On the origin of the gravitational quantization: The Titius–Bode law

Action at distance in Newtonian physics is replaced by finite propagation speeds in classical post-Newtonian physics. As a result, the differential equations of motion in Newtonian physics are replaced by functional differential equations, where the delay associated with the finite propagation speed is taken into account. Newtonian equations of motion, with post-Newtonian corrections, are often used to approximate the functional differential equations. In [Giné J. On the origin of quantum mechanics. Chaos, Solitons & Fractals 2006;30(3):532–41], a simple atomic model based on a functional differential equation which reproduces the quantized Bohr atomic model was presented. The unique assumption was that the electrodynamic interaction has finite propagation speed. Are the finite propagation speeds also the origin of gravitational quantization? In this work a simple gravitational model based on a functional differential equation gives an explanation of the modified Titius–Bode law.