42 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
V££
SaßxllgaigßkgXl8'Jm
aßXfJL
which,
according
to
known theorems
of
the
theory
of
determinants,
equals
Ä/m£
8aßXßg
^g ^g
Ug
which after
(11),
1
A
iklm
fg
[p. 1043]
With this
we
have
proven
that
Gm"
-
-p««,.
(21a)
4g
is
a
contravariant
anti-symmetrical
tensor.
In the
theory
of
general anti-symmetric
tensors,
finally, a
mixed tensor has
an
important
role;
it is formed from the fundamental tensor
of
the
guv
and its
components are
G*
=
£
fg8ikaßga,gßm =
£
-^8imaßgaigßk.
(22)
aß
*ß
\Jg
The tensorial character of these
two
expressions
is evident from what has been
said above and in
§4.
One has
to
prove only
that
they
are
equal.
According
to (21)
and
(19),
the
latter
one can
only
be
brought
into
the form
[11]
£
\f88xßPagugiUng pagoßgaigßk
Xfipoaß
and after summation
over a
and ß,
considering
(10),
one gets
£
Xji
This
expression
deviates from the first
one
in
(22)
only
in the notation
of
the
summation indices and in the
(irrelevant) sequence
of the
index-pairs
xu
and
ik
in
gyuik.
The mixed tensor
Gimik
is
anti-symmetric
in its indices
i,k
as
well
as
in
l,m, as can
be
seen
from
(22).
With the
help
of
the fundamental
tensor,
and
following
the rules
given
in
§5, we
can produce
from
any
tensor others
of
different character. For
example,
we
can
transform the covariant tensor
(Tuv)
into the contravariant tensor
(Tuv)
according
to
the rule