DOC. 30 FOUNDATION OF GENERAL RELATIVITY
171
Therefore,
in
order to demonstrate that
Auv
is
a
tensor if
any
covariant vector
is
inserted
on
the
right-hand
side
for
Au,
we only
need show
that this
is
so
for
the vector
Su.
But
for
this latter
purpose
it is
sufficient,
as a glance
at the
right-
hand
side of
(26)
teaches
us,
to
furnish the
proof
for
the
case
$
Now
the
right-hand
side
of
(25)
multiplied by
Y,
, *2t ,
,,*/ *
äiÄ
-
^
T+5r,
is
a
tensor.
Similarly
~XM ~Xv
being
the outer
product
of two
vectors,
is
a
tensor.
By
ad-
dition,
there
follows
the tensor character
of
Z)
-
1/h',
4
As
a glance
at
(26)
will
show,
this
completes
the demon-
stration
for
the
vector
YoQ/oxu
and
consequently,
from what
has
already
been
proved,
for
any
vector Au.
By
means
of
the extension
of
the
vector, we may easily
define
the
"extension"
of
a
covariant tensor
of
any
rank.
This
operation
is
a
generalization
of
the extension
of
a
vector.
We restrict
ourselves
to the
case
of
a
tensor of
the
second
rank,
since
this
suffices to
give a
clear
idea
of
the
law of
formation.
As
has
already
been
observed, any
covariant tensor
of
the
second rank
can
be
represented
*
as
the
sum
of tensors
of
the
* By
outer
multiplication
of
the vector with
arbitrary
components
A11, A12,
A13,
A14
by
the
vector with
components 1,
0, 0, 0,
we produce a
tensor
with
components
A11
A12
A13
A14
0
0 0
0
0
0
0
0
0 0 0 0.
By
the
addition
of
four tensors of this
type, we
obtain the tensor
Auv
with
any
ssigned components.