182
DOC.
13
GENERALIZED THEORY OF RELATIVITY
£hh
•••'/,
^
^
^r\r2--
rn
^i\r\^i2r2
*"
Yinrn9
r\r2•••
^
£'V2---'n
^1*2-'/.
^
Ylr,
Y2r2
Ynr/
r\r2
" •
rn
But since the determinant of the normalized subdeterminant
yik
is
1
IY,kN
g
it follows that
5,,
(39)
e,
;
'l
'2
"•
ln
ln
fg
The
significance
of the covariant
(contravariant)
discriminant
tensor
consists
in
the fact that its inner
multiplication by
a
contravariant
(covariant)
tensor
of rank
X
yields
a
tensor
of rank
(X
-
n)
of the
same
kind,
where the
tensor
will be of the
opposite
kind if
X
-
n
is
negative. (Complement
of the
tensor.)
If
n
=
4,
then there
are
only special
tensors
up
to
rank
four,
since
all
special
tensors
of
higher
rank vanish
identically.
The
nonvanishing
components
of
a special
tensor
of rank four
are
all
equal
to
one
another
or
equal
and
opposite. Complementation
(inner
multiplication
by
the contravariant discriminant
tensor)
yields a
scalar,
so
that the
differential
operations
that
may
be
carried
out
on a special
tensor
of the fourth rank
are
thereby
reduced
to
differential
operations
on a
scalar.
The
complement
of
a
special
covariant
tensor
of third rank is
a
contravariant
vector
of the first kind.
The
complement
of
a
special
covariant
tensor
of second rank
is
a
contravariant
special
tensor
of second rank.
Finally,
the
forming
of the
complement
of
a special
covariant
vector
of the first
kind leads
to
a
contravariant
tensor
of third rank.
The
investigation
of
the
influence of the
gravitational
field
on physical processes
(Part
I,
§6)
demands
a more
detailed
treatment
of the
special
tensors
of second rank
(six-vectors).
If
®uv
is
a
special
tensor
of second
rank,
then
its
divergence (formula
35)
-v'
1
a
Ivici
dx
VK~1~J
reduces,
because
of
©
=
-© 0 =0
WVK
UKV'
UVV
to