 ...An Exact Classical Mechanics leads toward Quantum Gravitation... Contents
4.0 Towards Quantum Gravitation (Quantum Compatible Gravity)
4.1 A preliminary formulation of basic equationsSince kinetic energy and kinetic mass have been shown equivalent it follows that kinetic energy as well as matter will couple with the gravitational force. The photons of light are pure kinetic energy: it follows that light must fall, just like matter, in a gravitational field. If a horizontal beam of light of depth dr is considered, bent by gravity, then clearly the photons will need to travel further on the outside of the bend than on the inside in order that its waves will always be normal to the direction of propagation. This is illustrated in Fig.4 showing how the change in light speed dc with dr arises due to gravitational acceleration g acting for time t. This results in the following equation:  | [27] |
(A totally rigorous deprivation yields exactly the same result.) Clearly as the distance from a ponderous object is increased, so c must increase. At some datum level, suffix D, its value will be cD and it will be c at another level. Then since rest-energy is a constant E0 = m0c2 = m0DcD2. In consequence it follows that:  | [28] |
The equivalence between energy and mass is now clearly restricted to constant gravitational potential, to be designated ψ. Rest mass now appears as a variable with respect to ψ! Clearly only energy can now be regarded as the constructional material of the universe. Mass now appears as only a convenient property of energy; something which helps govern motion by providing inertia. | Fig. 4 Gravitational light bending |
Inertial mass is now to be defined as:  | [29] |
In consequence, and since kinetic-energy as well as rest-energy feel gravity, total energy E must couple with any gravitational flux φ so that a force F = -φ E is produced. Furthermore, by simple Euclidean geometry φ must satisfy an inverse square law of intensity when emanating from some massive point object at distance d and so, with dD as some chosen datum distance, it must obey the expression:  | [30] |
By symmetry this flux must itself be proportional to the value ES the total energy of that massive object. So when a constant of proportionality GC is introduced the law for quantum gravity becomes:  | [31] |
Where, by comparison of equations[30] & [31] it can be seen that another useful constant, the "gravitational radius" r0 emerges, given by the identity:  | [32] |
The constant of gravitation is now GC whose value is 8.2615 X 10-45 Nm2J-2 when determined from the Newtonian constant G as shown by equation[31]. It differs from the Newtonian gravitational equation in two ways. Firstly GC has to be the true constant not G! The latter must vary as c4 varies so that GC remains constant. Secondly E varies with speed w and so replaces the constant rest-mass of the Newtonian. (w is the vector sum of u an v where u will be used for the field component and v the velocity component transverse to the field i.e. horizontal) These two effects go part way to giving the same predictions as general relativity. For example, F no longer obeys an exact inverse square law: it is slightly steeper. This difference produces precession in the orbits of planets. Next it is necessary to find how both c and E vary with gravitational potential ψ. It is best to obtain a general expression assuming ψ to have any profile, then:  | [33] |
Used with equation[27], putting φc2 for g, since g = -F/m = -Fc2/E, the variation in light speed can be found by integrating across an infinite number of elements such as that illustrated in Figure 4. A precise definition of ψ now emerges which is:  | [34] |
Since dE = Fdr then dE = - φE dr. So the variation in E with ψ becomes:  | [35] |
Equation[34] is universally valid, but [35] is valid for free-fall only. For the special case of a single massive attractor with φ defined by equations[30] & [32] the value of ψ becomes:  | [36] |
The radial distance r has replaced d since the massive attractor is assumed so large that its orbital motion can be ignored. | 