Page:Das Relativitätsprinzip und seine Anwendung.djvu/4

The expressions formed by the aid of proper time $\tau$

${\frac {d}{d\tau }}{\frac {dx}{d\tau }},\ {\frac {d}{d\tau }}{\frac {dy}{d\tau }},\ {\frac {d}{d\tau }}{\frac {dz}{d\tau }}{,}$

which are linear homogeneous functions of the ordinary acceleration components, are denoted by us as components of the "Minkowski acceleration". We describe the motion of a point by the equations

$m{\frac {d}{d\tau }}{\frac {dx}{d\tau }}={\mathfrak {K}}_{x}{\text{, etc.,}}$

where $m$ is a constant which we call the "Minkowski mass". Vector ${\mathfrak {K}}$ is denoted by us as "Minkowski force".

The transformation formulas for this acceleration and force can be easily derived; $m$ is left unchanged by us. Then one has

${\mathfrak {K}}'_{x}={\mathfrak {K}}_{x},\ {\mathfrak {K}}'_{y}={\mathfrak {K}}_{y},\ {\mathfrak {K}}'_{z}=a{\mathfrak {K}}_{z}-{\frac {b}{c}}({\mathfrak {v}}\cdot {\mathfrak {K}}).$

The essential thing is now as follows: The relativity principle requires, that (at an actual phenomenon) the Minkowski forces are in a certain way depending on the coordinates, velocities, etc. in one reference system, and the transformed Minkowski forces in the other reference system are depending in the same way on the transformed coordinates, velocities etc. That is a special property, which all forces of nature must have, when the relativity principle shall hold. If we presuppose this, then one can calculate the forces acting on moving bodies, when one know them for the case of rest. If e.g. an electron of charge $e$ is moving, then we imagine a reference system in which it is momentarily at rest. The Minkowski force is acting upon the electron in this system:

${\mathfrak {K}}=e{\mathfrak {d}};$

from that it follows by application of the transformation equations for ${\mathfrak {K}}$ and ${\mathfrak {d}}$, that the Minkowski force acting upon an electron moving with velocity ${\mathfrak {v}}$ in an arbitrary reference system, amounts to

${\mathfrak {K}}=e{\frac {{\mathfrak {d}}+{\frac {1}{c}}[{\mathfrak {v}}\cdot {\mathfrak {h}}]}{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}}$

This formula is not in agreement with the ordinary Ansatz of electron theory, due to the presence of the denominator. The difference stems from the fact, that one usually doesn't operate with our Minkowski force, but with the "Newtonian force" ${\mathfrak {F}}$, and we see, that (for an electron) these two forces are connected as follows:

${\mathfrak {F}}={\mathfrak {K}}{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}.$

One will assume that this relation holds for arbitrary material points.