82
DOC.
9 FORMAL
FOUNDATION OF RELATIVITY
[50]
d2Q
d2x.
The first
one
of
the
equations (88)
is written in
Newton's
theory
as
d2o d2o d2o
4fkpo
dx2
dy2
dz2
and thus
one
gets
£
=
4n*.
2
Utilizing
the second
as
the time
unit,
the constant K has the numerical value
6.7
•
108;
and
if
one
chooses
the
lightsecond as
the time
unit,
the value
is
6.7
•
108
(9
•
1020)
.
Consequently,
one
gets
K
=
8tt
6,7
•
10~8
=
1.87
•
1027. (89)
9
•
1020
For the
naturally
measured distance of
neighboring spacetime points,
the
Newtonian
approximation yields
*k2
=
£
8livdxlldxv
=

dx2

dy2

dz2
+
(1
+ 20)dt2.
pv
[p. 1084]
For
a
purely
spatial
distance
one
gets

ds2
=
dx2
+
dy2
+
dz2.
Coordinate
lengths
are
here also
equal
to
naturally
measured
lengths.
The Euclidean
geometry
of
distances is valid with the
accuracy we
considered here. For
purely
temporal
distance
we
had
ds2
= (1
+
2o)dt2
or
ds
=
(1
+
o)dt.
To the
naturally
measured duration
ds
belongs
the
time duration
ds/(1+Q).
The clock
rate is measured
by
(1
+
o)
and, therefore,
increases with the
gravitational potential.
One concludes
from this that
spectral
lines
of
light, generated
at
the
sun,
show
a
redshift relative
to
corresponding spectral
lines
generated
on
earth,
the shift
amounting
to
[51]
=
2
•
10"6.
X