DOC.
17
THE THEORY OF RELATIVITY
347
y',
z',
t' of the
same
event
referred
to
another
system
k', which
is
in
a
known,
uniform translational motion relative
to
k.
It
turns out
that the
customary
simple
solution of
this
problem is
based
on
the
two
assumptions
we
have
just
identified
as
arbitrary.
How to
put
kinematics
back
on
its
feet? The
answer
is
self-evident:
the
very same
circumstances
that
led
us
into
so
many
embarrassing
difficulties in
the
past
lead
us
to
a
negotiable path
now
that
we
have
gained more
room
to
maneuver by
putting
aside
the
arbitrary assumptions
mentioned
above.
For
it turns out
that
precisely
those
two
seemingly
incompatible
axioms,
which
were
imposed
on us by
experience,
namely
the
principle
of
relativity
and
the
principle
of
constancy
of the
velocity
of
light,
lead
us
to
a
perfectly
definite solution of the
space-time
transformation
problem.
One
arrives at
results
that, in
part,
run very
much counter to
our
customary conceptions.
The
mathematical considerations
leading
to
these results
are very simple;
this
is
not
the
place
to dwell
on
them.2
It
will
be better
if I
deal
with
the
most
important consequences
that
were
reached
in this
way by a
quite
logical
procedure,
without additional
assumptions.
First,
things
purely
kinematic. Since
we
defined the coordinates
and
the time
in
a
definite
way
in
physical
terms,
all
relationships
betwen
spatial
and
temporal
quantities
will
have
a
perfectly
definite
physical
content.
We obtain the
following:
If
we
have
a
solid
body
that
is
moving uniformly
with
respect
to
the coordinate
system k,
which
we
take
as
the
basis
for
our analysis,
then
this
body
appears
contracted
by
a
perfectly
definite
ratio
in
the direction of
its
motion, as
compared
with
the
shape
it has
when
it
is
in
a
state
of rest
with
respect
to this
system.
If
we
denote the
velocity
of
motion
of the
body
by
v
and the
velocity
of
light by c,
then each
length
measured
in
the direction of
motion,
and
equal to
l
when the
body
is
in
a
motionless
state,
will
be diminished because of
the
body's
motion relative
to
the
noncomoving
observer
to
the
length
1--
N
c2
If the
body
has
a spherical
shape
in
the
state
of
rest,
it will have
the
shape
of
a
flattened
ellipsoid
if
we move
it in
a
certain direction. When
its
velocity
reaches the
2If
x,
y,
z,
t,
and
x', y', z',
t'
are space
and time
coordinates
with
respect
to
the
two
reference
systems
k and
k',
then the
two
underlying principles
demand that
the
transformation
equations
be such
that
each
of the
two
equations
jc2
+y2
+z2
=
c¥
jc'2+/2+z'2
=
c'
V2
have
the other
equation as
its
consequence.
Since,
for
reasons
into
which I shall not enter
here,
the substitution
equations
must be
linear,
this
determines the transformation
law,
as a
brief
analysis
shows
(cf., e.g.,
Jahrbuch der
Radioaktivität
und
Elektronik 4
[1907]:
418ff).