170
DOC.
30
FOUNDATION OF GENERAL
RELATIVITY
and
since
by (23)
and
(21)
{uv,
t}
is
symmetrical
in
u
and
v,
it
follows
that the
expression
in
brackets
is
symmetrical
in
u
and
v.
Since
a
geodetic
line
can
be
drawn in
any
direction
from
a
point of
the
continuum,
and therefore
dxu/ds
is
a
four-
vector with the ratio
of
its
components arbitrary,
it
follows
from
the results
of
§
7
that
A*"
=
~
{flv'
t}ö£
. . .
(25)
is
a
covariant tensor
of
the
second
rank. We
have
therefore
come
to this result: from the covariant
tensor
of
the first
rank
_
d/
Au
=
we
can,
by
differentiation,
form
a
covariant
tensor of
the
second rank
Am" =
^t}At
. .
.
(26)
We
call
the tensor
Auv
the
"extension" (covariant derivative)
of
the
tensor
Au
In the
first
place we can
readily
show
that
the
operation
leads to
a
tensor,
even
if
the
vector
Au
cannot
be
represented
as a
gradient.
To
see
this,
we
first observe
that
"1$
is
a
covariant
vector,
if
yjr
and
j
are
scalars.
The
sum
of
four
such
terms
8,.
^+.+.+
is
also
a
covariant
vector,
if
yjr(1),
£(1)
. . .
\jr(4),
f(4)
are
scalars.
But it
is clear
that
any
covariant vector
can
be
represented
in the
form
Su.
For, if
Au
is
a
vector whose
components
are
any
given
functions
of
the
xv, we
have
only
to
put (in
terms
of
the
selected
system
of
co-ordinates)
i|r(1)
=
A1,
£(1)
= X1,
*(2)
-
A2,
f(2)
=
x2,
y/r(3)
=
A3,
*(4)
=
A4,
/(4)
=
x4,
in order
to
ensure
that
Su
shall
be
equal
to
Au.