 ...An Exact Classical Mechanics leads toward Quantum Gravitation... Contents
4.2 Compressibility of the i-therSo far only "flat" space has been considered, meaning that the i-ther is assumed to be of exactly uniform density. This is, however, incompatible with an expression from quantum theory given by Novikov(5) who says that space is filled with virtual particles which, on average, occupy cubes of side L given by the expression: L = (h/2π)/(m0c). Substituting from [28] this gives L = (h/2π)c/(m0DcD2): so L/LD = c/cD showing that the i-ther has non-uniform density. Then combining with [34] and [35] it follows that the general expression for motion in free-fall in a compressible fluid i-ther is:  | [37] |
4.3 The true speed of light cT in the compressible i-therThe propagation speed of light will be affected by the change of L with ψ as will now be shown. It is best to imagine the photons moving as if in instantaneous jumps of distance L followed by a dwell of time Δt before the next jump where c = LD/Δt and cD = LD/ΔtD. If then cT, the true light speed, is defined as cT = L/Δt then its ratio with cD becomes:  | [38] |
Series expansions show that for flat space the change in light speed in weak gravity will be almost equal to ψ whereas in the compressible i-ther it will be 2ψ. This means that the bending of light will be doubled from the original Newtonian prediction and so will accord exactly with that of general relativity. Velocities are also affected so that the energy equation becomes modified to:  | [39] |
4.4 The gravitational red-shift -Two Methods1. | The energy E of a photon is given by E = (h/2π)v where v is its frequency. Then from[35] E = ED EXP(ψ): ((h/2π) = Planck's constant). Hence the red-shift can be expressed as v/vD = EXP(-ψ): i.e. Δv ≈ -ψ × vD So the photon reduces in v as it rises from low level. | 2. | A pair of equal masses connected by a spring have a vibrational frequency inversely proportional to the square root of their rest-masses. It also follows by substituting for c/cD from equation[34] in [28] that rest mass m0 = m0DEXP(-2ψ). It follows that Δω/ωD ≈ ψ. This means that the object will vibrate at lower frequency when lowered in the gravitational field. This is consistent with method 1 and both are identical with general relativity. |
4.5 The conservation of angular momentumFor free-fall equation[37] shows m/mD = EXP(-3ψ) . However, all velocity ratios u/cD, v/cD & w/cD are altered in ratio L/LD i.e. in ratio EXP(ψ). These can be combined to yield the following expression for the new law of conservation of angular momentum. Exhaustive checks have been made using trajectories computed in Cartesians: an example is illustrated on the cover. The following equation has been found to hold exactly:  | [40] |
When equations[28],[37],[38],[39] and [40] are combined, exactly the same equation for the precession of the perihelion of planets arises as is given by general relativity. The original solution presented in the book(6) involved many pages of algebra but the mathematician John Day wrote to say he had found a simple solution which will now be presented. 4.6 The first contribution of John Day M.Sc. - The perihelion advance of planetsPlanets move in elliptical orbits about the Sun according to the original Newton inverse square law of gravity. The modified law causes the axes of the ellipse to rotate slowly and, measured from the line joining the point of closest approach to the Sun, this defines the perihelion advance. It is necessary to start, for the case of free-fall, by combining equations[37] and [39] and re-arranging to yield the following equation quoted from page 278 of the book(6): | [41] |
A "secondary datum", suffix 1 is used taken at the perigee and this replaces D. At this datum w1 is the speed of the planet and, being tangential, can be replaced with v1, which changes to w at potential ψ. John's first letter arrived dated May 13, 1993. He graduated with first class honours in mathematics at the University College in London in 1952 and held three M.Sc's. In this letter he stated that he had devised a much simpler method than the one I had used. This solution will now be presented based only on energy and angular momentum. Hence it starts from equations[40] & [41]. He defines u = 1/r, then, puts h = v1r1 to define dθ from the above equation[40] as:  | [42] |
Then he quotes from standard works:-  | [43] |
Now by substituting from equation[41] the result is:- The solution of this must be a precessing conic. Constants A + B govern only the size and shape, which can be found on evaluation to be only the conventional values with only minute correcting terms. The constant C alone governs the rate of precession (omitting from consideration of course the higher order evanescent terms of + ...). On evaluation the result is:-  | [44] |
This will give a precession per orbit of πC, i.e. 6π(GmS/(hc))2 The light-speed c varies by a negligible amount within the solar system and so an average value of c1 can be used without significant error in equation[44]. The value of v1 can be obtained from Newtonian mechanics to adequate accuracy. The radius of closest approach, at "perigee" is r1 and the radius at greatest distance is r2 at "apogee". A slightly more refined version of equation[44] is:  | [45] |
The result is identical with that given by General relativity. | 
