Sunday, May 13, 2007

First a short review of potential theory.

I. If the force f decreases with distance and the potential energy U is (positive) negative, then the force is (repulsive) attractive.

Example I.1

U = +e^2/r > 0

f = -dU/dr = +e^2/r^2 points toward r -> infinity, i.e. repulsion

note that (d/dr)(1/r) = - 1/r^2

the two - signs cancel

II. If the force increases with distance and the potential energy U is (positive) negative, then the force is (attractive) repulsive.

Example II.1 /\zpf is the vacuum zero point space curvature, assumed constant here.

Zero point energy, as mentioned by Andrei Sakharov in 1967, directly induces gravity because of the equivalence principle of Albert Einstein.

In the weak field low speed limit of general relativity, the universal zero point energy induced gravity potential energy per unit test particle is

V ~ -c^2/\zpf r^2

r < R

for a uniform sphere of isotropic zero point energy of radius R centered at r = 0, with vanishing /\zpf for r > R. This is same as drilling a straight hole all the way through the center of a sphere of constant mass density to the other side and dropping a test particle down the hole. This is a harmonic oscillator because the mass beyond the momentary position of the test particle makes no contribution to the force on the test particle.

Baron Munchausen on the geodesic test particle feels weightless of course, but from the POV of the non-inertial observer fixed to the non-geodesic surface of the sphere by non-gravity electrical and quantum forces, it's AS IF there is a force per unit test mass on the test particle

g = - dV/dr = +2c^2/\zpfr

When /\zpf > 0 this is repulsive.

This same formal result carries over into cosmology where r is replaced by the scale factor a(t) stretching space itself and what happens is that there is an extra acceleration of a(t) opposing the ordinary matter that tends to decelerate the stretching of the rubbery fabric of space itself, i.e. the 3Dim spacelike piece of the geometrodynamic field.

The cosmological equations are here

Therefore, in these sign conventions, /\zpf > 0 is the repelling dark energy and /\zpf < 0 is the attracting dark matter.

Repelling dark energy is isotropic w = -1 positive zero point energy density with equal but opposite negative pressure.

Attracting dark matter is isotropic w = -1 negative zero point energy density with equal but opposite positive pressure.

Adding torsion fields converts Einstein's cosmological constant /\zpf into a locally variable "quintessent" field. You get torsion with curvature by locally gauging the entire 10-parameter Poincare group of globally rigid special relativity.

Now what happens between quarks inside the hadronic "bag"? What we have is a bag of dark matter where the quintessent field is

/\zpf(quarks) = - 1/ar

Therefore, the constant attractive force per unit mass between the quarks is

g = -c^2/a ~ string tension

for strong short-range (Abdus Salam) ZPF induced gravity

We see exactly the same thing on the larger scale of the NASA Pioneer Anomaly where

g = -cH ~ 1 nanometer/sec^2

i.e. c^2/Hubble radius ~ 10^21/10^28 ~ 10^-7 cm/sec^2

i.e. a hollow sphere of dark matter centered at Sun beginning at about orbit of Saturn.

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