Page:IgnatowskiBemerkung.djvu/1

Some general remarks on the relativity principle.

W. v. Ignatowsky (Berlin)

When Einstein introduced the relativity principle some time ago, he simultaneously assumed that the speed of light $c$ shall be a universal constant, i.e. it maintains the same value in all coordinate systems. Also Minkowski started from the invariant $r^{2}-c^{2}t^{2}$ in his investigations, although it is to be concluded from his lecture "Space and Time"^[1], that he attributed to $c$ the meaning of a universal space-time constant rather than that of the speed of light.

Now I've asked myself the question, at which relations or transformation equations one arrives when only the relativity principle is placed at the top of the investigation, and whether the Lorentzian transformation equations are the only ones at all, that satisfy the relativity principle.

In order to answer this question, we again repeat what is given to us by the relativity principle per se.

If we have two coordinate systems $K$ and $K'$, being in translatory motion with respect to each other, then the relativity principle says that both systems can be seen as equally valid, i.e. any of them can be seen as at rest and the other one as in motion. In other words: we cannot determine absolute motion.

However, if $K$ and $K'$ are equally valid, and if we can express in system $K$ any physical quantity $E$ by a function of parameters $a_{1},a_{2},a_{3}\dots$, i.e. by writing

$E=\varphi \left(a_{1},a_{2},a_{3},\dots \right)$,

(1)

then it must be possible that the corresponding quantity $E'$ in system $K'$ can be expressed by the same function $\varphi$ of the corresponding parameters $a_{1}',a_{2}',a_{3}',\dots$, i.e. it will be

$E'=\varphi \left(a_{1}',a_{2}',a_{3}',\dots \right)$.

(2)

If it is assumed that we represent $E'$ by the unprimed parameters, for instance

$E'=f\left(a_{1},a_{2},a_{3},\dots \right)$

(3)

then, since $K$ and $K'$ are equally valid, the equation

$E=f\left(a_{1}',a_{2}',a_{3}',\dots \right)$

(4)

must be correct. Equations (1) to (4) form the mathematical formulation of the relativity principle.

Furthermore, if $q$ denotes the velocity of system $K'$ with respect to $K$ as measured by the latter, and $q'$ the velocity of system $K$ as measured from $K'$, then it evidently must be

$q'=-q$.

(5)

If we now consider a purely kinematic process, i.e., where only $x,y,z$ and $t$ come into consideration, then for instance we can write the following equation

$x'=\varphi (x,y,z,t,q)$

(6)

and similar ones for $y',z'$ and $t'$. Because $x,y,z$ and $t$ are to be considered as parameters by which (among other things) a physical phenomenon can be described, and from (1) to (4) we see that $a_{1}$ in general doesn't have to be equal to $a_{1}'$.

Although the following calculations are very elementary, I only will give the reasoning and the end results in order to save space, and allude for further details to an article of mine which will appear in the Archiv f. Math. u. Phys. soon.

We denote by ${\mathfrak {c}}_{0}$ the unit vector that gives the direction of the motion of $K'$ with respect to $K$, then we lay the $X$- or the $X'$-axis into this direction, and further assume for simplification that the $X'$-axis forms the elongation of the $X$-axis. Since space is to be assumed as being homogeneous and isotropic, it can be shown from that and from reasons of symmetry that $y$ and $z$ can only implicitly occur through $r$ in equation (6), where $r$ is the distance of a point from the $X$-axis. Furthermore it can be shown that $r=r'$, and consequently $x'$ cannot depend on $r$. Therefore we can write instead of (6)

$\left.{\begin{aligned}x'=&\varphi (x,t,q)\\t'=&f(x,t,q)\end{aligned}}\right\}$

(7)

and accordingly because of (3) and (4)

$\left.{\begin{aligned}x=&\varphi (x',t',q')\\t=&f(x',t',q')\end{aligned}}\right\}$.

(8)

If we take the complete differential of (7) and (8), then it is given

$\left.{\begin{aligned}dx'=&pdx+sdt\\dt'=&p_{1}dx+s_{1}dt\end{aligned}}\right\}$

(9)

and

$\left.{\begin{aligned}dx=&p'dx'+s'dt'\\dt=&p_{1}'dx'+s_{1}'dt'\end{aligned}}\right\}$

(10)

where $p,s,p',s'$ etc. denote the corresponding partial derivatives, which we preliminarily must be consider as unknown functions of $x,t,q$ and $x',t',q'$.

1. This journal 10, 104, 1909.