v.9 n.3 May

On the Noneuclidean Geometries of the Kepler Problem

Bernard Howard Lavenda

Parana J. Sci. Educ., v.9, n.3, (1-8), May 1, 2023.

DOI:    10.5281/zenodo.7884612

Abstract

Noneuclidean geometries of constant and non-constant curvature are treated within the framework of the Beltrami and warped metrics. An example of non-constant curvature is equivalent to the Schwarzschild metric which yields the Einstein modification of the orbital equation for the precession of the perihelion. The static nature shows that time does not enter at all into the Schwarzschild solution. Introducing a parameter that describes the geodesic motion and then identifying it with the azimuthal angle confuses energy and angular momentum conservation; the latter does not preserve its Euclidean form. Gravitational repulsion is a figment of the indefinite metric, and has nothing to do with reality. The introduction of two times and the condition between the frequency and coefficient of the square of the time increment in the indefinite metric compensates the erred assumption that the angular momentum is given by its Euclidean form. The origin of the perihelion advance is to be found in the unequal velocities at the apsides causing the ellipse to rotate forward; the rosette shape of the orbit is its manifestation. Although it contributes to the total energy, a quadrupole moment cannot affect the shape of the elliptic orbit. Specifically, it cannot cause the orbit to precess. The sun’s oblateness is treated as a rotational perturbation, derived from a constant curvature metric. Constant curvature rotational deformations are compared to non-constant tidal deformations. Both constitute completely static phenomena; neither warrants the introduction of time into the metric. The indefinite, hyperbolic, metric is a vestige of special relativity that is incompatible with the nature of the these perturbations.

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