}^{*}-{\frac {\partial }{\partial x^{(3)}}}\mathrm {F} _{124}^{*}+{\frac {\partial }{\partial x^{(4)}}}\mathrm {F} _{123}^{*}\equiv 0$

The disappearance of the $\mathrm {F} _{234}$ etc. gives us the reason for the introduction of the potentials, because it is consequently $F_{\alpha \beta }=\mathrm {F} _{\beta \alpha }$; we write

$F_{\alpha \beta }={\frac {\partial }{\partial x^{(\alpha )}}}\Phi _{\beta }-{\frac {\partial }{\partial x^{(\beta )}}}\Phi _{\alpha },$ (thus $\Phi _{\alpha }=-F_{\alpha }),$

where the potential vector is treated as covariant as well. We have

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

where ${\mathfrak {A}}$ is the vector potential, $\varphi$ the scalar potential. By substitution into the first quadruple it follows

${\mathfrak {Div}}^{(1)}\left(F_{12}\dots F_{34}\right)={\frac {\partial }{\partial x^{(1)}}}\left[\mathrm {Div} \left(\Phi _{1}\Phi _{2}\Phi _{3}\Phi _{4}\right)\right]-\square \Phi _{1}=\mathrm {P} ^{(1)}$ etc.

and here it is set

$\mathrm {Div} \left(\Phi _{1}\Phi _{2}\Phi _{3}\Phi _{4}\right)\equiv 0$

since the fields $F_{\alpha \beta }$ remain correct in consequence of the representation as $\mathrm {F} _{\alpha \beta }$, if an additional vector $\mathrm {F} _{\alpha }$ is added to $F_{\alpha }$; the arbitrariness in the values of the $\Phi _{\alpha }$ is resolved by the previous condition. It follows

$\square \Phi _{\alpha }=-\mathrm {P} ^{(\alpha )},\quad \alpha =1,2,3,4$

with the known solution of Herglotz^[2]

$4\pi ^{2}\Phi _{\alpha }(y)=\int dx^{(1)}dx{}^{(2)}dx^{(3)}dx^{(4)}{\frac {\mathrm {P} ^{(\alpha )}(x)}{R^{2}}}$

where

$R^{(\alpha )}=x^{(\alpha )}-y^{(\alpha )},\quad \alpha =1,2,3,4$

is the radius vector of the reference-point $y$ with respect to point $x$. It is known, as to how the integration in the complex $x^{(4)}$ plane is carried out by means of a loop, which is clock-wise circulating around the negative imaginary semi-axis. To a fixed value system $x^{(1)}x{}^{(2)}x^{(3)}$ belongs a pole $x^{(4)}$, for which we have

$R^{2}=\sum _{\alpha =1}^{4}\left(x^{(\alpha )}-y^{(\alpha )}\right)^{2}=0,$,

upon which the loop is to be drawn together; $x$ is then a light-point for $y$, that is, $R$ is a minimal vector and $x^{(4)}-y^{(4)}$ is negative imaginary or $x$ lies on the pre-cone of $y$ (Minkowski).

The Maxwell quadruple in generalized coordinates.

By means of the invariance of the integral forms it follows, if one uses

$ds^{2}={\underset {\alpha ,\beta }{\sum }}c_{\alpha \beta }dx^{(\alpha )}dx{}^{(\beta )}$

with $x^{(4)}$ as timelike coordinate, and if

$F_{\alpha _{1}\alpha _{2}}^{\ast }={\frac {1}{\sqrt {c}}}{\underset {\beta _{1},\beta _{2}}{\sum }}c_{\alpha _{1}\beta _{1}}c_{\alpha _{2}\beta _{2}}F_{\alpha _{3}\beta _{3}}$

is now defined as:

${\frac {1}{\sqrt {c}}}\mathrm {F} _{234}^{*}={\mathfrak {Div}}^{(1)}\left(F_{12}\dots F_{34}\right)={\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}F_{\alpha \beta /\gamma }={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta ,\gamma }{\sum }}c^{(1\beta )}c^{(\alpha \gamma )}F_{\beta \gamma }\right)=\mathrm {P} ^{(1)}$ etc.

${\frac {1}{\sqrt {c}}}\mathrm {F} _{234}={\mathfrak {Div}}^{(1)}\left(F_{12}^{\ast }\dots F_{34}^{\ast }\right)={\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}F_{\alpha \beta /\gamma }^{\ast }={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta ,\gamma }{\sum }}c^{(1\beta )}c^{(\alpha \gamma )}F_{\beta \gamma }^{\ast }\right)=0$ etc.

for which it follows with the aid of the potentials:

$F_{\alpha \beta }=-\mathrm {F} _{\alpha \beta }={\frac {\partial \Phi _{\beta }}{\partial x^{(\alpha )}}}-{\frac {\partial \Phi _{\alpha }}{\partial x^{(\beta )}}}=\Phi _{\beta /\alpha }-\Phi _{\alpha /\beta }$.

