DOC. 30 FOUNDATION OF GENERAL
RELATIVITY
195
(T
=
1,
2,
3)
.
.
(67)
This
is
the
equation of
motion
of
the material
point
accord
ing
to Newton's
theory,
in
which
1/2g44
plays
the
part
of
the
gravitational
potential.
What
is
remarkable in this
result
is that the
component
g44
of
the fundamental tensor alone
defines,
to
a
first
approximation,
the motion
of
the material
point.
We
now
turn to
the
field
equations
(53).
Here
we
have to
take into consideration that the
energytensor
of
"matter"
is
almost
exclusively
defined
by
the
density
of
matter
in the
narrower
sense,
i.e.
by
the
second
term
of
the
righthand
side of
(58) [or, respectively,
(58a)
or
(58b)].
If
we
form
the
approximation
in
question,
all
the
components
vanish with the
one exception
of
T44
=
p
=
T. On
the
left
hand
side
of
(53)
the
second
term is
a
small
quantity
of
second
order;
the
first
yields,
to
the
approximation
in
question,
i»"'1]
+
23
+
4^'33"
4].
For
u
=
v
=
4,
this
gives,
with
the
omission of
terms
differ
entiated with
respect
to
time,

*($•
+
IST
+ 1ZJ
iV9"
The last
of
equations
(53)
thus
yields
VV44
=
kp
. • • •
(68)
The
equations
(67)
and
(68)
together
are equivalent
to
Newton's
law of
gravitation.
By
(67)
and
(68)
the
expression
for the
gravitational
potential
becomes
pdr
si
(68a)
while Newton's
theory,
with the
unit
of
time which
we
have
chosen, gives
KJpir