DOC. 9 FORMAL FOUNDATION OF RELATIVITY 37
Mixed
tensors.
It is also
possible
to
form
tensors
of
second
and
higher
ranks
which
are
covariant with
respect
to
some
indices,
contravariant with
respect
to others.
They
are
called mixed tensors. A mixed tensor
of
rank two
is,
for
example,
A;
=
+ CßD\ (9)
Anti-symmetric
tensors.
Next to
symmetric
covariant and contravariant
tensors,
{2}
so-called
anti-symmetric
covariant and contravariant tensors
play
an
important
role.
It is
characteristic
of
them
that
components
whose
pair
of indices
are exchanged are
oppositely equal.
For
example, a
contravariant tensor
Auv
which satisfies the
condition
Auv
=
-Avu is
called
an anti-symmetric
contravariant tensor
of
rank
two, [p.
1038]
or
also
a
six-vector
(because
it has
12 non-zero components which,
in
pairs,
have
equal magnitudes).
A contravariant tensor
Auvy
of
rank three is
anti-symmetric
if it
satisfies the conditions
AuvA
=
-Auv
=
-AvzA
=
Avu
=
-Auv
=
Auv.
One realizes that
(in a
continuum
of
four
dimensions)
this
anti-symmetric
tensor
has
only
four
numerically non-zero components.
With formulas
(5a)
and
(8),
resp., one
can
easily
show that this definition has
a
meaning
which is
independent
of
the choice
of
the
system
of
reference.
According
to
(5a)
we
have, e.g.,
A,
U
K
*
Replacing
AaB
with
-Aßa
(which
is
permissible by
the
hypothesis made)
and
then
exchanging
the
summation indices
a
and ß
in
the
double-sum,
one gets
A
'
=
V
^XßA
=
-A'^
Vfl
"
U
K
dx'v
as
has been claimed. The
proof
for
contravariant
tensors
and
tensors
of rank three and
four
is
analogous. Anti-symmetric
tensors
of
higher
than fourth rank
cannot
exist in
a
four-dimensional continuum because all
components
with
two
identical indices
vanish.
§5.
Multiplication
of Tensors
The outer product of tensors.
We have
seen
that
by
multiplication
of
the
components
{3}
of
a
tensor
of rank
one,
one
obtains the
components
of
tensors
of
higher
rank
(see
equations 6,
8,
and
9).
In
analogy
we
can always
derive tensors
of
higher
rank from
those
of
lower rank
by multiplying
the
components
of
one
tensor
with those of
another.
If,
for
example,
(AaB)
and
(Byua)
are
covariant
tensors,
then
(AaB
.
Buyv)