146
ELECTRODYNAMICS OF MOVING
BODIES
The
origin of
one
of the
two
systems (k)
shall
now
be
imparted
a
(con-
stant) velocity
v
in
the direction of
increasing
x
of
the other
system
(K),
which
is
at rest,
and
this velocity shall also
be imparted
to
the
[14]
coordinate
axes,
the
corresponding
measuring
rod, and
the clocks.
To
each
time
t
of the
system
at rest
K
there
corresponds
then
a
definite
position
of
the
axes
of the
moving
system, and
for
reasons
of
symmetry we
may
right-
fully
assume
that the
motion
of
k
can
be such
that
at
time
t
("t"
always
denotes
a
time of the
system at
rest)
the
axes
of
the
moving
system
are
parallel
to
the
axes
of
the
system
at
rest.
We
now imagine
the
space
to be measured
both
from
the
system
at
rest
K
by
means
of the
measuring
rod at rest and from
the
moving
system
k
by
means
of
the
measuring
rod
moving
along
with
it, and
that
the coordinates
x, y,
z
and
E,
n,
(
are
obtained in this
way.
Further,
by means
of
the clocks
at
rest
in the
system at rest and using
light
signals
in the
manner
described in
§1,
the
time
t
of the
system
at rest
is determined for all its points
where
there is
a
clock;
likewise,
the time
T
of the
moving
system
is determined
for all the
points of
the
moving
system
having
clocks that
are
at rest
relative
to
this
system, applying
the
method of
light
signals
described
in
§1
between
the
points
containing
these clocks.
To every system
of
values
x, y, z,
t
that
determines
completely the
place
and
time
of
an
event
in
the
system
at rest,
there
corresponds
a
system
of
values
E,
n,
(,
T
that fixes this
event
relative
to
the
system
k, and
the
problem
to be
solved is
to
find the
system
of
equations connecting
these
quantities.
First of all, it is
clear that these
equations
must be
linear because
of
the properties
of
homogeneity
that
we
attribute
to
space
and
time.
If
we
put
x'
=
x-
vt,
then it is clear that
a
point at rest
in the
system
k
has
a
definite,
time-independent
system
of
values
x',
y,
z
belonging
to
it.
We
first determine
T
as
a
function
of
x',
y,
z,
and
t.
To
this
end,
we
must
express
in
equations
that
T
is
in
fact the
aggregate
of
the
readings
of the clocks
at rest
in the
system
k,
which
have been
synchronized according
to
the rule
given
in
§1.
Suppose
that
at
time
T0
a
light
ray
is
sent from
the
origin
of the
system k
along
the X-axis
to x' and
is reflected
from
there
at
time
T1
toward
the
origin, where
it arrives
at
time
T2;
we
then
must
have