{This was hard to write, Dear Reader. I've probably left some errors or
obscurities in, and I will be happy to explain what I mean about anything
below.]
David Bowman's posting leaves me somewhat puzzled. David knows much more
gravitation than I do, and I will try to read his explanation in some
detail when I get a chance. Presenting my derivation of the gravitational
field energy density (which led me to the conclusion that it is negative)
will take little time, so let me do that. My derivation is much simpler
(and less general) than his, but I hope he will agree that, *if* there is
a field energy which depends only upon gravity, then this derivation will
yield the correct formula. I may be able to do the assembly of point
masses version too, but it is somewhat longer.
Consider first an infinite plane sheet of mass density 2 mu (mu would be
expressed in kg/m^2 in SI). The gravitational field due to this sheet has
a magnitude g found easily using Gauss's law. Using the same units David
is using I find the field to have magnitude
g = 4 pi G mu
where G is the gravitational constant. The field fills all of space and
points inward, toward the sheet, everywhere. I will call this physical
arrangement System A. I choose the zero of energy to be the energy of
this configuration. This is an arbitrary choice, but I will only be
considering energy changes in my calculation.
Next, imagine the infinite sheet to be made up of two infinite sheets,
each having mass density mu. Call them the upper and lower sheets. These
sheets attract one another with a force per unit area (with units of a
pressure) P given by the product of the mass per unit area of one of the
sheets times the gravitational field due to the other sheet. (P is a
vector, but that will concern us only briefly.) This force density is
2
P = mu 2 pi G mu = 2 pi G mu
This force density is independent of the separation of the two sheets so
long as they remain mutually parallel. Now pull the plates apart so that
they are separated by a distance d. You will perform an amount of work
per unit area, w, given by
2
w = P d = 2 pi G mu d
We will call the system in this final configuration System B. The
gravitational potential energy per unit area, e, of System B is, with
our chosen zero, given by
e = w
It happens that, using our initial result, this can be written in terms
of the gravitational field strength g outside the plates as
2
g d
e = --------
8 pi G
The only difference between the gravitational fields of System A and
System B is in the region between the plates of System B. In that region
the field is zero in system B. Everywhere else the fields in System A
and System B are the same.
Now, if we are to ascribe the gravitational potential energy per unit
area to a gravitational field energy (volume) density term u(g'), then
e = e(B) - e(A) = e(u(g'(B))) - e(u(g'(A)))
where g' = g'(x,y,z) is the gravitational field strength. In the region
of space which lies outside the plates of System B we have
g'(B) = g'(A) = g ( z < 0 and z > d, say)
Thus the only difference we have to consider is what occurs between the
plates:
g'(B) = 0; g'(A) = g ( 0 < z < d )
Our difference equation becomes:
e = e(B) - e(A) = e(u(0)) - e(u(g))
Since we must evaluate the energy per unit area difference only between
the plates separated by d, we have
e(u(g')) = d u(g')
and
e = d u(0) - d u(g)
thus
2
g d
d u(0) - d u(g) = --------
8 pi G
Solving for u(g) we obtain
2
g
u(g) = u(0) - --------
8 pi G
I would like to be able to assume
u(0) = 0
Since I don't believe energy is anything real, it doesn't bother me that
this implies that
2
g
u(g) = - --------
8 pi G
All my derivations lead to this same result.
I will quite seriously ask David for two things beyond what he has done
already. Please criticize my derivation (my other derivations all give
this result as well). My second request is PLEASE venture into the area
of attribution of reality to concepts, especially to energy. We are all
entitled to dabble in metaphysics (where I feel at least as competent
or incompetent as any specialist), and your input would be especially
welcome, given your knowledge.