334
DOC.
42 SPECIAL
AND GENERAL RELATIVITY
Behaviour
of
Clocks
and
Rods 91
applied
to
the disc
in
the direction of the radius.
If,
then,
the
observer first
measures
the circumference of the disc with
his
measuring-rod
and then
the
diameter of the
disc,
on
dividing
the
one
by
the
other,
he
will
not
obtain
as
quotient
the
famil-
[47]
iar
number
n
=
3
.
14
...,
but
a
larger
number,1
whereas of
course,
for
a
disc
which
is
at rest
with
respect to
K,
this
op-
eration would
yield n exactly.
This
proves
that
the
proposi-
tions
of
Euclidean
geometry cannot
hold
exactly
on
the
rotating disc,
nor
in
general
in
a
gravitational field, at
least if
we
attribute the
length
1
to
the
rod
in
all
positions
and in
every
orientation. Hence the idea of
a
straight
line also
loses
its
meaning.
We
are
therefore
not
in
a
position to
define
ex-
[48]
actly
the co-ordinates
x,
y,
z
relative
to
the disc
by
means
of
the method used
in
discussing
the
special
theory,
and
as
long
as
the co-ordinates and times of
events
have
not
been
defined,
we
cannot
assign
an
exact
meaning
to
the natural
laws in
which these
occur.
[49]
Thus
all
our
previous
conclusions based
on
general
relativ-
ity
would
appear to
be called
in
question.
In
reality
we
must
make
a
subtle detour
in
order
to
be able
to
apply
the
postulate
of
general relativity exactly. I
shall
prepare
the reader
for
this
in the
following
paragraphs.
1 Throughout
this consideration
we
have
to use
the Galileian
(non-rotating) system K
as
reference-body,
since
we may only assume
the
validity
of the results of the
special theory
of
relativity
relative
to
K (relative to K'
a
gravitational field prevails).