52 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
continuum
with
a
given
line element is
Euclidean-in
other
words,
whether
or
not
it is
possible
to
select
a suitably
chosen substitution such that ds2 is
everywhere
equal
to the
sum
of
the
squares
of
the coordinate differentials.
By
twofold
extension,
we
form, according
to
(29),
from the covariant four-vector
Au
the tensor
of
rank three
(AMvi).
One obtains
&A
/xv
X
dxvdxx
-
E
dA
dx.
+
/XV
dA
dx
E
vA dA
dxr
a +
or
E
vA| \th\
a
E
a
dx
/xv
o
E
/xA
VT
o
From this follows
immediately
that
(A^v
-
A^vX)
is also
a
covariant tensor of
rank
three,
and
consequently
E
a
dx
/xv
a
d
dx
/xA
a
E
MV
AT
cr
/x
A
VT
CT
is
a
covariant tensor
cr
of
rank
three;
that
is,
the
square
bracket is
a
tensor
of
rank
four
(K°vX)
which is covariant in
n,
v,
A
and contravariant in
er.
All
components
of
this tensor
vanish when the
g^v
are
constants. This
vanishing happens always
when
it
occurs
relative
to
one suitably
chosen coordinate
system.
The
vanishing
of
the
bracketed
expression
for all index combinations is therefore
a
necessary
condition for
the
possibility
that the line element
can
be
brought
into Euclidean form.
However,
it
requires
further
proof
to show that the condition is
also
sufficient.
V-Tensors. A
glance
at
the formulas
(37), (39), (40), (41), (41a)
shows that tensor
components are
often encountered with the factor
y/g.
For this
reason
we
want
to
introduce
a
special
notation for tensor
components
which
are multiplied
with
yjg
(or
y/^g
in
case g
is
negative).
We write these
products
with German
(or Gothic)
letters and
set,
for
example
AM
=
«.
A:fg
=
{10}
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