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
energy-tensor
of
"matter"
is
almost
exclusively
defined
by
the
density
of
matter
in the
narrower
sense,
i.e.
by
the
second
term
of
the
right-hand
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