DOC.
13
GENERALIZED THEORY OF RELATIVITY
181
the discriminant
g
of the line
element.10
This discriminant transforms
according
to
the
equation
(36)
g'=P2.g,
where
P
=
I
Pile
I
=
dxi
dxk'
is
the functional determinant of
the substitution.
If
one assigns a specific sign
to
the
yfg
for the
original
reference
system
and
stipulates
whether this
sign
should
or
should
not
change
under
a
transformation,
depending
on
whether the substitution determinant
p
is
negative or
positive,
then the
equation
(37)
Jgr=P'fg
has
an
exact
meaning
with the inclusion of the
sign.
Now let the
8r,
be
zero
if
two
of the indices
are
identical with
one
another,
r\
r2...
rnr
but
±
1
if this
is not
the
case
and the
permutation
r1,r2,...rn
is
formed from the basic
permutation 1,2,
...
n by means
of
an even or
odd number of
interchanges
of
two
indices.
Then
(38)
e
=
8r
r r
'y/fl°
12"'
n
r\
r2
•
rn
are
the
components
of
a
special
covariant
tensor
of rank
n,
which
we
will call
the
covariant discriminant
tensor.
For
a
transformation
yields
first
[64]
e
r1r2
rn = Sr1r2...rn-\f¥ =
8Vl...r;PJS
but since
P
"
i2
• •
•
'/i
Pinn " ^rxr2...rn
^2
ln
'
Phr\
^2r2
'''
• • • • » •
lll2---
ln
l\l2-'-
ln
it
follows that
e
r{r2...rn
^
in
'
Pixrx
Pi2r2
*
* *
Pinrn'
hence, by
virtue of the definition
(38),
e
rxr2...rn
"
zl
'
P'\r\ Phr2
''' Pinrn '
• • •
'l
l2
*
•
•
ln
For the
reciprocal
contravariant
tensor
one
finds in
accordance with
(13)
eil\l2-"ln
i
i
=
Y
i
r
Y
i
r
Y
i
r
'
er
r
t-*
Z1
r\
'
l2r2
n n
r\r2
'
rnr
r\r2...
rn
10The
"system
e" of
Ricci and
Levi-Civita, l.c.,
p.
135.
[63]