Page:Das Relativitätsprinzip und seine Anwendung.djvu/3

of $P$, a moment $t'$ is corresponding in another reference systems according to the transformation equations, and every point lying in $dS$ at time $t'$, has certain $x',y',z'$ for this definite value of $t'$. Points $x',y',z'$ satisfy a space element $dS'$, which is connected with $ds$ as follows:

$dS'={\frac {dS}{\omega }}.$

If we imagine an agent (matter, electricity etc.) as connected with these points, and if we assume that observer $B$ has reason to connect the same amount of that agent with every point as observer $A$, then the space density must be inversely proportional to the volume elements, i.e.,

$\varrho '-\omega \varrho .$

All of these relations are reciprocal, i.e., one can permute the primed and unprimed letters, when one simultaneously replaces $b$ by $-b$.

The fundamental equations of the electromagnetic field retain their form at the transformation, when one introduces the following magnitudes^[1]:

${\begin{array}{ccc}{\mathfrak {d}}'_{x}=a{\mathfrak {d}}_{x}-b{\mathfrak {h}}_{y},&{\mathfrak {d}}'_{y}=a{\mathfrak {d}}_{y}+b{\mathfrak {h}}_{x},&{\mathfrak {d}}'_{z}={\mathfrak {d}}_{z},\\{\mathfrak {h}}'_{x}=a{\mathfrak {h}}_{x}+b{\mathfrak {d}}_{y},&{\mathfrak {h}}'_{y}=a{\mathfrak {h}}_{y}-b{\mathfrak {d}}_{x},&{\mathfrak {h}}'_{z}={\mathfrak {h}}_{z};\end{array}}$

between these ones, and the transformed space density $\varrho '$, and the transformed velocity ${\mathfrak {v}}'$, the following equations hold in system $x',y',z',t'$:

${\begin{array}{l}\operatorname {div} \ {\mathfrak {d}}'=\varrho ',\\\operatorname {div} \ {\mathfrak {h}}'=0,\\\operatorname {rot} \ {\mathfrak {h}}'={\frac {1}{c}}\left({\mathfrak {\dot {d}}}'+\varrho '{\mathfrak {v}}'\right),\\\operatorname {rot} \ {\mathfrak {d}}'=-{\frac {1}{c}}{\mathfrak {\dot {h}}}'.\end{array}}$

In so far, the field equations of the theory of electrons satisfy the relativity principle; though we have to bring the equations of the electron themselves into accordance with this principle.

We will (somewhat more general) consider the motion of an arbitrary material point. At this occasion, the introduction of the concept of "proper time" (a nice invention of Minkowski) is useful. According to this, every point is so to speak connected with its own time which is independent of the reference system chosen; its differential is defined by the equation:

$d\tau ={\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}dt.$

1. Regarding the notations, see Mathematische Encyklopädie Vol. 14.