Albert Einstein: Relativity

Part I: The Special Theory of Relativity

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On the Relativity of the Conception of Distance

Let us consider two particular points on the train ^{1)}
travelling along the embankment with the
velocity v, and inquire as to their distance apart.
We already know that it is necessary to have a body of
reference for the measurement of a distance, with respect
to which body the distance can be measured up. It is
the simplest plan to use the train itself as reference-body (co-ordinate system). An observer in the train
measures the interval by marking off his measuring-rod
in a straight line (*e.g.* along the floor of the carriage)
as many times as is necessary to take him from the one
marked point to the other. Then the number which
tells us how often the rod has to be laid down is the
required distance.

It is a different matter when the distance has to be judged from the railway line. Here the following method suggests itself. If we call A^{1} and B^{1} the two points on the train whose distance apart is required, then both of these points are moving with the velocity v along the embankment. In the first place we require to determine the points A and B of the embankment which are just being passed by the two points A^{1} and B^{1} at a
particular time t — judged from the embankment. These points A and B of the embankment can be determined by applying the definition of time given in Section 8. The distance between these points A and B is then measured by repeated application of thee measuring-rod along the embankment.

*A priori* it is by no means certain that this last
measurement will supply us with the same result as
the first. Thus the length of the train as measured
from the embankment may be different from that
obtained by measuring in the train itself. This
circumstance leads us to a second objection which must
be raised against the apparently obvious consideration of Section 6. Namely, if the man in the carriage
covers the distance w in a unit of time — *measured from
the train,* — then this distance — *as measured from the
embankment* — is not necessarily also equal to w.

Next: The Lorentz Transformation

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Footnotes

^{1)} *e.g.* the middle of the first and of the twentieth carriage.