DOC.
30 FOUNDATION OF
GENERAL RELATIVITY
161
Auv
=
-
Auv,
....
(15)
or
respectively,
Auv =
-
Avu
....
(15a)
Of
the
sixteen
components
Auv,
the four
components
Auv
vanish;
the
rest
are
equal
and
of
opposite sign
in
pairs,
so
that there
are
only
six
components numerically
different
(a
six-vector).
Similarly we see
that the
antisymmetrical
tensor
of
the third rank
Auv
has
only
four
numerically
different
components,
while the
antisymmetrical
tensor
Auvor
has
only
one.
There
are no
antisymmetrical
tensors
of
higher
rank
than the fourth
in
a
continuum
of
four
dimensions.
§
7.
Multiplication
of Tensors
Outer
Multiplication of
Tensors.-We obtain from the
components
of
a
tensor
of
rank
n
and
of
a
tensor
of
rank
m
the
components
of
a
tensor of
rank
n
+
m by
multiplying
each
component
of
the
one
tensor
by
each
component
of
the
other. Thus, for
example,
the
tensors T
arise
out
of
the
tensors A and
B
of
different kinds,
Tuvo
=
AuvBo,
Tuvor
=
AuvBor
Toruv
= AuvBov.
The
proof
of
the
tensor character
of
T is
given
directly
by
the
representations
(8), (10),
(12), or
by
the
laws of trans-
formation
(9), (11), (13).
The
equations
(8), (10),
(12)
are
themselves
examples
of
outer
multiplication
of
tensors
of
the
first
rank.
"Contraction" of a
Mixed
Tensor.-From
any
mixed
tensor
we
may
form
a
tensor whose
rank
is less
by two,
by
equating
an
index
of
covariant with
one
of
contravariant
character,
and
summing
with
respect
to
this index
("con-
traction").
Thus,
for
example,
from
the
mixed
tensor
of
the
fourth rank
Aoruv,
we
obtain the
mixed tensor of
the
second
rank,
Arv
=
Auruv
(=
ZAuruv),
and from this,
by
a
second
contraction,
the
tensor of
zero
rank,
A
=
Avv
=
Auvuv.