 ...An Exact Classical Mechanics leads toward Quantum Gravitation... Contents
2.3 Method 2: Rest-energy Assumed to BE Kinetic Energy (Independent of Method 1)Note: Readers are advised to jump to section 4.0 on a first reading It seems possible that the universe could be constructed entirely from kinetic energy EK. For example, two particles connected by a weightless chain can be imagined in orbit about one another. Each could be made of pure kinetic energy so that it moved at the speed of light, as proved in method 1. The pair would, however, appear as a single stationary particle possessing a rest energy corresponding to the sum of the kinetic energy of the components. A pion might be modelled in this way. The quantum model will not be the same but the hope is to see if this (incorrect) classical model can provide the required classical/quantum interface; just as the classical Bohr radius of the hydrogen atom forms such an interface for electromagnetism. Bohr proposed a model in which the electron orbited the nucleus like a planet going round the Sun in a circle. This was later replaced by the quantum model due to Schrödinger, in which electrons existed in a spherical "orbital" with a random distribution. It had a probability of being found at any radius which increased with distance from the nucleus up to a certain value and then fell off again. However, the peak of the probability curve coincided with the Bohr radius. In this way it seems reasonable to consider the Bohr radius as a classical/quantum interface because a residue of classical theory can be said to have a useful existence. A pion consists of a pair of quarks, to be imagined rotating about one another like a spinning dumbbell, each at radius r about a common centre. Then the orbiting pair will be observed as a single stationary particle having rest mass m0. In a first simplified model, to be refined later, each quark will be assumed made entirely of photon-like orbiting kinetic energy EK, where EK = mKc2, moving at the speed of light. The pair is now assumed accelerated to a speed v by a force directed along the axis of rotation. Modelled as a rotating dumb-bell the orbital speed of each quark (shown as small spheres) falls from c, when the pion is at rest, to vorb as the linear speed is increased to v. The condition needing to be satisfied is that each member moves at unchanged speed c. (This is for the unrefined model in which each quark has no rest-energy) The combined rest energy will be:  | [15] |
If the pion is accelerated bodily to a linear speed v, with the axis of rotation in line with the direction of acceleration, then the orbital speeds will fall to to vorb as shown in FIG.1. It will be assumed that the orbital radius remains constant, a condition which Method 1 will later show to be justified.
 | [16] |
This derives purely from the geometrical theory of Pythagoras. Each member of the pair will follow a helical path at a speed c, directed along this path, since as shown in Method 1, when m0 = 0 absolute speed = c. Conservation of angular momentum pr given by pr = mvr then dictates that, assuming r to remain constant:  | [17] |
Then substituting for vorb from [16] in [17] yields:  | [18] |
Since this is identical with a re-arrangement of equation [13] derived by method 1, the assumption that r remains constant is justified. Furthermore an expression for the angular velocity ω for the orbiting quarks can be compared with the rest value ω0 and, since r is constant becomes, since ωr = vorb:  | [19] |
A more refined model would make the orbital speed of each quark of the stationary pion equal to ηXc where η<1 to allow for non-orbital components of energy, such as a spinning motion of each quark about its own axis. For example, a quark could be made up of sub-quarks of pure kinetic energy orbiting at speed vq about the centre of gravity of the quark. Then:  | [20] |
This simplifies to:  | [21] |
Hence η will appear on both sides of equation [17] and so will cancel leaving the value of m/m0 unchanged. A quantum description would show that, instead of orbiting, the quarks would seem to jump about over the surface of a sphere of radius r. The pion would then be spherical instead of disc-shaped but the foregoing classical approach should yield the required classical/quantum interface. | 