Translation:The Lorentz-Einstein transformation and the universal time of Ed. Guillaume

by Dmitry Mirimanoff

Session of 17 March 1921.

D. Mirimanoff. – The Lorentz-Einstein transformation and the universal time of Ed. Guillaume.

In a series of communications and articles, Ed. Guillaume sought to introduce a mono-parametric representation of time in the theory of relativity. He succeeded in giving to this problem an interesting solution, in the case where the number of reference systems is equal to two. This solution involves, as we know, a simple geometric interpretation.

I'd like to propose to give a new interpretation. I'll show that the $t$-parameter of Guillaume only differs by a constant factor of time $\tau$ of a particular Einstein system which I call median system.^[1] Each pair of reference systems correspond to a median system and a $t$-parameter of Guillaume. Then one realizes better why the procedure of Guillaume is no longer successful if the number $n$ of reference systems is greater than two. Indeed, for $n>2$, the number of reference systems and consequently of $t$ parameters is greater than one, and these parameters are generally distinct.

1. Median system. $S_{1}$ and $S_{2}$ are two Einstein reference systems, animated to move uniformly with respect to one another along axes $o_{1}x_{1},o_{2}x_{2}$. I assume that the Lorentz-Einstein transformation is applicable to these systems and therefore coordinates $x_{1},x_{2}$ and times $x_{1},x_{2}$ are linked by the relations

${\begin{aligned}x_{1}&=\beta (x_{2}+\alpha c\tau _{2})&,&&x_{2}&=\beta (x_{1}-\alpha c\tau _{1}),\\c\tau _{1}&=\beta (c\tau _{2}+\alpha x_{2})&,&&c\tau _{2}&=\beta (c\tau _{1}-\alpha x_{1}).\end{aligned}}$

(1)

where $\alpha ={\tfrac {v}{c}}$, $\beta ^{2}={\tfrac {1}{1-\alpha ^{2}}}$, $v$ is the velocity of $S_{2}$ with respect to $S_{1}$.

Now a third system $S$ parallel to $S_{1}$, $S_{2}$ also conducts a motion of translation along $ox_{1}$. Let $v_{0}$ be the velocity with respect to $S_{1}$. The Lorentz transformation still applies

${\begin{aligned}x_{1}&=\beta _{0}(x+\alpha _{0}c\tau )&,&&x&=\beta _{0}(x_{1}-\alpha _{0}c\tau _{1}),\\c\tau _{1}&=\beta _{0}(c\tau +\alpha _{0}x)&,&&c\tau &=\beta _{0}(c\tau _{1}-\alpha _{0}x_{1}).\end{aligned}}$

(2)

where $x$ $\tau$ is the abscissa and the corresponding time in $S$, $\alpha _{0}={\tfrac {v_{0}}{c}}$, etc.

We assume that the velocity of $S_{2}$ relative to $S$ is also $v_{0}$. I would say that $S$ is the corresponding median system. How can we express $v_{0}$, $\alpha _{0}$, $\beta _{0}$ as functions of $v$, $\alpha$, $\beta$? In order to find it, it is sufficient to express $x_{1}$, $\tau _{1}$ as functions of parameters $x$, $\tau$ (form. (2)) and the latter as functions of $x_{2}$, $\tau _{2}$ and identify the resulting formulas with (1), which gives

${\frac {2\alpha _{0}}{1+\alpha _{0}^{2}}}=\alpha ,\quad \alpha _{0}={\frac {\beta -1}{\alpha \beta }},\quad \beta _{0}^{2}={\frac {\beta +1}{2}},\quad \left(1-\alpha \alpha _{0}\right)\beta =1.$

(3)

2. Contraction. Consider two points $P'$ and $P''$. Let $x'_{1},x'_{2},x'$; $x''_{1},x''_{2},x''$ be their coordinates in $S_{1}$, $S_{2}$, and $S$ at the same moment $\tau$ (Einstein time of the median system). By virtue of (2)

$x'_{1}=\beta _{0}\left(x'+\alpha _{0}c\tau \right),\quad x''_{1}=\beta _{0}\left(x''+\alpha _{0}c\tau \right).$

Therefore

$x''_{1}-x'_{1}=x''_{2}-x'_{2}$

(4)

So there is no contraction, provided that $P'$ and $P''$ are considered at the same moment $\tau$.

The converse is true, in other words: If the contraction does not take place by adopting time $\tau$ of an Einstein system, this system is the median system.

3. Another relation. Let P be a point of the abscissas $x_{1}$ and $x_{2}$ in $S_{1}$ and $S_{2}$. There, by replacing parameter $\tau _{2}$ by its expression as function of $x_{2}$ and $\tau$ in the first formula (1), it is given

$x_{1}=\beta \left\{\left(1-\alpha \alpha _{0}\right)x_{2}+{\frac {\alpha }{\beta _{0}}}c\tau \right\}=x_{2}+{\frac {\beta }{\beta _{0}}}v\tau$

(5)

by virtue of (3).

4. The universal time of Guillaume. Let $k$ be an arbitrary function of $v$. As $v$ is const., $k$ is constant. Suppose $k>0$ and put $t=k\tau$. If we adopt time $t$ instead of Einstein time $\tau$, simultaneity is not altered. Equality (4) remains true, therefore no contraction, equality (5) is written $x_{1}=x_{2}+{\tfrac {1}{k}}{\tfrac {\beta }{\beta _{0}}}vt$. In particular we assume that $k={\tfrac {\beta }{\beta _{0}}}$, where $t={\tfrac {\beta }{\beta _{0}}}\tau$. Equation (5) is written

$x_{1}=x_{2}+vt$

(6)

Multiplication of the second equation of the second group (2) by $k={\tfrac {\beta }{\beta _{0}}}$, gives in virtue of (3):

$c\tau _{1}={\frac {c}{\beta }}t+{\frac {\beta -1}{\alpha \beta }}x_{1}$

We come, as seen, to the equation that defines the universal time $t$ of Guillaume ^[2]. Therefore the time $t$ defined by $t={\tfrac {\beta }{\beta _{0}}}\tau$ is the parameter of Guillaume. It only differs from time $\tau$ of the median system by the constant factor $k={\tfrac {\beta }{\beta _{0}}}$.

5. Case of three systems. Imagine three systems $S_{1},S_{2},S_{3}$ conducting a uniform translatory movement parallel to the axes of $x$. Let $v_{12},v_{13},v_{23}$ be the relative velocity of $S_{2}$ with respect to $S_{1}$, of $S_{3}$ with respect $S_{1}$, of $S_{3}$ with respect $S_{2}$, and $t_{12},t_{13},t_{23}$ the parameters of Guillaume. Then we have in virtue of (6)

$x_{1}=x_{2}+v_{12}t_{12};\quad x_{1}=x_{3}+v_{13}t_{13};\quad x_{2}=x_{3}+v_{23}t_{23};$

for example, the abscissa $x_{1}$ of $O_{2}$ is given by $x_{1}=v_{12}t_{12}$, that of $O_{3}$ by $x_{1}=v_{13}t_{13}$. Parameters $t_{12},t_{13},t_{23}$ should not be confused with each other.

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1. This term was suggested to me by Plancherel.
2. Guillaume, Ed. La théorie de la relativité en fonction du temps universel, Arch. Sc. phys. et nat. (4), 46, p. 809.