DOC.
9
FORMAL
FOUNDATION
OF RELATIVITY
73
looking
for will have
a
strong
correlation between the
tensors Suv
and
Zvo
because
we
already saw, following
after
(42a),
that the
energy
tensor Zvo
is
decisive for the
action of the
gravitational
field
upon
matter. It is therefore natural
to
assume
the
desired
equations
as
Sot=KEot.
(74)
k
is here
a
universal
constant
and
Zot
is
the
symmetric
covariant
V-tensor,
associated
with the mixed
energy
tensor Zvo
by
the relation
and
%ar =
E
Äir
$aV
=
E$-^VT
resp.
(75)
The
determination of
the
function
H.
The
equations we are
looking
for
are
not
yet completely given
insofar
as we
have not
yet
determined the function
H.
Presently,
we only
know H
to
depend
solely
upon
the
guv
and the
guvo,
and to be
a
scalar
under
linear
transformations.7 A further condition that H must
satisfy
is found in the
following
manner.
The V-four-vector
(Ro)
of
the force
density
vanishes in
(42a)
if
Zvo
is
the
energy
tensor of all the material
processes
in the domain under consideration.
Equation (42a)
then states that the
divergence
of
the
energy
tensor Zvo
of
the material
processes
vanishes;
and the
same applies-according
to (74)-to the tensor
Sor
or resp.
for the
mixed
V-tensor
Svo
to
be formed from
it.
Consequently, every gravitational
field
must
satisfy
the relation
(see (41b)
and
(34)):
[p. 1075]
j^(gTV®J +
|E
vr
OXv
2
aguv
E
By means
of
(73)
and
(65a),
this relation
can
be
brought
into the form
"
dx..
-B=0,
(76)
where
s:
-
e
\LT
8
vr
dg07
g
vr
dH\pg
1
err
-
i«r
dg!r
(76a)
7We
would not have found
the
expression (65a)
for
Bu
without the latter
limitation,
which
we
introduced in
§14.
The consideration
given
in the
following
text would fail
if
we
dropped
this limitation. This fact is the
justification
for its introduction.