162
DOC.
30
FOUNDATION
OF GENERAL RELATIVITY
The
proof
that the result
of contraction
really
possesses
the
tensor
character
is
given
either
by
the
representation of
a
tensor
according
to
the
generalization
of
(12)
in
combination
with
(6),
or
by
the
generalization
of
(13).
Inner
and
Mixed
Multiplication
of
Tensors.-These
consist
in
a
combination
of
outer
multiplication
with contraction.
Examples.-From
the covariant
tensor
of
the
second
rank
Auv
and the contravariant tensor
of
the first rank
Bo
we
form
by
outer
multiplication
the
mixed
tensor
Douv =AuvBo.
On
contraction with
respect
to the
indices
v
and
o,
we
obtain
the
covariant four-vector
Du
=
Dvuv =
This
we
call
the
inner
product
of
the
tensors
Auv
and Bo.
Analogously we
form
from
the
tensors
Auv
and
Bor, by
outer
multiplication
and double contraction,
the
inner
product
AuvBuv.
By
outer
multiplication
and
one
contraction,
we
obtain from
Auv
and
Bor
the mixed
tensor
of
the
second
rank
= AuvBVT.
This
operation may
be
aptly
characterized
as
a
mixed
one,
being
"outer" with
respect
to
the
indices
u
and
t,
and
"inner"
with
respect
to
the
indices
v
and
o.
We
now prove a
proposition
which
is often useful
as
evi-
dence of
tensor
character.
From
what has
just
been
ex-
plained,
AuvBuv
is
a
scalar
if
Auv
and Bor
are
tensors.
But
we
may
also
make the
following
assertion: If
AuvBuv
is
a
scalar
for
any
choice
of
the tensor
Buv,
then
Auv
has
tensor
character.
For,
by hypothesis,
for
any substitution,
A'orB'or
= AuvBuv.
But
by
an
inversion
of
(9)
-dmu'
Buv=
dx'o
dx'r.
This,
inserted in the
above
equation,
gives
(A'or
-
0.
This
can only
be satisfied
for
arbitrary
values
of
B'or if
the