DOC.
13
GENERALIZED THEORY OF RELATIVITY
165
Each of these
operators yields again
a
tensor
of the
same
kind
(w. resp.
to
linear
transformations).
With the
application
of these abbreviations the
identity (12)
assumes
the form
(12a)
E
{v^
V
}
=
|E
{-
Amv
(Y) +
KÖ^V
},
/xv
v
/xv a
or
also
(12b)
E
'Y/iv
}
=
|E ^{"^v
fe) +
*-tMV
}.
/XV
/xv ^"*7
If
we
write the conservation law
(10)
for
matter
and the conservation
law
(12a)
for the
gravitational
field
in
the form
(10)
=
0
/XV
UAV
/XV
UAcr
Ei^-vV.)
-
fE^'lf'»
/XV
^v
^
jU-V
UA/x
/XV
2k
%^~8'
dxDo-
/xv
(12c)
then
one recognizes
that the
stress-energy
tensor
üuv
of the
gravitational
field
enters
the conservation law for the
gravitational
field in
exactly
the
same
way
as
the
tensor
0uv
of the material
process
enters
the conservation law for this
process;
this
is
a
noteworthy
circumstance
considering
the difference
in
the derivation of the
two
laws.
From
equation (12a)
follows the
expression
for the differential
tensor
entering
into the
gravitational equations
(17)
rMv
=
MY)
"
K'V
Thus,
the
gravitational
equations
(11) are
of the form
(18)
MY)
= K(@,v+
V)-
These
equations satisfy
a
requirement that,
in
our
opinion,
must
be
imposed
on
a
relativity theory
of
gravitation;
that is
to
say, they
show that the tensor
Uuv
of the
gravitational
field
acts
as a
field
generator
in the
same
way
as
the
tensor
0uv
of the
material
processes.
An
exceptional position
of
gravitational energy
in
comparison
with
all
other kinds of
energies
would lead
to
untenable
consequences.
Adding
equations
(10)
and
(12a)
while
taking
into
account
equation (18),
one
finds
[32]
[33]