414(1): Summary of the Orbit Shrinking Equations

Hello Horst, I just wanted to check that my understanding below of your numerical discovery of shrinking is correct. I understand that you solved Eqs. (1) and (12) simultaneously using the numerical integrator, and that this gave a shrinking and precessing orbit.

From: Myron Evans <myronevans123>
Date: Mon, Sep 3, 2018 at 1:25 PM
ct: 414(1): Summary of the Orbit Shrinking Equations

From Horst’s communication received this morning the numerical integration of equations (1) and (12) produces a shrinking and precessing orbit, with the spin connection defined by Eq. (8). The orbit produced by the numerical integration should be Eq. (17), which is obtained from considerations of the hamiltonian (16). Considerations of the lagrangian give equation (18), in which L is a constant of motion obeying dL / dt = 0. The hamiltonian is a constant of motion obeying dH / dt = 0. This equation leads back to Eqs. (1) and (12) self consistently. It is clear from Eq. (19) that r must go to zero as the time t goes to infinity, i.e. as phi’ goes to infinity. So Eq. (19) is the neatest form of the shrinkage equation. The equivalent of Eq. (19) in the Newtonian limit is Eq. (34). For a given alpha and epsilon or H and L, the Newtonian ellipse remains stable, and does not precess or shrink, but as soon as the de Sitter frame transformation (38) is used, the orbit precesses and shrinks. In the next note the relativistic hamiltonian and lagrangian, Eqs. (41) and (42), will be considered in order to begin the development of the completely relativistic theory. The de Sitter rotation is considered to be due to torsion. The latter is zero in Newtonian dynamics, and so is the curvature.


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