98
DOC.
3
STATICS
OF GRAVITATIONAL
FIELD
Since ß' vanishes and
we can assume
that
x
increases with
E,
it
follows from the first
equation
of the
first line that
A'
=
1,
hence,
if
we
ought
to
have
x
=
0 for
t
=
0
and
£ =
0,
then
A
=
x.
Finally,
from the third
equation
of the first line and the second
equation
of
the third
line,
we
obtain,
with the
help
of the relations
already found,
the differential
equations
2a'
-c'2
=
0,
2a
-
cc'
=
0.
From them
we
obtain,
if
we
denote the
integration
constants
by
c0
and
a,
c
= c0
+
ax,
2a
=
a(c0
+ ax)
=
ac.
Thus
we
have
the substitutions that
we were
seeking
for
sufficiently
small values
of
t.
If
we
neglect
the third and
higher
powers
of
t,
the
equations
1
=
*
+
T
(4)
\7)
= y
C
=
z,
T
=
Ct
hold,
where the
velocity
of
light
c
in
system
K,
which
can
depend only
on x
but
not
on t,
is
given by
the
relation
we
have
just
derived,
(5)
c
= c0
+
ax.
The
constant
c0
depends
on
the
rate
of the clock with which
we measure
the time
at
the
origin
of
K.
The
meaning
of the
constant
a
is obtained in the
following way.
The
first and fourth of
equations
(4)
yield
in
light
of
(5)
the
following equation
of motion
for the
origin
(x
=
0)
of K
2cq
Thus,
a/c0
is the
acceleration of the
origin
of K with
respect
to
2,
measured in the
units of time
in
which the
velocity
of
light
is
equal
to
1.
§2.
The
Differential Equation
of
the
Static Gravitational
Field, the
Equation
of
Motion
of
a
Material Point
in
the Static Gravitational Field
It
already emerges
from the
previous
paper
that there exists
a
relationship
between
c
and the
gravitational potential
in the static
gravitational
field,
or,
in other
words,
[10]