174
DOC.
13
GENERALIZED THEORY OF RELATIVITY
a
contravariant
tensor of
rank
A
if
these
quantities transform according
to the
formulas
(7) ®r\
r r
= TT:
r
TTl
r
...
TT:
r
•©•
, ,
.
4
2'"rX
'
llrl
l2r2 lXrX
lll2-"lX
lll2--lx
III. The
totality
of
a system
of
functions
2^
/
/k]k2
k
of the
variables
x
shall
be
called
a
mixed tensor, covariant
of
rank
u,
contravariant
of
rank
v,
if these
quantities
transform according
to the
formulas
(8)
^
W
-r,,/.,
,,
• •
^
Ph
r,
Pi,
r, •
• •
Pifl
rfl
'
^3,
V"*Vv'
tx
f
a
...
iß
• • •
Jc
^
From these definitions and
equations
(4)
and
(5)
it
follows
that:
The
quantities guv
form
a
covariant
tensor
of the second
rank,
and the
quantities
Yuv
a
contravariant tensor of the second
rank;
in
the
case n
=
4, they
form the
fundamental
tensors of the
gravitational field.
According
to
equation
(3),
the
quantities
dxi
form
a
contravariant
tensor
of rank
one.
Tensors of rank
one are
also called
vectors of
the
first
kind
or
four-vectors
for
n
=
4.
The
following algebraic
tensor
operations
follow
immediately
from the definition
of
the
tensors:
1.
The
sum
of two tensors of the
same
kind of rank
X
is
again
a
tensor
of the
same
kind of rank
A,
the
components
of
which
are
formed
by
the addition of the
corresponding components
of the two
tensors.
2. The outer
product
of
two
covariant
(contravariant) tensors
of rank
A
or
u
is
a
covariant
(contravariant)
tensor
of rank
A
+ u
with the
components
(9)
T
ib b -
A B
,
i , ^2'" 'a
I
2
* *' /x
'l'
'A
1
2
*
* *
fi
or,
respectively,
(9')
ixklk2".
kß
~ ®ixi2~.ix
^k{k2...
k/
3.
We
designate
as
the inner
product
of two tensors
(a)
the covariant
tensor
[52]
4Thus,
our
covariant
(contravariant)
tensors of rank
A
are
identical with the "covariant
(contravariant)
systems
of
Ath
order"
of
Ricci and Levi-Civita and
are
denoted
by
these
r
T v
authors
by
Xr1r2...r
and
Xr1r2..r,
respectively.
However
many advantages
the latter
notation
may
offer,
complications
in
more complex equations
have, nevertheless,
forced
us
to
choose the above notation and thus
to
denote covariant
tensors
by
Latin
letters,
contravariant
tensors by
Greek
letters,
and mixed
tensors
by
Gothic letters. Covariant and
contravariant
tensors
are
special
cases
of the mixed tensors.