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 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 $$V^{3}$$

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 $$x, y, z, u$$:

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

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:

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