Page:AbrahamMinkowski2.djvu/6

When projecting into the space of three-dimensional ($x,y,z$), then the first six components form a three-dimensional tensor ($T^{3}$), which are transforming as the squares and products of ($x,y,z$); the following three components of $T^{4}$ are forming a $V^{3}$, the tenth a scalar $S^{3}$.

The four components of the operator "lor" transform as components of $V_{I}^{4}$, we can deduce – from a four-dimensional scalar $\varphi$, given as a function of $x,y,z,u$ – a $T^{4}$ which is twice differentiable with respect to $x,y,z,u$:

${\frac {\partial ^{2}\varphi }{\partial x^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial y^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial z^{2}}},\ {\frac {\partial ^{2}\varphi }{\partial y\ \partial z}},\ {\frac {\partial ^{2}\varphi }{\partial z\ \partial x}},\ {\frac {\partial ^{2}\varphi }{\partial x\ \partial y}};\quad {\frac {\partial ^{2}\varphi }{\partial x\ \partial u}},\ {\frac {\partial ^{2}\varphi }{\partial y\ \partial u}},\ {\frac {\partial ^{2}\varphi }{\partial z\ \partial u}};\quad {\frac {\partial ^{2}\varphi }{\partial u^{2}}}.$

A $S^{4}$ is given, being a quadratic homogeneous function of $x,y,z,u$:

(13)	${\begin{cases}\varphi (x,y,z,u)&={\frac {1}{2}}c_{11}x^{2}+{\frac {1}{2}}c_{22}y^{2}+{\frac {1}{2}}c_{33}z^{2}\\\\&+c_{23}yz+c_{31}zx+c_{12}xy\\\\&+c_{14}xu+x_{24}yu+c_{34}zu+{\frac {1}{2}}c_{44}u^{2},\end{cases}}$,

the 10 coefficients:

$c_{11},\ c_{22},\ c_{33},\ c_{23},\ c_{31},\ c_{12};\ c_{14},\ c_{24},\ c_{34};\ c_{44},$

form a four-dimensional tensor.

In the electrodynamics of moving bodies, the equations of the momentum and energy apply:^[1]

(14)

${\begin{cases}&{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}},\\\\&{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}},\\\\&{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}},\\\\\\c{\mathfrak {qK}}+&Q=-{\frac {\partial {\mathfrak {S}}_{x}}{\partial x}}-{\frac {\partial {\mathfrak {S}}_{y}}{\partial y}}-{\frac {\partial {\mathfrak {S}}_{z}}{\partial z}}-{\frac {\partial \psi }{\partial t}}.\end{cases}}$

In order to give to these four equations a more symmetrical from, we put:

(15)	$u=ict,\ {\mathfrak {K}}_{u}=i{\mathfrak {qK}}+i{\frac {Q}{c}},\ U_{u}=\psi ;$

(15a)

$X_{u}=-ic{\mathfrak {g}}_{x},\ Y_{u}=-ic{\mathfrak {g}}_{y},\ Z_{u}=-ic{\mathfrak {g}}_{z};$

(15b)

$U_{x}=-{\frac {i}{c}}{\mathfrak {S}}_{x},\ U_{y}={\frac {-i}{c}}{\mathfrak {S}}_{y},\ U_{z}={\frac {-i}{c}}{\mathfrak {S}}_{z}.$

1. M. Abraham, l. c. equations (6) and (7).