The Mannheim-Kazanas solution, the conformal geometrodynamics and the dark matter

TL;DR

This paper demonstrates that the Mannheim-Kazanas solution, originally from conformal gravity, also solves conformal geometrodynamics equations with a nonzero Weyl vector, suggesting a Weyl geometric approach to galactic dynamics.

Contribution

It shows that the Mannheim-Kazanas metric is a solution to conformal geometrodynamics equations with a nonzero Weyl vector, extending its applicability beyond conformal gravity.

Findings

Mannheim-Kazanas metric solves conformal geometrodynamics equations.

Supports the idea of Weyl geometry describing galactic scales.

Provides a geometric explanation for flat galactic rotation curves.

Abstract

Within the framework of the Einstein's standard equations of the general theory of relativity, flat galactic rotational curves of galaxies cannot be explained without hypothesis attracting the dark matter, the particles of which had not yet been identified. The vacuum central-symmetric solution of the equations of conformal gravitation is well known as metrics of Mannheim-Kazanas, on the basis of which these curves receive purely geometrical explanation. We show in our work that the metrics of Mannheim-Kazanas is the solution of not only Bach equations received from conformal-invariant Weyl Lagrangian, but also the solution of equations of the conformal geometrodynamics at a nonzero vector of Weyl. In this connection the hypothesis is formulated that the space on galactic scales can be described not only by Riemannian geometry, but with geometry of Weyl.