# Dark Matter-Baryonic Matter Radial Acceleration Relationship in Conservation Group Geometry

## TL;DR

This paper introduces a new geometric theory extending spacetime structure with a conservation group, deriving solutions that naturally incorporate dark matter and energy, and explaining galaxy rotation curves through a novel acceleration relationship.

## Contribution

It develops a conservation group-based geometric framework that naturally accounts for dark matter, dark energy, and galaxy rotation curves, and relates these to observed acceleration laws.

## Key findings

- Derived spherically symmetric solutions with nonzero stress-energy tensors.
- Model explains flat galaxy rotation curves without dark matter.
- