Page:AbrahamMinkowski2.djvu/9

They give to empty space (where ${\mathfrak {D}}$ and ${\mathfrak {E}}$, ${\mathfrak {H}}$ and ${\mathfrak {B}}$ become identical) the known values ​​of the Maxwell pressure, the current, and the energy density. To ponderable bodies in a resting state, the values ​​(21a) and (21c) of the pressure and energy density are acceptable, yet not the value (21b), because it is

${\mathfrak {D}}=\epsilon {\mathfrak {E}},\ {\mathfrak {B}}=\mu {\mathfrak {H}},$,

then the energy current would be

${\mathfrak {S}}=c{\mathfrak {f}}=\left({\frac {\epsilon \mu +1}{2}}\right)c[{\mathfrak {EH}}],$

which differs from the current given by the Poynting vector

${\mathfrak {S}}=c[{\mathfrak {EH}}]$

by

$\left({\frac {\epsilon \mu +1}{2}}\right)c[{\mathfrak {EH}}]$.

So we must subtract from the invariant $\varphi$ (given by equation (20)), another $S^{4}$, which contains $(\epsilon \mu -1)$ as a factor, and which is equal to zero for empty space.

To obtain such a $S^{4}$, we consider two $V_{I}^{4}$; first the $V_{I}^{4}$-"velocity"

${\mathfrak {r}}_{1}=k^{-1}{\mathfrak {q}},\ u_{1}=ik^{-1},$

then the "rest ray", given by equations (12):

${\mathfrak {R}}=k^{-1}{\mathfrak {f}}'+k^{-3}{\mathfrak {q}}({\mathfrak {qf}}'),\ U=ik^{-3}({\mathfrak {qf}}').$

We introduce the $V^{3}$

(22)	${\mathfrak {W}}=(\epsilon \mu -1)k^{-1}{\mathfrak {R}}=(\epsilon \mu -1)\left\{k^{-2}{\mathfrak {f}}'+k^{-4}{\mathfrak {q}}({\mathfrak {qf}}')\right\}$

with $(\epsilon \mu -1)$ being a $S^{4}$,

${\begin{array}{l}{\mathfrak {r}}_{2}=(\epsilon \mu -1){\mathfrak {R}}=k{\mathfrak {W}},\\u_{2}=(\epsilon \mu -1)U=ik({\mathfrak {qW}}),\end{array}}$

which forms a $V_{I}^{4}$.

Now we compose, according to scheme (2), two $S^{4}$:

${\begin{array}{l}{\mathfrak {rr}}_{1}+uu_{1}=k^{-1}\{({\mathfrak {rq}})+iu\},\\{\mathfrak {rr}}_{2}+uu_{2}=k\{({\mathfrak {rW}})+iu({\mathfrak {qW}})\},\end{array}}$

which are both linear in $x,y,z,u$, and we multiply them. Thus a $S^{4}$ is given, being a homogeneous second-order function of $x,y,z,u$:

(23)	$2\chi =({\mathfrak {rq}})({\mathfrak {rW}})+\left(iu{\mathfrak {r}},\ {\mathfrak {W}}+{\mathfrak {q}}({\mathfrak {qW}})\right)-u^{2}({\mathfrak {qW}}).$

By adding $S^{4}$, $\varphi$ and $\chi$, which are given by (20) and (23), we form the new $S^{4}$

(24)	$f=\varphi +\chi$

and we are using this instead of $\varphi$ as a characteristic invariant, which determines the pressures, the current, and the electromagnetic energy density, by setting:

$f(x,y,z,u)=\Phi (x,y,z)-iu({\mathfrak {rf}})+{\frac {1}{2}}u^{2}\psi .$