100
DOC. 21 GENERAL RELATIVITY
But
according
to
(24)
l.c. and
(24a)
l.c.
s1/2Eas
ST 'dgsa
+
^8ra
dg:
dx"
dx.
d&r
)
_
Í
csa
^£sa
_
d(lgyf-g)
(6)
dxaj
2^8
dxT
dxT
'
And this
quantity
has the characteristics
of
a
vector,
due
to
(3).
Consequently,
the last
term
on
the
right-hand
side
of
(5)
is
itself
a
contravariant
tensor
of
rank
l.
We
are
therefore entitled
to
replace (5)
by
the
simple
definition
of
divergence,
viz.,
[ST]
A
a,...a,s
A
«!•••«/
_
V"' OA
+
'
S
dxs
TLN
and
we
shall do
so
throughout.
For
example,
the definition
(37)
l.c.
T»2-”«|S
+
....
ST
a,
A
_
jTS
(5a)
$
yR»
K
has
to
be
replaced
by
the
simpler
definition
dA**
9
=
E
M
ÔV
(7)
and
equation (40)
l.c. for the
divergence
of
the
contravariant six-vector
by
the
simpler
dA^
^
=
E
dx"
(8)
In
place
of
(41a)
l.c.
we
have,
due
to
our assumption,
A*
=
E
dx"
-ÍE*
~
/1VT
T/I
dx"
'/IV. (9)
[p.
781]
A
comparison
with
(41b)
reveals that under
our assumption
the law
of
divergence
is
the
same as
that for the
divergence
of
V-tensors
in
the
general
differential calculus.
This remark
applies
to
any divergence
of
tensors, as can
be derived from
(5)
and
(5a).
3. Our
limitation to transformations
of
determinant
1
brings
the
farthest-reaching
simplification
for those covariants which
are
formed
only
from the
gmv
and their
derivatives. It is shown in mathematics that these covariants
can
all be derived from
the
Riemann-Christoffel
tensor
of
rank
four,
which
(in
its covariant
form)
reads: