The Lorentz Transformation
The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity ([chapter:VII]Section VII) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as
follows:
(1) The time-interval (time) between two events is
independent of the condition of motion of the body of reference.
(2) The space-interval (distance) between two points
of a rigid body is independent of the condition of motion of the body of reference.
If we drop these hypotheses, then the dilemma of [chapter:VII]Section VII disappears, because the theorem of the addition of velocities derived in [chapter:VI]Section VI becomes invalid. The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question arises: How have we to modify the considerations of [chapter:VI]Section VI in order to remove the apparent disagreement between these two fundamental results of experience? This question leads to a general one. In the discussion of
[chapter:VI]Section VI we have to do with places and times relative both to the train and to the embankment. How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment? Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity? In other words: Can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission relative to the embankment and relative to the train? This question leads to a quite definite positive answer, and to a perfectly definite transformation law for the space-time magnitudes of an event when changing over from one body of reference to another.
Before we deal with this, we shall introduce the following incidental consideration. Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line. In the manner indicated in [chapter:II]Section II we can imagine this reference-body supplemented laterally and in a vertical direction by means of a framework of rods, so that an event which takes place anywhere can be localised with reference to this framework. Similarly, we can imagine the train travelling with the velocity to be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework. Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing