130
DOC. 2
RELATIVITY AND ITS
CONSEQUENCES
with
respect
to
the
system
S
is
identical
with
the
geometric configuration
of
the
system
S'.
If
one
abandons the
ordinary
kinematics
and
builds
a new
kinematics
based
on
the
new
foundations,
one
arrives
at
transformation
equations
different from those
given
above.
And
now,
we are going
to show10
that based
on
1.
The
principle
of
relativity
and
2.
The
principle
of
the
constancy
of the
velocity
of
light,
we
arrive at
transformation
equations
that
allow
us
to
see
that Lorentz's
theory is compatible
with
the
principle
of
relativity.
The
theory
based
on
these
principles we
shall call the
theory
of
relativity.
Let
S
and S' be
two
equivalent
coordinate
systems,
i.e.,
systems
in which
lengths
are
measured
in
the
same unit,
and
each
of
which
possesses a
group
of
clocks
that
run
in
synchrony
when
the
two
systems are
at
relative
rest with
respect
to each
other.11
According
to
the
principle
of
relativity, physical
laws must be
identical
for
the
two
systems
regardless
of whether the
systems
are
at relative rest
or
in uniform translational
motion with
respect
to each
other.
Thus,
in
particular,
the
velocity
of
light
in
a vacuum
must
be
expressed
by
the
same
number
in
the
two
systems.
Let
t,
x,
y, z
be
the
coordinates of
an
elementary
event with
respect
to
S,
and t', x',
y',
z' the
coordinates of
the
same
event with
respect
to
S'. We seek
to find
the relations that
link
these
two
groups
of coordinates. It
can
be shown
that these relations
must be
linear
because of the
homogeneity
of time and
space,12
and time
t is
therefore
linked with
time
t'
by
a
formula of the form
(2)
t'
=
At
+
Bx
+ Cy +
Dz.
Furthermore,
for
an
observer linked
to S it follows from
this,
in
particular,
that the
three coordinate
planes
of S'
are planes
in
uniform
motion; but,
in
general,
these three
planes
will not form
a
rectangular
triad
even
though
we assume
that the
system
S'
is
rectangular
for
an
observer connected
with this
system. However,
if,
referring
to
the
system
S,
we
have
chosen the
position
of the
x'-axis
parallel
to
the direction of the
motion
of
S',
it
will follow
for
reasons
of
symmetry
that the
system
S'
will
appear
as
rectangular.
In
particular, we
may
choose the
relative
position
of the
two
coordinate
systems
in such
a way
that the
x-axis will
permanently
coincide with
the
x'-axis,
and
the
10A.
Einstein,
Ann.
der
Phys.
17
(1905):
891-921,
and
Jahrbuch
der
Radioaktivität und Elektronik
4
(1907):
411-462.
11It
should be noted that
we
will
always implicitly
assume
that the
fact
of
a measuring
rod
or a
clock
being
set in motion
or
brought
to rest does not
change
the
length
of
the
rod
or
the
rate
of
the
clock.
12Cf.
footnote
15.
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