164 DOC.
30
FOUNDATION OF GENERAL
RELATIVITY
The
Contravariant Fundamental Tensor.-If
in
the deter-
minant
formed
by
the elements
guv we
take the
co-factor
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
four-vector,
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-
Previous Page Next Page