On the Relativity of the Conception of DistanceOnTheRelativityOfTheConceptionOfDistance2006-03-11 23:24:202006-03-11 23:24:20DefinitionRelativity: The Special and General Theory% this is the default PlanetMath preamble. as your knowledge
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% define commands here\subsection{On the Relativity of the Conception of Distance}
Let us consider two particular points on the train\footnotemark 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 ({\it 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$'$ and B$'$ 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$'$ and B$'$ 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---\emph{measured from the
train},---then this distance--\emph{as measured from the embankment}---is
not necessarily also equal to $w$.
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\footnotetext{{\it e.g.} the middle of the first and of the hundredth carriage.}