144
ELECTRODYNAMICS OF MOVING
BODIES
velocity
=
light
path
time
interval ,
where
"time interval" should
be
understood in the
sense
of
the definition in
S1.
Let there be
given
a
rigid
rod
at
rest;
its
length, measured
by a
measuring
rod
that is also
at rest,
shall
be
l.
We now
imagine
that the axis
of
the rod
is
placed along
the X-axis
of
the
coordinate
system
at rest,
and
that the
rod
is then
set
in
uniform
parallel translational
motion (velocity
v)
along
the X-axis in
the direction
of increasing
x.
We
now
seek
to
determine
the
length
of the
moving
rod, which
we imagine
to be
obtained
by
the
following
two
operations:
(a) The
observer
co-moves
with the
above-mentioned
measuring
rod
and
the rod
to be measured, and
measures
the
length
of the rod directly,
by
applying
the
measuring
rod
exactly
as
if
the rod
to be measured,
the observer,
and
the
measuring
rod
were
at rest.
(b) Using
clocks
at rest
that
are
set
up
in the
system
at rest and
are
synchronous
in the
sense
of
§1,
the observer determines the points
of
the
system
at rest
at
which
the
beginning
and
the
end
of
the
rod to be measured
are
found at
some
given
time
t.
The
distance
between
these
two
points,
measured
by
the
rod used
before,
which
in the
present
case
is
at
rest,
is also
a
length, which
can
be
designated
as
the
"length
of
the rod."
According
to
the principle of
relativity,
the
length
to be found
in
operation
(a),
which
we
shall call "the
length
of the
rod
in the
moving
system,"
must
equal
the
length
l of the
rod at rest.
We
will
determine
the
length
to be
found
in
operation (b),
which
we
shall call "the
length
of the
(moving)
rod
in the
system at
rest,"
on
the
basis
of
our
two
principles,
and
will
find it
to
be
different
from
l.
The
commonly
used
kinematics
tacitly
assumes
that the
lengths
determined
by
the
two
methods mentioned
are
exactly
identical,
or,
in
other
words,
that
in
the
time
epoch t
a moving
rigid
body
is
totally
replaceable,
as
far
as
geometry
is
concerned,
by
the
same
body
when
it is
at rest
in
a
particular
position.
Further,
we
imagine
that the
two
ends
(A
and
B)
of the
rod
are
equipped
with clocks that
are
synchronous
with the clocks of
the
system
at rest,
i.e.,
whose readings
always
correspond
to
the
"time of
the
system
at
rest"
at
the