DOC.
13
GENERALIZED THEORY OF RELATIVITY
177
rs rs
[
and
\
f
are
the Christoffel three-index
symbols
of the first and second
kind,
u
respectively; solving equations
(17), one
finds that
(19)
6
rs
u
If
one
replaces
the covariant
tensors
in
equation
(16)
by
the contravariant
tensors
reciprocal to them,
one
obtains
as
the "contravariant extension"
(20)
€1
r1r2~• r~a
0;
_________ __
rik~
iilcj
S
&
?ai
+1rI
iT!
J
J
ørzrs...k)
II.
We define
as
the
divergence
of
a
covariant
(contravariant) tensor of
rank
X
the
covariant
(contravariant)
tensor of
rank
X
-
1
obtained
by
inner
multiplication
of
the
extension
by
the
contravariant
(covariant) fundamental
tensor.
Thus,
the
divergence
of the covariant
tensor
T
"
is
the
tensor
(21)
T
sr.
and the
divergence
of the contravariant
tensor
©r
"
is
the
tensor
°
r1
r2
. . .
rX
(22)
srx
The
divergence
of
a
tensor
does
not
follow from this
uniquely;
in
general,
the result
changes
if
r1
in
equations
(21)
and
(22)
is
replaced
by one
of the indices
r2,
r3
...
rx.
III.
We designate
as
the
generalized
Laplacian operation
on a
tensor
the
successive
forming of
the
extension and
the
divergence.
Hence, the
generalized
Laplacian operation
makes
it possible
for
a
tensor of the
same type
and rank
to be
derived
from
a
given
tensor.
Of
special
interest
are
the
cases
X
=
0,
1,
2.
(a)
X =
0.
The
starting
tensor is
a
scalar
T,
which
we
can
consider
as a
covariant
or
contravariant
tensor
of rank
0.
(23)
Tr =
oxr
is
the covariant extension of the scalar
T,
i.e.,
a
covariant
tensor
of rank
one
(covariant
four-vector for
n
=
4),
which
is
called the
gradient
of the scalar.
The
invariant
6On
the basis
of
these formulas
one can
easily prove
that the
extension
of
the
fundamental
tensor
vanishes
identically.