DOC.
30
FOUNDATION OF GENERAL
RELATIVITY
197
For
a
unit-measure
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
unit-measure
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
measuring-rod
thus
appears
a
little shortened in
relation
to
the
system
of co-ordinates
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)