XXXII

ISimple Derivation of the Lorentz Transformation[Supplementary to [chapter:XI]Section XI]

For the relative orientation of the co-ordinate systems indicated in [fig:2]Fig. 2, the $x$-axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the $x$-axis. Any such event is represented with respect to the co-ordinate system $K$ by the abscissa $x$ and the time $t$, and with respect to the system $K^{\prime }$ by the abscissa $x^{\prime }$ and the time $t^{\prime }$. We require to find $x^{\prime }$ and $t^{\prime }$ when $x$ and $t$ are given.

A light-signal, which is proceeding along the positive

axis of $x$, is transmitted according to the equation $x=ct$ or $x-ct=0.(1)$ Since the same light-signal has to be transmitted relative to $K^{\prime }$ with the velocity $c$, the propagation relative to the system $K^{\prime }$ will be represented by the analogous formula $x^{\prime }-c{t}^{\prime }=0.(2)$ Those space-time points (events) which satisfy eqn:(1) must

also satisfy eqn:(2). Obviously this will be the case when the relation $(x^{\prime }-c{t}^{\prime })=\lambda (x-ct)(3)$ is fulfilled in general, where $\lambda$ indicates a constant; for, according to eqn:(3), the disappearance of $(x-ct)$ involves the disappearance of $(x^{\prime }-c{t}^{\prime })$.

If we apply quite similar considerations to light rays which are being transmitted along the negative $x$-axis, we obtain the condition $(x^{\prime }+c{t}^{\prime })=\mu (x+ct).(4)$

By adding (or subtracting) equations eqn:(3) and eqn:(4), and introducing for convenience the constants $a$ and $b$ in place of the constants $\lambda$ and $\mu$, where

$a\&=\frac{\lambda +\mu }{2}\mathrm{\backslash intertext}andb\&=\frac{\lambda -\mu }{2},$

we obtain the equations $\left.

x' &= ax - bct, \ ct' &= act - bx.

\right} (5)$

We should thus have the solution of our problem, if the constants $a$ and $b$ were known. These result from the following discussion.

For the origin of $K^{\prime }$ we have permanently $x^{\prime }=0$, and hence according to the first of the equations eqn:(5) $x=\frac{bc}{a}t.$

If we call $v$ the velocity with which the origin of $K^{\prime }$ is moving relative to $K$, we then have $v=\frac{bc}{a}.(6)$

The same value $v$ can be obtained from equation eqn:(5), if we calculate the velocity of another point of $K^{\prime }$ relative to $K$, or the velocity (directed towards the

negative $x$-axis) of a point of $K$ with respect to $K^{\prime }$. In short, we can designate $v$ as the relative velocity of the two systems.

Furthermore, the principle of relativity teaches us that, as judged from $K$, the length of a unit measuring-rod