Page:AbrahamMinkowski2.djvu/8

 $$\mathfrak{g}$$:

finally, the tenth part of $$T^{4}$$ which is derived from (18), $$\psi$$, determines the density of electromagnetic energy.

To arrive at a suitable four-dimensional scalar, which is a second-order homogeneous function of coordinates $$x, y, z, u$$, with bilinear coefficients in the components of the electromagnetic vectors, we form the first one according to scheme (6), the radius vector $$\{\mathfrak{r,u}\}$$ in a space of four dimensions, and $$V_{I}^{4}$$ from $$V_{II}^{4}\{\mathfrak{a,b}\}$$:

$\mathfrak{R}=u\mathfrak{a}+[\mathfrak{rb}],\ U=-(\mathfrak{ra})$

Similarly, from another $$V_{II}^{4}\{\mathfrak{a',b'}\}$$ and from $$V_{I}^{4}\{\mathfrak{r,a}\}$$ which is comprised of $$V_{I}^{4}$$:

$\mathfrak{R}'=u\mathfrak{a}'+[\mathfrak{rb}'],\ U'=-(\mathfrak{ra}')$

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

$S=\mathfrak{RR}'+UU'=u^{2}\mathfrak{aa}'+u\mathfrak{a}[\mathfrak{rb}']+u\mathfrak{a}'[\mathfrak{rb}]+[\mathfrak{rb}][\mathfrak{rb}']+(\mathfrak{ra})(\mathfrak{ra}')$

that can be written:

As it follows from (6a), we can permute $$\mathfrak{a}$$ with $$\mathfrak{b}$$ and $$\mathfrak{a'}$$ with $$\mathfrak{b'}$$, and obtain in a corresponding way another $$S^{4}$$:

Putting

$4\varphi=S-S^{*}$

it is given:

{{MathForm1|(20)|$$\left\{ \begin{array}{c} 2\varphi=(\mathfrak{ra})(\mathfrak{ra}')-\frac{1}{2}\mathfrak{r}^{2}[\mathfrak{aa}']-(\mathfrak{rb})(\mathfrak{rb}')+\frac{1}{2}\mathfrak{r}^{2}[\mathfrak{bb}']\\ \\+u\mathfrak{r}[\mathfrak{b'a}]+u\mathfrak{r}[\mathfrak{b'a}]+\frac{1}{2}u^{2}\left\{ (\mathfrak{aa}')-(\mathfrak{bb}')\right\} .\end{array}\right.$$}}

Now, by identifying the homogeneous second-order function $$\varphi$$ of $$x, y, z, u$$ which is invariant under the Lorentz transformation, with $$S^{4}$$ as given by (18), we find the expressions:

We introduce the electrodynamic $$V_{II}^{4}$$ of, by setting

By taking into account (18a), the following expressions are resulting now: