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71
Relativity: The Special and General Theory
Albert Einstein: Relativity
Appendix
Appendix
I
Simple Derivation
of the Lorentz
Transformation
(Supllementary
to Section
11)
For the relative orientation of the co?ordinate systems indicated in Fig. 2, the x?axes of both
systems pernumently 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 K¹ by the abscissa x' and the time t'. We require to find x' and t' 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 K¹ with the velocity c, the propagation
relative to the system K¹ will be represented by the analogous formula
x' ? ct' = O
.
.
.
(2)
Those space?time points (events) which satisfy (x) must also satisfy (2). Obviously this will be the
case when the relation
(x' ? ct') = » (x ? ct)
.
.
.
(3).
is fulfilled in general, where » indicates a constant ; for, according to (3), the disappearance of (x ?
ct) involves the disappearance of (x' ? ct').
If we apply quite similar considerations to light rays which are being transmitted along the negative
x?axis, we obtain the condition
(x' + ct') = µ(x + ct)
.
.
.
(4).
By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants
a and b in place of the constants » and µ, where
and