138
DOC.
2
RELATIVITY AND
ITS CONSEQUENCES
(1)
x'2
+
y'2
+
z'2
=
x2
+ y2
+
z2
is
identically
satisfied.
2.
Uniform motion
(translation)
of the
system
E'
with
respect
to
the
system
S.
This
second
transformation
is
characterized
by
the
equations
x'
=
x
+
at
y'
=y
+
ßt
z/
=
z +
yt,
(2)
where
a, ß,
y
are
constants.
For these
two kinds
of
transformation,
the condition
(3)
must
be
satisfied.
In other
words,
time
is
an
invariant
under these
two
transformations.
Combining
the transformations
(1)
and
(2),
we
obtain the
most
general
transforma-
tion
by means
of
which
one can
transform the
equations
of
mechanics
without
altering
them.
This
transformation
is
characterized
by
the
equation
(3)
and
by
three
equations
that
express
x',
y',
z'
as
linear
functions
of
x, y, z,
t.
The
coefficients
of these three
equations
are
connected
with
each other
by
relations
that,
for
t
=
0,
satisfy
condition
(1)
identically.
Let
us now
consider the
most
general
coordinate transformation
compatible
with
the
theory
of
relativity.
From
what
we
have
seen,
this
transformation
is
characterized
by
the
fact
that
x',
y',
z',
t'
must
be linear
functions
of
x,
y,
z,
t,
such
that the condition
(a)
x'2
+/
2
+z'2
-
cH'2
=
jc2 +
y2
+
z2
-
cH2
will
be
satisfied
identically.
It should
be
noted that the transformations
compatible
with
Newtonian mechanics
can
be
obtained
at
once
by
setting
c
= »
in
condition
(a). Thus,
if
we
take the
same
route
as
before,
we
arrive
at
the
equations
of
ordinary
kinematics
if,
instead
of
the
principle
of the
constancy
of the
velocity
of
light,
we
assume
the
existence
of
signals
whose
propagation
does not
require
any
time.
The
group
characterized
by
equation (a)
contains the transformations that
correspond
to
a change
in
the orientation of the
system.
These
are
the transformations
compatible
with
the condition
t
=
t'.
The
simplest
transformations
compatible
with
condition
(a) are
those for
which two
of the four coordinates of
an
elementary
event
remain
invariant.
Let
us consider,
for