As I indicated in my original derivation of the gravitational field
energy density, I had more than one way to derive it. I chose what
I considered to be the mathematically and conceptually simplest of
them since all gave me the same final result. David Bowman taught
me two valuable things in response. The first was that my result
is not the standard one. Being relatively naive about gravitation I
assumed that the derivation of a gravitational field energy density
was something I could get from simple Newtonian first principles, a
result which, as you have seen, is very easy to derive. The second
thing David taught me is that there is something called "the surface
at infinity" of which I can only vaguely recall hearing before, and
I certainly didn't understand it at the time.
I am not particularly adept at mathematics; I prefer physical models
which are grounded in simpler rather than more sophisticated
concepts. While I recognize that there may exist True Laws of Nature,
the endeavor we call physics is only concerned with making better
descriptions of how Nature is. If a description is more exact in its
description of Nature it is certainly better in an important sense.
Special relativity is better than Galilean relativity in its ability
to describe Nature in temporal and spatial terms, and its mathematics
is simple, but I don't need it to describe phenomena in which the
relativistic effects are swamped by other real world nonidealities,
and I wouldn't use it to describe those phenomena. Similarly I prefer
semiclassical descriptions (explanations) of physical phenomena
whenever they are appropriate. That has been my preference since long
before my qualifiers when I was ambushed with a request to derive the
dispersion relation for ferromagnetic spin waves - and I succeeded in
doing so, to the great relief of my senior supervisor. My derivation
was semiclassical, extemporaneous, and my examiner liked it.
Placing this topic in my frame of reference, I know that energy is
not substantial, and that only differences in system energy are
affected by work and heat acting on an isolated system. I also know
that there is sometimes more than one way to calculate an energy
difference, and that sometimes one can calculate it by an algorithm
that asserts a field energy density. This works well in electromag-
netism and I expected it would work well in gravitation. I claim that
it does, but my algorithm involves attributing a negative definite
(?) energy density to a gravitational field. So far as energy
differences go my homebrew method works fine, and I don't see why it
should be modified, especially in the way that David suggests.
Now of course energy density has another function that has nothing to
do with energy differences; it is part of a gravitational source term
which must be internally consistent to be correct. In other words,
unlike the case discussed in which an electromagnetic wave gave rise
to a gravitational field because of an inherent mass density u/c^2,
the gravitational field, if it posseses energy, must give rise to -
a gravitational field. If *my* idea of gravitational field energy
density is correct (and I'm certainly not claiming it is) then the
self-consistent gravitational field of a spherically symmetric mass
will fall off *slightly faster* than 1/r^2 for small values of r,
asymptotically approaching 1/r^2 at large distances in a flat space,
but with a very much smaller central mass as the central body gets
down toward black hole size. Will this suffice to replace the
accounting for the potential term David says is missing? On the other
hand, if the energy density in the field is positive, them my simple
view would suggest that the field falls of less rapidly than 1/r^2
for small values of r because the spherically symmetric source term
increases with increasing radius. The formal calculation of these
relations is left as an exercise for the reader (is anyone still
reading?).
Of course all of this is in the Newtonian limit, and perhaps I am way
off base in even suggesting it. MTW is silent on this matter of field
energy density if I can go by the index. So are Burke, Kenyon and
Rindler. I don't have a more extensive gravity library (and I don't
know GR), and I can't follow David's development*. My field energy
formula works in the admittedly limited scope I claim for it. Is
there a known phenomenon which must be explained in terms of this
positive field energy density?
Leigh
*I would suggest a pdf file generated from Latex, but given that I
got zero responses when I posted one last week I doubt that the mode
is going to get popular. I can't think in ASCII, and I'm afraid the
comments about even-integer-spin field mediators of interactions are
wasted on a dull fellow like me. Sorry, David; you have certainly
given me insight in the past, but you're firing way over my head
this time. Is there an accessible text that treats the energy
density problem?