48 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
furthermore,
Wv =
Av
(again
in the
special
coordinate
system
under
consideration).
The tensorial character
of
the
quantities Auv
in
(28a)
can
be understood
if
we
show
that
Auv
is
a
tensor,
provided
we
set in
(28a)
Au = WdQ/dx,
while
W
and
Q
are
scalars.
According
to
(28),
the
*
d2(f
dxßxv
"
E
are components
of
a
tensor,
due
to
(26)
and
(6);
likewise
are
the
9i|x
dcj)
a*m
dxv'
After
addition,
we
have the tensorial character
of
a
dx
*
df)
dx..
-E
fLV
*
dcj)
dxT/
Equation
(28a)
makes
a
tensor also from the four-vector
dQ/dxu
and
therefore,
as
proven
before,
from
any
covariant four-vector
Au.
This
concludes the desired
proof.
It is
easy
to
find the extension
of
covariant
tensors
of
any
rank after
we
derived
already
the extension of
a
tensor
of
rank
one.
According
to
(6)
and
(6a),
we can
represent any
covariant tensor
as
a
sum
of tensors
of
the
type
A
"
=
A^A^.-.A
OL^...
Ct[
«1 «2
(0
»/'
where the
Aav(v)
represent
covariant four-vectors. Due to
(28a)
we
first
have
.
!
-
£
lv
(V)
avs
dx
as a
covariant
tensor
of rank
two.
By
the rules of
outer
multiplication we multiply
it
[p. 1050]
with all
of
Aau(u),
except
the
componentAav(v),
and
thus
get
a
tensor
of
rank l
+
1
in
whose formation the index
v was
privileged.
In this
manner one can
form l tensors
by merely privileging
the indices
v
=
1,
v
=
2,
...
v
=
l
in
sequence. Finally, adding
them all
together, one
obtains the tensor of rank
(l +
1),
[18]
dA
aT...at
I-j
a{...a[S dx
axs
A
+
«2s
A
+
a]ra-i...al
• • •
(29)
This formula has
already
been found
by
Christoffel
and
produces, as
has been
said,
from
any
covariant
tensor
of
rank
l
another
one
of rank
(l +
1),
which
we
call the
"extension"
of
the former. All
differential
operations
on
tensors
can
be derived from
this
operation.