The Simplest Derivation of the Lorentz Transformation
By Wolfgang G. Gasser
Several attempts have been made to demonstrate errors in concrete derivations of the Lorentz transformation. Yet even if all existing derivations turned out to be erroneous, the ideal space-time relations expressed by the Lorentz transformations would remain unaffected. (Whether the Lorentz transformation leads to contradictions in case of accelerations, or whether the Lorentz transformation is somehow "implemented" in our empirical reality, are different questions.)
In order to derive the Lorentz transformation in one spatial dimension, one has to start with the Galilean transformation:
x' = x - ß c t
Here the factor ß is used to express the velocity v of coordinate system S' relative to system S according to v = ß c (with c as light-speed). The requirement that the speed of light (i.e. length per time) remains unchanged in system S' can easily be satisfied by transforming time t' analogously to the space coordinate x':
t' = t - ß/c x
By using concrete values for x and t, one can easily recognize that light moving at c in coordinate system S also moves at c in S'. For instance in case of ß = 0.3 we get: To 1 light-second per 1 second of S correspond either 0.7 LS per 0.7 sec (forward direction) or 1.3 LS per 1.3 sec (backward direction). The asymmetry between the transformation to S'
x' = (x - ß c t)
t' = (t - ß/c x)
and the transformation back to coordinate system S
x = (1 - ß2) -1 (x' + ß c t')
t = (1 - ß2) -1 (t' + ß/c x')
can easily be removed:
x' = (1 - ß2) -1/2 (x - ß c t)
t' = (1 - ß2) -1/2 (t - ß/c x)
x = (1 - ß2) -1/2 (x' + ß c t')
t = (1 - ß2) -1/2 (t' + ß/c x')
This is an extract from: Verteidigung der Relativitätstheorie vor unberechtigter Kritik