202 DOC. 32 INTEGRATION OF FIELD
EQUATIONS
{1}
+
-
-k|^v
+
=
E
a
/XV
«
^«la
=
a2 log yg
_
dxßdxv
E
aß
fjboc
ß
v/3
a
E
a
I
a
i
a
log
^
dx.
(1)
The braces denote here the well-known
Christoffel
symbols
(of
the second
kind),
Tuv
the covariant
energy
tensor of
matter,
T its associated scalar. The
equations
(1)
produce,
in
the
approximation
that interests
us,
and after
expansion, directly
the
equations
E
a2y\ia
«
dxvdxa
+
E a2Y
va
«
dxßdxa
_
£
ay
_
a2
a
dxa
Ey
dxßdxv
1
aa
~ 2kITmv -
^v£rl(2)
The last
term
on
the left-hand side
originates
from the
quantity
Suv,
which
can
be
solved
by
Yuv
=
Y'uv
+
W8uv,
(3)
when the
y'uv obey
the additional condition
£
=
0.
v
a*v
(4)
Substituting (3)
into
(2)
one
gets
for the left-hand side
[4]
{2}
ay
mv
_
a2
a
dxa
dxßdxv
xa
Ey'aa, a2^
+2
dx.dx..
^
a2^
_
4
a2^
M*
v
a
v
The contributions
from the
second,
third,
and fifth
terms
vanish
if
W
is chosen from
the
equation
Ey'ao
+
2W
=
0,
(5)
which
we
will
do.
Considering
this, one gets
in
place
of
(2)
[p. 690]
E^|yMv
a
dxa
or