DOC. 30 FOUNDATION OF GENERAL RELATIVITY 177
This
expression suggests
forming
the
tensor
Auot
-
Auro.
For, if
we
do
so,
the
following
terms
of
the
expression
for
Auot
cancel
those
of
Auto,
the
first,
the
fourth,
and the
member
corresponding
to the last
term
in
square
brackets;
because
all
these
are
symmetrical
in
o
and
t.
The
same
holds
good
for
the
sum
of
the
second and
third
terms.
Thus
we
obtain
.
(42)
where
=
-
b;t{1U7~ p}
+
~1
-
{tw, a}~ar,
p}
+
{fir,
z}{ar,
p}
(43)
The essential
feature
of
the result
is that
on
the
right
side of
(42)
the
Ap
occur
alone,
without their
derivatives.
From the
tensor
character
of
Auot
-
Auro
in
conjunction
with the
fact
that
Ap
is
an
arbitrary vector,
it
follows,
by
reason
of
§
7,
that
Bpuot
is
a
tensor
(the
Riemann-Christoffel
tensor).
The mathematical
importance of
this tensor
is
as
follows
:
If
the continuum
is
of
such
a
nature
that
there
is
a
co-ordinate
system
with
reference to
which the
guv
are constants,
then
all
the
Bpuor
vanish.
If
we
choose
any
new
system
of
co-
[21]
ordinates
in
place
of
the
original ones,
the
guv
referred
thereto
will
not
be constants,
but
in
consequence
of
its
tensor
nature,
the transformed
components
of
Bpuot
will
still vanish
in the
new system.
Thus the
vanishing
of the Riemann
tensor is
a
necessary
condition
that,
by
an
appropriate
choice
of
the
system
of
reference,
the
guv
may
be constants. In
our
problem
this
corresponds
to the
case
in
which,*
with
a
suitable
choice of
the
system
of
reference,
the
special
theory
of
relativity
holds
good
for
a
finite
region
of
the
continuum.
Contracting
(43)
with
respect
to
the
indices
t
and
p
we
obtain
the
covariant
tensor
of second
rank
*
The mathematicians have
proved
that
this
is
also
a sufficient
condition.