For the $\Phi _{\alpha }$ the equations hold (see § 2, end):

$\sum c^{(1\alpha )}\sum c^{(\beta \gamma )}\Phi _{\beta /\gamma \alpha }-\sum c^{(1\alpha )}\sum c^{(\beta \gamma )}\Phi _{\alpha /\beta \gamma }=\mathrm {P} ^{(1)}$ etc.

or, if as before the condition

${\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}\Phi _{\beta /\gamma }=0$

is imposed on them:

$\sum c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}\Phi _{\alpha /\beta \gamma }=-\mathrm {P} ^{(1)}$ etc.

or by the introduction of the vector $\mathrm {P} _{\alpha }$ reciprocal to $\mathrm {P} ^{(\alpha )}$:

$\mathrm {P} ^{(1)}={\underset {\alpha }{\sum }}c^{(1\alpha )}\mathrm {P} _{\alpha }$:

${\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}\Phi _{\alpha \beta /\gamma }=-\mathrm {P} _{\alpha },\quad \alpha =1,2,3,4$

Maybe it is not superfluous to mention, that for instance

$\square B={\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}B_{\alpha /\beta \gamma }$

can be transformed into the form (see § 2)

${\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta }{\sum }}c^{(\alpha \beta )}{\frac {\partial B}{\partial x^{(\beta )}}}\right)$,

though this cannot be done with our previous four equations, therefore it will be avoided to denote them by $\square \Phi _{\alpha }$.^[3] On the other hand, the conditional equation between the $\Phi _{\alpha }$ has the form:

$\sum c^{(\beta \gamma )}\Phi _{\beta /\gamma }=\mathrm {Div} \left(\Phi _{1}\Phi _{2}\Phi _{3}\Phi _{4}\right)={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta }{\sum }}c^{(\alpha \beta )}\Phi _{\beta }\right)$

By that, we have derived everything that is needed for the purpose of our investigation; before we proceed, however, a remark may have its place here:

On the polar nature of ${\mathfrak {E}}$ and the axial nature of ${\mathfrak {H}}$.

The most general direction-changing transformations ${\frac {\partial (x)}{\partial ({\bar {x}})}}<0$ can namely always be traced back to a transformation ${\frac {\partial (x)}{\partial ({\bar {x}})}}>0$ multiplied by a change of direction of the coordinate axes. Now, because of the irreversibility of time (permutation of pre- and after-cone would indeed be a permutation of cause and effect), in Minkowski's $S_{4}$ we always have:

${\frac {\partial x^{(4)}}{\partial {\bar {x}}^{(4)}}}>0$,

therefore we can confine ourselves to the change of direction

${\bar {x}}^{(1)}=-x^{(1)},\quad {\bar {x}}^{(2)}=-x^{(2)},\quad {\bar {x}}^{(3)}=-x^{(3)},\quad {\bar {x}}^{(4)}=x^{(4)}$

and here we see, that ${\mathfrak {H}}_{x}=F_{23}$, ${\mathfrak {H}}_{y}=F_{31}$, ${\mathfrak {H}}_{z}=F_{12}$ transform as a vector of second kind, i.e.

${\mathfrak {\bar {H}}}_{x}={\mathfrak {H}}_{x},\quad {\mathfrak {\bar {H}}}_{y}={\mathfrak {H}}_{y},\quad {\mathfrak {\bar {H}}}_{z}={\mathfrak {H}}_{z}$,

while ${\mathfrak {E}}_{x}=iF_{14}$, ${\mathfrak {E}}_{y}=iF_{24}$, ${\mathfrak {E}}_{z}=iF_{34}$ transform as a vector of first kind, i.e.

${\mathfrak {\bar {E}}}_{x}=-{\mathfrak {E}}_{x},\quad {\mathfrak {\bar {E}}}_{y}=-{\mathfrak {E}}_{y},\quad {\mathfrak {\bar {E}}}_{z}=-{\mathfrak {E}}_{z}$.

In the dual system

$F_{\alpha _{1}\alpha _{2}}^{\ast }={\frac {1}{\sqrt {c}}}\sum c_{\alpha _{1}\beta _{1}}c_{\alpha _{2}\beta _{2}}F_{\alpha _{3}\beta _{3}}$

the transformations are exactly reversed due to the change of sign of the ${\sqrt {c}}$ in accordance with the rule given in § 1:

${\bar {F}}_{23}^{\ast }=-F_{23}^{\ast },\quad {\bar {F}}_{31}^{\ast }=-F_{31}^{\ast },\quad {\bar {F}}_{12}^{\ast }=-F_{12}^{\ast }$

and

${\bar {F}}_{14}^{\ast }=F_{14}^{\ast },\quad {\bar {F}}_{24}^{\ast }=F_{24},\quad {\bar {F}}_{34}^{\ast }=F_{34}$.

In Cartesian coordinates, where $F_{23}^{\ast }=-i{\mathfrak {E}}_{x}$ etc. and $F_{14}^{\ast }={\mathfrak {H}}_{x}$, this is of course self-evident. From this different nature of the two vectors ${\mathfrak {E}}$ and ${\mathfrak {H}}$ the known advantage can be drawn, for instance, with respect to the known theorem of optics: “When standing waves are reflected, the electrical vector has a vibration node at the the mirror surface, the magnetic vector has a vibration antinode”,^[4] or with respect of the analogous theorem of the phase change by half a wave-length in the case of reflection.

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1. Bateman, London Math. Soc. Proc. 1910 (8), p. 227 ff.
2. Herglotz, Göttinger Nachrichten, 1904.
3. Therefore the absolute differential calculus cannot be circumvented. One should notice though, how easy the general form of an invariant, covariant or contravariant emerges from the Cartesian one by means of its methods. It is sufficient to replace the ordinary differential quotients by co- or contravariant ones, besides the connection with certain factors $c_{\alpha \beta }$, $c^{(\alpha \beta )}$. See Wright, l.c. p. 27.
4. Drude, Optik, 1. Aufl., p. 264.