The variant of post-Newtonian mechanics with generalized fractional derivatives

TL;DR

This paper explores a novel variant of post-Newtonian mechanics incorporating generalized fractional derivatives, leading to modified gravitational equations with unique effects on large scales and solutions exhibiting space diffusion of gravitational waves.

Contribution

It introduces a mathematically rigorous generalization of post-Newtonian mechanics using fractional derivatives, extending classical gravitational equations and analyzing their solutions.

Findings

Potential higher gravitational effects on large scales.

Diffusion of gravitational waves due to fractional derivatives.

Tiny secular perihelion shift observed.

Abstract

In this article we investigate mathematically the variant of post-Newtonian mechanics using generalized fractional derivatives. The relativistic-covariant generalization of the classical equations for gravitational field is studied. The equations match to (i) the weak Newtonian limit on the moderate scales, (ii) deliver a potential higher, that Newtonian, on certain, large-distance characteristic scales. The perturbation of the gravitational field results in the tiny secular perihelion shift and exhibits some unusual effects on large scales. The general representation of the solution for fractional wave equation is given in form of retarded potentials. The solutions for the Riesz wave equation and classical wave equation are clearly distinctive in an important sense. The Riesz wave demonstrates the space diffusion of gravitational wave at the scales of metric constant. The diffusion leads to the blur of the peak and disruption of the sharp wave front. This contrasts with the solution of D'Alembert classi-cal wave equation, which obeys the Huygens principle and does not diffuse.