DOC.
13
GENERALIZED THEORY OF RELATIVITY
175
(10)
T
•
=
V
t
,
-A.
, , ,
, l\l2
•••
lk kxk2
l\k\ki
kßy
klk2...
k^
(b)
the contravariant
tensor
(11)
0
•
=
A
b t /o /
,
hl2",lx
k^k2...
k^
»i
i2
• •
•
ixk^k2...
k^
kik2...
k^
(c)
the mixed
tensor
(12)
/j
r2...
'''
5v
k^ ^2
•
*
r\*2"'^n k\
'''
k\
'''
S\
'
k\k2...
kx
or,
with
complete generality, subsuming
the
three
cases
(a)
to
(c)
d)
^r,
• • •
r"
*«,
• • •
,oWl
=
2
'
V**''"'h
V
••*!*%•••
««/* V'r-
^1
The
designations
"inner and
outer
product,"
which
are
taken
from
ordinary
vector
analysis,
are
justified
because,
when
all is
said and
done,
those
operations prove
to
be
special
cases
of
the
operations
considered here.
If
in
cases (a) or (b)
the rank
A
is
equal
to
zero,
then the
inner
product
is
a
scalar.
4.
Reciprocity
of
a
covariant and
a
contravariant
tensor.
From
a
covariant
tensor
of
rank
A
one
forms the
reciprocal
contravariant
tensor
of rank
A
through
A-fold
inner
multiplication by
the contravariant
fundamental
tensor:
(13)
®hh-x=
E
v
k\k2...
kx
from which
we
obtain
(14)
THh-ix =
.
£
W2V"-V,"0
k\k2...
kx'
^1
&2
•
• *
^
X
Hence,
one
obtains
a
scalar from
a
tensor
by
multiplying
the latter
by
its
reciprocal
tensor
according
to
the formula
(15)
Y
T
. .
•©.
v
1i]i2...tk
wi,i2...ir...
A
covariant
(contravariant)
tensor
of
rank
one
(four-vector
for
n
=
4)
has the
invariant
£y*
T,Tk
ik
or, respectively,
£g*
0A-
ik
In the
customary theory
of
relativity,
contravariance
is
identical with
covariance,
and the above invariant becomes the
square
of the absolute value of the four-vector
T2
j2
+
T2
+
T}.I
x x 1
y
x
z
A
covariant
(contravariant) tensor
of rank
two
has the invariant
[53]