DOC.
30
FOUNDATION OF GENERAL
RELATIVITY
197
For
a
unitmeasure
of
length
laid
"parallel"
to
the
axis
of
x,
for
example, we
should
have to set
ds2
=

1;
dx2
=
dx3
=
dx4
=
0.
Therefore

1
= g11dx21.
If,
in
addition,
the
unitmeasure
lies
on
the
axis
of x,
the
first of equations
(70)
gives
g11
=

(1
+ a/r)
From these
two
relations
it
follows that,
correct to
a
first
order
of small
quantities,
dx
=
1

^
. . . .
(71)
The unit
measuringrod
thus
appears
a
little shortened in
relation
to
the
system
of coordinates
by
the
presence
of
the
gravitational
field,
if the
rod is laid
along a
radius.
In
an
analogous
manner
we
obtain the
length
of
co
ordinates in
tangential
direction
if,
for
example, we
set
ds2
=

1;
dx1
=
dx3
=
dx4
=
0;
x1 =
r,
x2
=
x3
=
0.
The result
is

1
= g22dx22 =
 dx22
. . .
(71a)
With the
tangential
position,
therefore,
the
gravitational
field
of
the
point of
mass
has
no
influence
on
the
length
of
a
rod.
Thus Euclidean
geometry
does
not hold
even
to
a
first
ap
proximation
in
the
gravitational
field,
if
we
wish to
take
one
and
the
same
rod, independently
of its
place
and
orientation,
as a
realization
of
the
same
interval;
although,
to be
sure,
a
glance
at
(70a)
and
(69)
shows
that the
deviations to be
ex
pected
are
much
too
slight
to be noticeable
in measurements
of
the
earth's
surface.
Further,
let
us
examine the
rate
of
a
unit
clock,
which
is
arranged
to be at rest in
a
static
gravitational
field.
Here
we
have
for
a
clock
period
ds
=
1;
dx1
=
dx2
=
dx3
=
0
Therefore
1
=
g44dx24;
dXi
=
Vg7*
= V(i
+
(9«

i))=
1
"
"
X)