DOC. 9 FORMAL FOUNDATION OF RELATIVITY 43
=
£
Taßg^
(23)
aß
while vice
versa
one gets
TßV
=
£
T^8ailgßv.
(23a)
aß
The mutual
equivalence
of
equations (23)
and
(23a)
follows
easily by using
(10).
The
tensors
(Tuv)
and
(TuV)
are
called
"reciprocal."
If
one
of
two
reciprocal
tensors
is
symmetrical or anti-symmetrical,
then
so
is the
other;
this follows from
(23) and/or
(23a).
It also
applies
to tensors
of
any
rank.
Dual six-vectors.
If, furthermore,
(Fuv)
is
an anti-symmetric
tensor
(of
rank
two),
[p. 1044]
we can
form
a
second
anti-symmetric
tensor
Fuv*
with the
equation
FMV*
=
G%
Faß.
(24)
2
aß
Fuv*
is called the contravariant six-vector "dual" to Fuv. Vice
versa, Fuv
is dual
to
Fuv*. Because,
if
we
multiply (24) by
Gotuv
and
sum over u
and
v,
we get
\Y,g^F^
=
±£
G°ß
paß-
Z
aß/iv
But,
since
according
to
(22)
E
-
E
M»»« ss "
^
v
s,-. s,;
-
2(6: s;
-8? K
/iv /xvAkAV
yg
[see
footnote
3]
one finally gets
\
£
G$
Faß
=
\(F"
-
FTO) =
FOT
4aßiv
Z
from which
our
claim follows.
An
analogous
result
applies
to
covariant six-vectors. One
can
also
easily prove
that six-vectors that
are
reciprocal
to two dual
ones are
dual
themselves.
[13]
3The second
formulation
is
based
upon
the
fact
that
guvyk
differs from
zero only
when
all indices
are
different. This leaves
only
two
possibilities,
(y
=
y',
k
=
k')
and
(y
=
k',
k
=
y').
Considering this, one gets first, by summing over u
and
v,
2£f8Xo8KT8xagaß
-
8Xa8KT8iß8J,
Ak
where further summation needs to consider
only
the index combinations
(Ak)
with
A
#
K.
But,
since the
parenthesis
vanishes for
A
=
k anyway, one can go over
all
combinations.
Considering
(1),
one
then
gets
the
expression
given
in the text.
[12]