DOC. 30 FOUNDATION
OF GENERAL RELATIVITY
159
Levi-Civita,
we
denote the contravariant
character
by placing
[13]
the
index
above,
the covariant
by placing
it
below.
§
6.
Tensors
of
the Second and Higher Ranks
Contravariant Tensors.-If
we
form
all
the sixteen
pro-
ducts
Auv
of
the
components
Au
and
Bv
of two
contravariant
four-vectors
Auv
=
AuBV
....
(8)
then
by (8)
and
(5a)
Auv
satisfies
the law
of
transformation
A'or
=
.
. .
(9)
dxu dxv
We
call
a
thing
which
is
described
relatively
to
any system
of
reference
by
sixteen
quantities, satisfying
the
law of trans-
formation
(9),
a
contravariant
tensor of
the
second rank. Not
every
such tensor allows
itself
to be formed
in
accordance
with
(8)
from
two
four-vectors,
but
it
is
easily
shown
that
any
given
sixteen
Auv
can
be
represented
as
the
sums
of
the
AuBv of
four
appropriately
selected
pairs
of four-vectors.
Hence
we can
prove
nearly
all
the
laws
which
apply
to
the
tensor
of
the
second
rank
defined
by (9)
in the
simplest
manner
by
demonstrating
them
for
the
special
tensors
of
the
type
(8).
Contravariant
Tensors
of
Any
Rank.-It
is clear that,
on
the
lines
of
(8)
and
(9),
contravariant tensors of
the
third
and
higher
ranks
may
also be defined
with
43 components,
and
so
on.
In
the
same
way
it
follows
from
(8)
and
(9)
that
the
contravariant
four-vector
may
be
taken in this
sense as a
contravariant
tensor
of
the
first rank.
Covariant Tensors.-On the other
hand,
if
we
take the
sixteen
products
Auv
of two covariant four-vectors
Au
and
Bv,
Auv
=
AuBv,
....
(10)
the
law of
transformation
for
these
is
A'
~
day
. . .
(11)
This law
of
transformation
defines
the covariant tensor
of
the
second
rank.
All
our previous
remarks
on
contravariant
tensors
apply
equally
to
covariant
tensors.