DOC.
9
FORMAL FOUNDATION OF
RELATIVITY
39
the
guv
the minors for
every
guv and divides them
by
the determinant
g
=
|guv|
of
those
guv,
one
obtains
certain
quantities
guv(=
gvu)
for which
we
want to
prove
that
they
form
a
contravariant
symmetrical
tensor.
From this definition it follows
next
that
=
s;,
a
where
guv
signifies
the
quantity
1
or
0
resp., depending upon u
=
v or u
#
v.2
[p.
1040]
Furthermore,
£
SaßdXcfaß
aß
is
a
scalar
which
we can
equate-according
to (10)-with
E
S^a^a^ß
aß\i
and also
equal
to
E
S^g^dxjlxp.
aß)±\
However, according
to the
previous
paragraph,
the
d^
=
£
#/A
ß
are
the
components
of
a
covariant
vector,
and
similarly
of
course
also the
dSv
=
EW*V
a
Our
scalar, therefore,
takes the form
E«MVv
fiv
It
can
be
easily proven
that
guv
is
a
contravariant
tensor;
this is based
upon
the
facts
that the
sum
above is
a
scalar,
the
dEu are
by
nature
arbitrarily
selectable
components
of
a
covariant
four-vector,
and
guv
=
gvu.
Note.
According
to
the
multiplication
theorem
of
determinants,
IE^"V|
=
18J
|gav|.
a
On the other
hand,
(10)
{5}
2According
to
the
previous paragraph,
guv
is
a
mixed tensor
("mixed
fundamental
tensor").