DOC. 30 FOUNDATION OF
GENERAL RELATIVITY
173
From
(28),
it
follows
that
Vjll
.
4J2fl^Z
|2=r
.
(29)
-
g bxr bxr
*
bxa ^X
Further,
from
guogvo
=
gvu,
it
follows
on
differentiation that
g^dg"
- -
g'"'dg»*
V
= _
9^-^z
9
dXA
* dxA
j
(30)
From
these, by
mixed
multiplication
by
got
and
gvA
re-
spectively,
and
a
change
of
notation for
the
indices, we
have
dg*v
=
-
g"ag^
dgaß
V
=
_
gliagvß*
.
(31)
bXr
*
7Xa
and
dg^
=
-
g^gvß dgaß
Vi-
• • •
(32)
"^7
g"a9"ß""
The relation
(31)
admits
of
a
transformation,
of which
we
also
have
frequently
to
make
use.
From
(21)
-
[*r, ß]
+
[ßr,
a]
.
(33)
Inserting
this
in
the
second
formula
of
(31), we
obtain,
in
view of
(23)
*9liv
dx,
-
v}
-
g"{rcr,
/*}
. .
(34)
Substituting
the
right-hand
side of
(34)
in
(29),
we
have
^
--2
-
fc«r,.
.
(29a)
J
-
g
Ct
The
"Divergence" of
a
Contravariant
Vector.-If
we
take the inner
product
of
(26) by
the contravariant funda-
mental
tensor
guv,
the
right-hand side,
after
a
transformation
of
the
first
term,
assumes
the
form
S^ra.)
-
a,}£
-
+
&
-
[16]
[17]
[18]