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Relativity: The Special and General Theory
Albert Einstein: Relativity
Part I: The Special Theory of Relativity
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 (Section 7) has been derived by means of a
consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as
(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 Section 7 disappears, because the theorem of
the addition of velocities derived in Section 6 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 Section 6 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 Section 6 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 c 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 Section 2 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 v 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,