John Denker has put his finger on the problem with my derivation, as has
David Bowman. It apparently involves looking at the difference between
two infinities which, I must assume, involves what David calls "the
surface at infinity". I hope that the self energy is not also involved,
mostly because I hate self energy arguments. They make my head hurt! I
may also bother John a bit with what follows because I will use few, if
any, equations. I do this partly to show that good physics *can* be done
with rigorous use of language and partly because it is really hard for
me to do equations in ASCII and maintain a coherent line of reasoning.
I am an experimentalist and I have no illusions about my prowess as a
mathematical physicist. Sophisticated mathematical arguments are often
difficult for me; I much prefer physical arguments for which I think I
have a more robust talent. For that reason alone, and not because I do
not accept the likelihood that my result is wrong (after all, it differs
from a standard result accepted by folks cleverer than I), I would like
to present a second derivation which, while still three dimensionally
infinite, does not suffer from the fringing field problem that John
worries about. I will argue that it demonstrates physically that the
gravitational field energy density has the just value I proposed. I hope
that David, John, and others in this group can bash at my *physical*
argument to show me my error in a nonmathematical way. I had never seen
the standard result and I had always assumed it must be the same as mine
because I don't know GR, as I said before. I thank David for once again
getting me past a lacuna in my education and knowledge.
In what follows I will try to analyze the problem from another point of
view, one which, I think, eliminates the surface at infinity problem on
a physical basis. I will then sketch an attack on the self enegy aspect
in a way that perhaps David Bowman can help me on. When I started this
it was 3:30 am, so please forgive my occasional goofs. I'm learning.
Consider the gravitating system consisting of two identical thin
spherical dust shells of radius R, each of which has mass M uniformly
distributed over its surface. I will say at the outset that this
derivation also works for two shells of finite thickness and uniform
volume charge density, but apart from a dimensionality difference which
may become important in reckoning the self energy term, there's no new
physics, and there are many more terms. Initially the spherical shells
are concentric (and coincident). I will hold one of these shells at its
initial position (with some sort of rigid frame separating the dust
particles) and I allow the other to relax (under its own interparticle
self-gravitation) to a radius R - dR. The standard result that the
gravitational field of the rigid sphere has no effect on the collapse
of the other sphere inside it will be accepted uncritically for the
moment, but we will revisit this question later and, I think, show that
it is a valid assumption.
After the infinitessimal collapse the system has a lower gravitational
potential energy, and the electric field differs from the initial
condition only in the region between the spheres. The field in that
region, initially zero, is now finite and equal to the field due to a
point mass M at a distance R. I will leave it as an exercise for the
reader to show that this gives exactly the same result for the field
energy density I obtained before*. It has that same annoying problem:
there's less energy after the infinitessimal collapse, but there's
more gravitational field, so the field has negative energy density!
Now let's think physically about what goes on outside the spheres. The
fields before and after collapse are both due to spherically symmetric
mass distributions of equal mass 2M. The external fields should be
identical. Physically I see nothing at all different about the universe
outside the outer sphere. In this manner I hope I put to rest any
specter of a "surface at infinity" problem; physically there can't be
one.
Unfortunately that leaves me with that other problem, the one for which
I need some ibuprofen, the self energy of the dust sphere that shrank
by dR. The dust particles (which I will steadfastly resist treating as
point masses) in the inner sphere are now somewhat closer together than
they were before the collapse (by dR/R, relatively). I must now
consider in detail the gravitational fields between dust particles.
First, I will note that in an infinite three dimensional hexagonal
close packed (hcp) array of such particles the gravitational field at a
given distance r from any particle would not be expected to change by
very much, if at all. I infer this somewhat counterintuitive result by
applying Gauss's law, and also by noting that the fields due to
neighbor particles cancel pairwise by the symmetry of the lattice.
(Here I may be slightly in error; I may have to make this argument in
an fcc lattice instead. It obviously works in a simple cubic lattice.)
I have put some of my thoughts on this in another posting (The rabbi's
copout) and I will continue this thread later. I'm afraid I fell asleep
at my keyboard, and I need a nap.
Leigh
*This should come as no surprise. All I've done differently is to
separate two finite planes by an infinitessimal amount. I will say that
this also works if one separates the spheres by a finite amount, so the
fringing field argument is moot (and was probably irrelevant). The more
serious reader who is puzzled might perform the finite dR exercise for
completeness.