Since,
\(F_v=-\cfrac{d\,\psi}{d\,r}\)
\(\int{F_v}\,d\,r=-\psi\)
and since,
\(F=-\psi\) --- (*)
\(F=\int{F_v}\,d\,r\)
for a particle.
The formulation,
\(F=F_{v}\cfrac{1}{12\pi r}\)
is wrong. However, \(\psi\) does not go on forever, at some point along \(r\), \(\psi=0\) and remains zero thereafter. In addition,
\(\int^\infty_0{\psi}\,d\,r\nrightarrow\infty\)
The expression (*) is then valid up to \(r\) where \(\psi\ge 0\), thereafter, for a point particle, \(\psi\) remains zero, the change in \(\psi\) is zero and so \(F=0\). This is different from Newton's Law, Coulomb's Law and the constant flux derivation, all of which suggest that \(F\) is nonzero as \(r\to\infty\).
It is classic in physics to assert that there cannot be infinite amount of energy. It follows then, that a force field as a result of a spread of energy with an associated energy density distribution around a particle must not be infinite too. And the equivalent mass (\(E=mc^2\)) cannot be infinite.
What happened?
If \(\psi=\cfrac{D}{r^2}\)
at \(r=0\), \(\psi=C=\cfrac{D}{r_o^2}\), (ie the graph is shifted left)
then \(\int^\infty_0{\psi}\,d\,r\)
is actually finite but \(\psi\to0\) as \(r\to\infty\); \(\psi\) has a infinite tail and the force field extends to infinity.
There is still a problem with \(F\). \(F\) is derived from \(F_v\) which is obtained by differentiating \(\psi\) along \(r\).
\(F_v=-\cfrac{d\,\psi}{d\,r}\)
\(\psi\) is the energy density of a wave travelling along a time dimension at light speed \(c\). It is not a wave at light speed in space. In the latter case, \(\psi\) in along the wave. In the case of a wave with light speed in time, \(\psi\) was taken to be around a point particle. The two big assumptions here are that the time wave manifest itself in space as a point particle and that the energy density, \(\psi\) associated with such wave is distributed uniformly around the point mass. The post "Standing Waves, Particles, Time Invariant Fields" dated 20 Nov 14 shows that \(\psi\) is a standing wave around the point particle.
Is an explicit transformation needed between a time wave and a space wave? A transform that will morph \(\psi\) from a cylinder to a sphere? No, the post "Standing Waves, Particles, Time Invariant Fields" dated 20 Nov 14 shows that \(\psi\) is a standing wave around the point particle.
Furthermore, in the post "Difficult To Correct Oneself", it was proposed that instead of a wave oscillating in two space dimensions and travelling down a third time dimension at light speed, the wave in time that manifest itself as a particle is oscillating between one space and one time dimension. Fortunately, the wave equation is the same for both cases, from a later post "Photon, More Lights, Big Mistake".
These new particles are one dimensional entities; 1D particles not at light speed in space. These particles can provide potential energy (electrical, gravitational and heat) to other particles that they collide with. These particles could be responsible for induction/radiation. Are they also responsible for the force field around a mass or charge? It is possible that an oncoming mass be radiated with gravitational 1D particles and so gain gravitational potential energy and be bounced off.
Next stop, wave equations for photons.
