DOC. 30 FOUNDATION
OF GENERAL RELATIVITY
175
by
their
values
as given by (34),
there results from
the
right-
hand
side
of
(27)
an
expression
consisting
of
seven
terms, of
which four
cancel,
and
there remains
k°
=
+
{ry'
a}AV"
+
{ay
^
•
(38)
This
is the
expression
for
the extension
of
a
contravariant
tensor
of
the
second rank,
and
corresponding expressions
for
the
extension of
contravariant tensors
of
higher
and lower
rank
may also be formed.
We note that in
an analogous way
we
may
also
form the
extension
of
a
mixed tensor
:-
Kr
=
^
-
Wp
+
{ffT,
a}kl
.
.
(39)
Ö2/(f
On
contracting (38)
with
respect
to the indices
ß and
a
(inner
multiplication by
$£),
we
obtain the
vector
A"
=
^
+
{ßy,
/3}A"
+
{ßy,
a}A.yß.
On account of
the
symmetry
of
{ßy, a}
with
respect
to the in-
dices
ß
and
y,
the third term
on
the
right-hand
side vanishes,
if
Aaß
is,
as we
will
assume,
an
antisymmetrical
tensor.
The
second term allows itself to be
transformed in accordance
with
(29a).
Thus
we
obtain
A«
=
~
gA'P)
. . .
(40)
J.
-
g *xß
This
is
the
expression
for
the
divergence
of
a
contravariant
six-vector.
The
Divergence
of
a
Mixed Tensor of
the
Second
Rank.-
Contracting
(39)
with
respect
to the
indices
a
and
a,
and
taking
(29a)
into
consideration,
we
obtain
J -gK-
Xy/~x?K)
-
{*?
rW
-
gK
•
(41)
If
we
introduce the contravariant tensor
Aoa
=
gptAoT
in the
last
term,
it
assumes
the form
-
[oyx,
pW
-
gA".