Yes, it was a typo. I simply forgot to be consistent with the factors of
4pi. I fixed it last night for the book version. Remember, I was
actually deriving it for the first time as you saw it happening in real
time. What is important is the concept. I am not worried about factors
of 4pi. In fact, it is not there I think when you define the Green's
function with as [(4pi)(r - r')]^-1. I am only interested right now in
order of magnitude. I think I am on right track here. Carlos's
explanation of the direction and even the magnitude of the effect is
incomprehensible to me. Maybe it will make more sense after I study his
paper, which is interesting, but which will take me a long time to
really understand. I do not have that time. This Pioneer Anomaly is too
important and I am rushing it into conventional publication as a NASA
document with an important scientist at NASA as co-author. I of course
will be giving acknowledgments to Carlos and you in that paper. The
Pioneer result is extremely important and my model is mainstream needing
only Einstein's general relativity and the current theory of inflation
consistent with WMAP, Type 1a Supernovae, dark matter gravity lensing
etc. The NASA guy will connect it to all that and he is able to get
conventional papers published no problem on the arXiv as well as in
journals. He knows the drill how to please the Victorian Station
Masters! :-)


On Oct 15, 2004, at 10:22 PM, Tony Smith wrote:

Jack, you say:
"... the exotic vacuum effective gravity potential V(r) per unit test
particle mass from non-vanishing zero point energy density is the
integral of the static Newtonian Green's function G(r - r') for
the Poisson equation for spherical symmetry in the simplest model
to begin with
V(r) = ... = 4 pi c^2 Integral[ ... ] r'^2 dr'...". (eq. 0.1)

In another place you write
"... V(r) = c^2 Integral[ ... ] d3r' ...". (eq. 0.6)

Your final result is "... a_P = -c^2/LpR(t) ~ 10^-7 cm/sec^2 ...",
but
it would be different by an order of magnitude if your
factor 4 pi were present in your eq. 0.10 as it is in eq. 0.1,
so
I am confused about the factor of 4 pi.

Don't be, it's trivial not a physical worry. I was a bit sloppy that's
all. It's a minor detail. Obviously, the 4pi is simply a conventional
normalization factor. It is not a big deal. Forget these twigs look at
the forest.


Further, I thought that the c^2 factor would appear
as 1/c^2 rather than c^2 in an equation for
a Green's function solution for the wave equation
as shown on page 893 of Morse and Feshbach,
where the factor is shown as - 1 / (4 pi c^2 ).

That's the WRONG Green's function for the WRONG equation!

I am not doing the wave propagation equation. I am doing the STATIC
Poisson equation. Newton's gravity works fine as first approximation. No
retardation needed!

No you do not understand the physics of the c^2 factor in my model.

I understand that completely.

The c^2 factor comes from replacing the on mass shell G(mass density)
source by the virtual exotic vacuum source c^2/\zpf. It's one of my
"Feynman rules of thumb" I have developed since 2002.

V(t,r) has dimensions of (velocity)^2.

A similar factor appears with respect to the Helmholtz
equation on page 837 of Morse and Feshbach.

I have, in fact, been using Morse and Feshbach. I have both huge Green
volumes in front of me on my desk. They cost I think at least $300 and
they are not well printed like the original ones.


Maybe I am misunderstanding exactly what Green's function
you are using, and how it differs from those in Morse and Feschbach ?

Obviously the static Poisson eq has no c^2 factor. In effect c is
infinite in Galilean relativity. My c^2 factor has an entirely different
origin. All forms of energy density both real and virtual gravitate (or
anti-gravitate) in Einstein's theory. The problem is how to deal with
the virtual zero point energy density - that's the cosmological constant
problem I have solved with the proper way to understanding how the
pre-inflationary Dirac Sea phase space volume collapses into the
smaller phase space volume of the post-inflationary Higgs Ocean to
create Einstein's gravity. Rest masses of particles as "Mass without
mass"/"Charge without charge" extended micro-geon hidden variables
require the smooth non-random Higgs Ocean (macro-quantum Goldstone phase
rigidity), which soaks up the choppy random troublesome zero point
energy. The soaking up hiding it under the rug is not perfect and that
is why there is both dark energy and dark matter forming the
post-inflationary WMAP seeds for stellar and galaxy formation via the
exotic vacuum halo landscape attractors, i.e. like vortex cores, defects.

Note my initial exotic vacuum field at the Big Bang itself where t - 0
is simply

/\zpf(t = 0) = Lp^-2

i.e. the incoherent value when there is NO HIGGS OCEAN! Try to grok the
Big Picture here Tony. :-)