164 DOC.
30
FOUNDATION OF GENERAL
RELATIVITY
The
Contravariant Fundamental Tensor.If
in
the deter
minant
formed
by
the elements
guv we
take the
cofactor
of
each
of
the
ghV
and divide
it
by
the determinant
g
=
 guv

,
we
obtain certain
quantities guv(=
guv)
which,
as
we
shall
demonstrate,
form
a
contravariant
tensor.
By
a
known
property
of
determinants
guogvo =
....
(16)
where the
symbol
guv
denotes
1 or 0, according as u
=
v or
u
+
v.
Instead
of
the
above
expression
for
ds2 we
may
thus write
guodovdxudxv
or,
by
(16)
guyrg^dxfj.dxy.
But,
by
the
multiplication
rules
of
the
preceding paragraphs,
the
quantities
dsa
=
guodxu
form
a
covariant
fourvector,
and in fact
an
arbitrary
vector,
since
the
dxu are
arbitrary. By introducing
this into
our ex
pression we
obtain
ds2
=
gordsodsr.
Since this,
with the
arbitrary
choice of
the vector
dso,
is
a
scalar,
and
gor by
its definition
is
symmetrical
in the
indices
o
and
r,
it
follows
from the results
of
the
preceding
paragraph
that
gor
is
a
contravariant tensor.
It further
follows from
(16)
that
du
is also
a
tensor,
which
we
may
call
the
mixed
fundamental
tensor.
The
Determinant
of the
Fundamental
Tensor.By
the
rule
for
the
multiplication
of
determinants
I
guagav
I
=
I
ft«
1
*
I
9a"
I
.
On
the other
hand

guagav
 =

dvu

=1.
It therefore
follows
that
I
9r
I
x

gr
 =
1
...
(17)
The Volume
Scalar.We
seek first
the law
of
transfor