128
DOC. 2
RELATIVITY AND
ITS CONSEQUENCES
particular
coordinates
always
involves
the idea of
a
well-defined
experiment
concerned
with
the
position
of
solid bodies.9
Let
us
now
make
an
important
remark: In order
to
define the
physical
time with
respect
to
a
coordinate
system, we
used
a group
of
clocks in
a
state
of rest relative to that
system.
According
to
this
definition,
the
time
readings or
the establishment of the
simultaneity
of
two events have
meaning
only
if the
motion of
the
group
of
clocks
or
that
of the coordinate
system
is
known.
Consider
two
nonaccelerated coordinate
systems
S
and
S'
in
uniform translational
motion
with
respect
to
one
another.
Suppose
that
each
of
these
systems
is provided
with
a
group
of
clocks
invariably
attached
to
it,
and
that
all clocks
belonging
to
the
same
system are
in
phase.
Under
these
conditions,
the
readings
of the
group
attached to S
will define
the
physical
time
with
respect
to
S; analogously,
the
readings
of the
group
attached to
S' define the
physical
time
with
respect
to
S'. Each
elementary
event will
have
a
time coordinate
t
with
respect
to
S,
and
a
time
coordinate
t
with
respect
to
S'.
But,
we
have
no
right
to
assume
a
priori
that
the clocks of
the two
groups
can
be
set
in such
a
manner
that
the
two
time coordinates of
the
elementary
event
would be the
same,
or
in other
words,
in such
a
way
that
t would be
equal
to
t'.
To
assume
this would
mean
to
introduce
an
arbitrary
hypothesis.
This
hypothesis
has
been introduced
into kinematics
up
to
the
present
time.
The second
arbitrary
hypothesis
introduced
in kinematics
concerns
the
configuration
of
a body
in motion.
Consider
a
bar
AB
moving
in
the direction of
its axis
with
velocity
v
with
respect
to
a
coordinate
system
S not in
accelerated
motion.
What should
we
understand
by
the
"length
of the bar"? One
is
at first inclined to
believe
that
this
concept
does
not
require
any special
definition.
However, we
will
immediately
see
that
nothing
of the
sort is true if
we
consider the
following
two
methods of
determining
the
length
of the
rod:
1.
One accelerates
the motion
of
an
observer furnished
with
a measuring
rod
until
he
attains the
velocity
v,
i.e.,
until he
is at
relative
rest with
respect
to
the bar. The
observer then
measures
the
length
AB
by successively applying
the
measuring
rod
along
the bar.
2. Using
a group
of
clocks in
phase
with
each other and
at rest with
respect
to
the
system
S,
one
determines the
two
points
P1
and
P2
of
S
where
one
finds
the
two ends
of the bar
at
the instant
t;
after
that, one
determines the
length
of the
straight
line
9We
do not claim
that the
time
and
space
coordinates
must
necessarily
be defined
in such
a way
that their definitions could
serve as
the
basis
of
measurement
methods that
permit
the
experimental
determination of
these
coordinates-the
way
it has
been done
above.
But
whenever
the
quantities t,
x,
y,
z are
introduced in the
capacity
of
purely
mathematical
variables, equations
in
physics
will have
meaning
only
if
they
allow
the elimination of
these
quantities.