178
DOC.
13
GENERALIZED THEORY OF RELATIVITY
(24)
£y
dT dT
rs
dxr dxt
[59]
is
the first Beltrami differential
parameter
of the scalar
T.
To form the
divergence
of the
gradient,
one
has
to
form from its extension
T
=
d2T
E
O
-
rs
dxdx
k
l*
r
s
the scalar
£
yrs
Trs
rs
which
can
be
given
the form
[60]
(25)
1
y-s
9
/
v/gy
dT7rJ
Vg
«
dx
rs
dx
\
[61]
The
divergence
of the
gradient
is
the result of the
generalized Laplacian
operation
carried
out
on
the scalar
T,
and
is
identical with the second Beltrami
differential
parameter
of the scalar
T.
(b)
A
=
1.
Let the
starting
tensor
be
a
covariant
four-vector,
though
it
could
just
as
well be
a
contravariant four-vector.
According
to
(16),
the covariant extension
is
(26)
Trs
=
-IdT,
dx.
The
divergence
is
(27)
(
£y
rsTrs =
Ey
dT,
rs.
rs
dx_
rs
rsk
\
to
which
we
give,
in accordance with
(17),
the
form
(28)
yv
T
-yfj-iv
T)-^-T
-~v
V
(dSri,^9.i ^9r*\
rp
\
rs
rtkl
dsn)2-*)
If
one
eliminates
dy
rs
dx
with the
help
of the formula8
7See,
e.g., Bianchi-Lukat, Vorlesungen
über
Differential
geometrie
(1st ed.),
p.
47;
or
also the calculation
of
the
divergence
of
a
four-vector in the next
case (b).
8This
formula,
which
we
also
apply
in
establishing
the differential
equations
of the
gravitational
field in
§4,
is
proved
in
the
following way:
We have
E
8i
8u
=
8k
(0
or
O.
/
hence,