176
DOC. 30 FOUNDATION OF GENERAL
RELATIVITY
If, further,
the
tensor
Apo
is
symmetrical,
this reduces to
*9,
-
w
-
Had
we
introduced,
instead
of
Apo,
the covariant
tensor
Apo
=
gpagoßAaß,
which is also
symmetrical,
the last
term,
by
virtue
of
(31),
would
assume
the
form
i
/
W
\
^
9
lxß
In the
case
of
symmetry
in
question, (41)
may
therefore
be
replaced by
the two forms
(41a)
+
.
(41b)
which
we
have
to
employ
later
on.
§
12.
The Riemann-Christoffel Tensor
We
now
seek
the
tensor
which
can
be
obtained from the
fundamental tensor
alone,
by
differentiation.
At
first
sight
the solution
seems
obvious. We
place
the fundamental
tensor
of the
guv
in
(27)
instead
of
any
given
tensor
Auv,
and
thus have
a
new
tensor,
namely,
the
extension
of
the funda-
mental tensor. But
we easily
convince ourselves
that this
extension
vanishes
identically.
We reach
our goal,
however,
in
the
following
way.
In
(27)
place
K*
=
^
-
{/*?,
p}k",
i.e.
the extension
of
the four-vector
Au.
Then
(with
a some-
what
different
naming
of the
indices)
we
get
the
tensor of
the
third rank
VK
,
,7A0
,DAa
.
^A
M
+ [
~
p)
+
W,
a)
{aa,
p}
+
{tt,
a}{a/t,
p}]ap.