198
DOC. 30 FOUNDATION
OF GENERAL
RELATIVITY
or
**
1
+
£
J
"r
•
(72)
Thus the
clock
goes more slowly
if
set
up
in the
neighbour
hood of
ponderable masses.
From this it
follows
that the
spectral
lines of
light reaching
us
from
the
surface of
large
stars
must
appear
displaced
towards
the
red
end
of
the
spectrum.*
We
now
examine the
course
of lightrays
in the static
gravitational field. By
the
special theory
of
relativity
the
velocity
of
light
is
given by
the
equation

dx21 
dx2

dx23
+
dx24
=
0
and
therefore
by
the
general theory
of
relativity
by
the
equation
ds2
=
guvdxudxv
=
0
. .
.
(73)
If
the
direction,
i.e.
the ratio
dx1
:
dx2
:
dx3
is
given,
equation
(73) gives
the
quantities
dx1 dx2 dx3
dx4
dx4
dx4
and
accordingly
the
velocity
(dx1/dx4)2
+
(dx2/dx4)2
+
(dx3/dx4)2
=
Y
[35]
[34]
defined
in the
sense
of
Euclidean
geometry.
We
easily
recognize
that the
course
of
the
lightrays
must
be
bent with
regard
to
the
system
of
coordinates,
if the
guv
are
not
con
stant. If
n
is
a
direction
perpendicular
to
the
propagation
of
light,
the
Huyghens principle
shows
that the
lightray,
en
visaged
in the
plane
(Y,
n),
has
the curvature

dy/dn.
We examine the
curvature undergone
by
a
ray
of
light
passing by a
mass
M
at the
distance
A.
If
we
choose
the
system
of coordinates
in
agreement
with the
accompanying
diagram,
the total
bending
of
the
ray (calculated positively
if
*
According
to
E.
Freundlich, spectroscopical
observations
on
fixed
stars
of
certain
types
indicate
the existence
of
an
effect of
this
kind,
but
a
crucial
test
of
this
consequence
has not
yet
been made.