Page:Das Relativitätsprinzip und seine Anwendung.djvu/2

Provided that there is an aether, then under all systems $x,y,z,t$, one is preferred by the fact, that the coordinate axes as well as the clocks are resting in the aether. If one connects with this the idea (which I would abandon only reluctantly) that space and time are completely different things, and that there is a "true time" (simultaneity thus would be independent of the location, in agreement with the circumstance that we can have the idea of infinitely great velocities), then it can be easily seen that this true time should be indicated by clocks at rest in the aether. However, if the relativity principle had general validity in nature, one wouldn't be in the position to determine, whether the reference system just used is the preferred one. Then one comes to the same results, as if one (following Einstein and Minkowski) deny the existence of the aether and of true time, and to see all reference systems as equally valid. Which of these two ways of thinking one is following, can surely be left to the individual.

In order to discuss the physical side of the question, we have to state the transformation formulas first, where we confine ourselves to the special form in which they were already used in the year 1887 by Voigt at investigations concerning Doppler's principle, namely:

$x'=x,\ y'=y,\ z'=az-bct,\ t'=at-{\frac {b}{c}}z;$

there, the constants $a>0,\ b$ satisfy the relation

$a^{2}-b^{2}=1{,}$

which cause the identity

$x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=x^{2}+y^{2}+z^{2}-c^{2}t^{2}\,$

The origin of system $x',y',z'$ moves towards system $x,y,z$ in the $z$-direction with velocity ${\tfrac {b}{a}}c$, which is always smaller than $c$. Generally, any velocity has to be assumed as being smaller than $c$.

All state variables of any phenomenon, measured in one or the other system, are connected by certain transformation formulas. They read, e.g. for the velocity of a point:

${\mathfrak {v}}'_{x}={\frac {{\mathfrak {v}}_{x}}{\omega }},\ {\mathfrak {v}}'_{y}={\frac {{\mathfrak {v}}_{y}}{\omega }},\ {\mathfrak {v}}'_{z}={\frac {a{\mathfrak {v}}_{z}-bc}{\omega }}{,}$

where

$\omega =a-{\frac {b{\mathfrak {v}}_{z}}{c}}$

We furthermore consider a system of pints, whose velocity is a steady function of the coordinates. Let $dS$ be a space element surrounding point $P(x,y,z)$ at time $t$; to this value $t$ and the coordinates