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The variant of post-Newtonian mechanics with generalized fractional derivatives

### 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 (i) match the weak Newtonian limit on the moderate scales and (ii) deliver a potential higher than 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 the fractional wave equation is given in the form of retarded potentials. The solutions for the Riesz wave equation and classical wave equation are clearly distinctive in an important sense. The hypothetical gravitational Riesz wave demonstrates the space diffusion of the 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 the D’Alembert classical wave equation, which obeys the Huygens principle and does not diffuse.

© 2006 American Institute of Physics

Received 06 September 2006
Accepted 09 October 2006
Published online 07 December 2006

Lead Paragraph:
The present work continues the development of post-Newtonian mechanics using generalized fractional derivatives. Fractional derivatives have recently found applications in studies of scaling phenomena. The purpose of the generalizations for fractional derivatives is the formulation of the appropriate equations, which could describe scaling processes, preserving other natural fundamental symmetries, as conservation of momentum and energy and Lorenz invariance. We proceed with the idea of an additional Riesz potential term, which allows constructing the covariant equations of scale-dependent gravity with a sole metric parameter with dimension of length. The minimal mathematical apparatus for handling of fractional derivatives in the differential geometry is introduced. The motive of this phenomenological hypothesis is the attempt to fit high-precision data of Solar System experiments to observable astronomic phenomena on galactic scales. Namely, the astronomic date witnesses much higher levels of gravitation potentials on the galactic scales than those predicted by Newton's law. We try to explain the distribution of gravitational potential, which follows from the solution of the inverse problem from galaxy dynamics, without implication of dark matter. The hypothesis explains maximal sizes of galaxies and predicts the belts of stability in the outer periphery of the Galaxy, inhabited by dwarf galaxies, and matter-free belts of instability. This appearance proved to be conforming to the spatial distributions of neighboring members of the Local Group. The observed peculiarities of the gravitational lensing by galaxies are also discussed. The oscillatory behavior of the Riesz potential delivers an alternative explanation for the multiplicity of images for distant galactic sources. The solution for the fractional wave equation is given in the form of retarded potentials. The solutions for the Riesz wave equation and classical wave equation are clearly distinctive in one important sense. The Riesz wave diffuses and, therefore, its amplitude decays much stronger than the amplitude of the classical D’Alembert-type gravitationwave.

Article outline:

I. INTRODUCTION
II. MATHEMATICAL PRELIMINARIES: TENSOR ANALYSIS WITH FRACTIONAL DERIVATIVES
A. Semicovariant and semicontravariant fractional derivatives
B. Variational principles with fractional partial derivatives
C. The fractional Riemann operator
III. GOVERNING EQUATIONS OF RIESZ GRAVITATION
A. Symmetries and conservation laws of fractional partial differential equations
B. Variational principles of Riesz gravitation
C. Variational principles for weak Riesz gravitation
D. Equation of weak gravitations field
IV. GREEN FUNCTIONS FOR WEAK GRAVITATIONAL FIELD
A. Green functions and asymptotics of weak Riesz gravitational field
B. Gravitational potential of a point mass
V. ON THE POSSIBLE EXPERIMENTAL TESTS OF POST-NEWTONIAN GRAVITY
A. The Solar System orbital effects induced by Riesz acceleration
B. The Riesz effects in the Galaxy dynamics
VI. DEFLECTION OF LIGHT AND GRAVITATIONAL LENSING
A. Deflection of light
B. Equations of gravitational lensing and multiplicity of images
VII. THE PROPAGATION AND DIFFUSION OF GRAVITATIONAL WAVES
A. Fundamental solution of the Riesz wave operator
B. Diffusion of wave front for gravitational waves
VIII. CONCLUSIONS

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2006-12-07

2016-09-26

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