DOC.
2
RELATIVITY AND
ITS CONSEQUENCES
139
example,
the transformations under
which
x
and
t
do not
change.
Instead of the
general
condition
(a), we
will have
the
special
condition
t'
=
t
(a1)
*'
=*
y'2
+
z'2
=
y2
+
z2.
To
this
condition
corresponds
a
rotation of the
system
about the
x-axis.
If,
on
the other
hand,
we
consider transformations under
which two
of the
spatial
coordinates,
for
example
y
and
z,
remain
invariant, we
will have
instead of the
general
condition
(a)
the
special
condition
y'
=y
(a2)
z'
=z
x'2
-
c2t'2
=
x2
-
c¥
These
are
the transformations
we
have
encountered
in
the
preceding
section
while
investigating a system
in
uniform motion
parallel
to
the
x-axis
of
an
identically
oriented
system
at rest.
The
formal
analogy
between
the
transformations
(a1)
and
(a2)
is immediately
evident.
The
two
systems
of
equations
differ
only by a
change
of
sign
in
the third condition.
But
even
this
difference
can
be made
to
disappear
if
one chooses,
with
Minkowski,
to
take
ict
instead of
t
as a variable,
where
i
is
the
imaginary
unit.16
In that
case
this
imaginary
temporal
coordinate
plays
the
same
role
in
the transformation
equations as
the
spatial
coordinates. If
we
set
x
y
z
ict
"*i
=
*2
=
X3
=
X
and consider
x1, x2, x3, x4
as
the coordinates of
a
point
in
a
four-dimensional
space
such
that
to
each
elementary
event
there
corresponds
a
point
in this
space, we
reduce
everything
that
happens
in
the
physical
world to
something
static in
the four-dimensional
space.
In that
case
the condition
(a)
will
be written
as
X' *
+
/
22
+
Jt'
32
+
X'
42
=
x\
+
x\
+
x\
+
x\.
16H. Minkowski,
Raum
und
Zeit.
Leipzig,
1909.
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