DOC. 30 FOUNDATION
OF GENERAL RELATIVITY
163
bracket vanishes. The result then
follows
by
equation (11).
This rule
applies correspondingly
to tensors
of
any
rank and
character, and
the
proof
is
analogous
in
all
cases.
The
rule
may
also be
demonstrated
in
this
form
:
If
Bu
and
Cv are
any vectors,
and
if,
for all values of
these,
the
inner
product
AuvBuCv
is
a
scalar,
then
Auv
is
a
covariant
tensor. This latter
proposition
also holds
good even
if
only
the
more
special
assertion
is
correct,
that with
any
choice of
the
four-vector
Bu
the inner
product
AuvBuBv
is
a
scalar,
if
in addition it
is
known that
Auv
satisfies
the
condition of
symmetry
Auv
=
Auv.
For
by
the method
given
above
we
prove
the
tensor
character
of
(Auv
+
Auv),
and from
this the
tensor
character
of
Auv
follows
on
account of
symmetry.
This
also
can
be
easily generalized
to
the
case
of
covariant
and contravariant tensors
of
any
rank.
Finally,
there
follows
from what has
been
proved,
this
law,
which
may
also be
generalized
for
any
tensors:
If
for
any
choice
of
the
four-vector
Bv
the
quantities
AuvBv
form
a
tensor of
the
first rank,
then
Auv
is
a
tensor of
the
second
rank.
For,
if
Cu
is
any four-vector,
then
on
account
of
the
tensor
character
of
AuvBv,
the inner
product
AuvBvCu
is
a
scalar for
any
choice of the
two four-vectors
Bv
and
Cu.
From
which the
proposition
follows.
§
8. Some
Aspects
of
the
Fundamental
Tensor
guv
The
Covariant Fundamental Tensor.-In the
invariant
expression
for
the
square
of
the linear
element,
ds2
=
guvdxudxv,
the
part played
by
the
dxu
is
that
of
a
contravariant
vector
which may
be chosen at
will. Since further,
guV =
gvu,
it
follows
from the considerations
of
the
preceding paragraph
that
guv
is
a
covariant tensor
of
the
second
rank. We
call
it
the
"fundamental
tensor." In
what
follows
we
deduce
some
properties
of
this tensor
which,
it
is true,
apply
to
any
tensor of
the
second
rank. But
as
the fundamental
tensor
plays a special
part
in
our
theory,
which has
its
physical
basis
in the
peculiar
effects of
gravitation,
it
so
happens
that
the
relations
to be
developed
are
of
importance
to
us only
in the
case
of
the fundamental
tensor.
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