DOC. 9 FORMAL FOUNDATION OF RELATIVITY 47
a
-
df
"
dx
(26)
are, according
to
§3,
components
of
a
covariant four-vector
(tensor
of rank
one),
which
we
can suitably
view
as
the "extension"
of
the
scalar
Q
(a
tensor of rank
zero).
Furthermore,
we get
from
(25),
d2(f)
ds
=
E
a2^ dx^dx^
^
dx^dx^
ds ds
E
dP
d2xT
dx"
ds2
We
now
specialize
our
analysis by assuming
the line
S
to be
geodesic,
a
choice
which is
independent
of
the
adopted system
of reference.
According
to
(23b)
we
then
get
d2(j)
ds2
E
/XV
av
dxßdxv
E
/XV d(j)
dxß
dxv
dxr
«
ds ds
(27)
Now
we
focus
on
the
quantities
A
d24
E
MV
d(j)
dx'
(28)
which
satisfy
the conditions of
symmetry
Auv
=
Avu
according to
(24)
and
(24a).
Due to this and due to
(5c),
one
derives from
(27)
and from the
scalar character of
d2Q/ds2
that
Auv
is
a (symmetric)
covariant
tensor
of rank
two. We
can
view
Auv
as
the
extension of
the
covariant
tensor
of
rank
one,
Au
=
dQ/dxu,
and
therefore write
(28)
in
the
form
/XV
£
-
?
r
(28a)
One
can now expect
that not
only a
four-vector
of
type
(26)
but rather
any
covariant four-vector will become
a
covariant
tensor
of rank
two
when differentiation
(extension)
is
applied according
to
(28a).
We
will
verify
this next.
First,
it
is
easy
to
see
that the
components
Au
of
any
covariant four-vector in
a [p. 1049]
four-dimensional continuum
can
be
represented
in the form
A
=
+ +
*3-
+
-
¥la*"
¥2a*,
¥sa^
V4av
where the
quantities
WA
and
QA
are
scalars. In
order
to
satisfy
the
equation, we
equate arbitrarily
Qv
=
xv
(in
the
special
coordinate
system
under
consideration) and,