Page:AbrahamMinkowski2.djvu/8

${\mathfrak {g}}$:

(18b)

${\mathfrak {f}}=c{\mathfrak {g}}={\frac {1}{c}}{\mathfrak {S}};$

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:

(19)	$S=({\mathfrak {ra}})({\mathfrak {ra}}')-({\mathfrak {rb}})({\mathfrak {rb}}')+{\mathfrak {r}}^{2}({\mathfrak {bb}}')+u{\mathfrak {r}}[{\mathfrak {ba}}']+u{\mathfrak {r}}[{\mathfrak {b'a}}]+u^{2}({\mathfrak {aa}}')$

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 $S^{4}$:

(19a)

$S^{*}=({\mathfrak {rb}})({\mathfrak {rb}}')-({\mathfrak {ra}})({\mathfrak {ra}}')+{\mathfrak {r}}^{2}[{\mathfrak {aa}}']+u{\mathfrak {r}}[{\mathfrak {ab}}']+u{\mathfrak {r}}[{\mathfrak {a'b}}]+u^{2}({\mathfrak {bb}}')$

Putting

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

it is given:

(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:

(20a)

$2\Phi =({\mathfrak {ra}})({\mathfrak {ra}}')-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {aa}}')-({\mathfrak {rb}})({\mathfrak {rb}}')+{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {bb}}')$

(20b)

$2{\mathfrak {f}}=i[{\mathfrak {b'a}}]+i[{\mathfrak {b'a}}]$

(20c)

$2\psi =({\mathfrak {aa}}')-({\mathfrak {bb}}')$

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

(21)	${\begin{cases}{\mathfrak {a}}={\mathfrak {H}},&{\mathfrak {b}}=-i{\mathfrak {D}};\\{\mathfrak {a}}'={\mathfrak {B}},&{\mathfrak {b}}'=-i{\mathfrak {E}}.\end{cases}}$

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

(21a)

${\begin{cases}X_{x}x^{2}+Y_{y}y^{2}+Z_{z}z^{2}+2Y_{z}yz+2Z_{x}zx+2X_{y}xy\\=({\mathfrak {rE}})({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {ED}})+({\mathfrak {rH}})({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {HB}}),\end{cases}}$

(21b)

$2{\mathfrak {f}}=[{\mathfrak {EH}}]+[{\mathfrak {DB}}]$

(21c)

$2\psi ={\mathfrak {ED}}+{\mathfrak {HB}}.$