DOC. 9 FORMAL FOUNDATION OF RELATIVITY
35
§4.
Tensors of
Second
and
Higher
Ranks
[9]
Covariant
tensors
of
second
and
higher
ranks. Sixteen functions
Auv
of the
coordinates
are
called the
components
of
a
covariant tensor
of
rank two
if
the
sum
Y^A^dx®
=
3 (5)
/XV
is
a
scalar;
here
dxu(1)
and
dxv(2)
denote the
components
of
two
arbitrarily
chosen line
elements.
This
yields
the relation
Y,^W
-
E
/xv

aßfiv
OX
^
CfX
v
and
considering
that it is
supposed
to hold for
arbitrarily
selected
dxu(1)
and
dxv(2)',
one
also
gets
the sixteen
equations
sbrrb
(5a)
This
equation
is
again equivalent
to the
definition
above.
Obviously,
covariant tensors
of
third and
higher
ranks
can
be defined in
an
[p.
1036]
analogous manner.
The
symmetric
covariant tensor.
If
a
covariant tensor satisfies for
one
coordinate
system
the condition that two of its
components
which differ
by
a
mere exchange
of
indices
are equal
(AaB =
Aßa),
then
this
holds-as
a
glance
at
(5a)
shows-for
every
other coordinate
system.
In this
case,
the sixteen transformation
equations
of
a
covariant
tensor
of
rank two reduce to ten.
If
Auv
=
Auv,
the
tensorial
character of
(Auv)
can
be
proven
merely by showing
that
E^/^AAv
=
®
(5c) {1}
/IV
is
a
scalar.
Taking
(5a)
into
account,
this follows from the
identity
=
£Aaßdxadxß
=
E
A«ßib^}dx'»dxlv-^"*vi±®
/xv

aßfiv
Symmetric
covariant tensors
of
higher
rank
can
be defined in
complete analogy.
The
covariant
fundamental
tensor.
The
quantity
= ESMvV**v
has
a
special
role in the
developing theory;
we
shall call this
quantity
the
square
of
the
line element. From what has been
previously said,
it follows that
guv
is
a
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