118 DOC.
4
THEORY OF STATIC GRAVITATIONAL
FIELD
[27]
(3b)
cAc
-
1
(grade)2
=
kc2o,
which
can
also
be
brought
into the form
(3b')
A(c)
=
k
fco,
where
o
denotes the
density
of the
ponderable
matter,
or
the
density
of the
ponderable
matter
multiplied
by
the
energy density as
measured
by pocket
instruments. From these
equations
it
follows that
«
de dxx
dxy
dxz
=
"*3-
=
+ etc.,
dx dx
dy
dz
[28] (5)
•
where
c*x
=
11"
i
(gradc)2,
ckx
=
i
f
•
«
=
i
etc.
Thus,
the
reaction
principle
is indeed satisfied. The
term
added
to
equation
(3b)
in
order
to
satisfy
the reaction
principle
wins
our
confidence thanks
to
the
following
argument.
If each
and
every energy density
(oc)
produces
a
(negative) divergence
of the
gravitational
lines
of
force,
then this
must
also hold for the
energy density
of
gravitation
itself.
If
one
writes
(3b)
in the form
Ac
=
kico
+
1
-grad2_c
2k c
one immediately
sees
that the second
term
within the bracket is
to
be viewed
as
the
energy
density
of
the
gravitational
field.4
It remains
now
to
show that
only
this
term
denotes the
energy density
of the
gravitational
field
according
to
the
energy principle
as
well.
To that
end,
we
imagine
a
spatial arrangement
of
ponderable
masses
(density o)
situated in
a
finite
region
and
enclosed
by
an
infinitely
distant
surface;
let
c
tend
to
a
constant
value
at
infinity,
insofar
as
this
is
permitted by equations
(3b)
and
(3b').
We then have
to
prove
that the work
öA
to be
supplied
to
the
system
for
an arbitrary
infinitesimal
displacement
of the
masses
(dx, dy, dz)
is
equal
to
the increase
ÖE
of
the
integral (which
is
extended
over
the entire
space)
of the total
energy density,
which is
given
within the brackets in the above
equation.
One first obtains
by
virtue of
(4)
[29]
4It has to be
pointed
out
that it attains
a
positive value, as
with Abraham.