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Relativity: The Special and General Theory

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 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 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, we can disregard the fact that in
reality these frameworks would continually interfere with each other, owing to the
impenetrability of solid bodies. In every such framework we imagine three surfaces
perpendicular to each other marked out, and designated as " co-ordinate planes " (" co-
ordinate system "). A co-ordinate system K then corresponds to the embankment, and a
co-ordinate system K' to the train. An event, wherever it may have taken place, would be
fixed in space with respect to K by the three perpendiculars x, y, z on the co-ordinate
planes, and with regard to time by a time value t. Relative to K1, the same event would be
fixed in respect of space and time by corresponding values x1, y1, z1, t1, which of
course are not identical with x, y, z, t. It has already been set forth in detail how
these magnitudes are to be regarded as results of physical measurements.
 
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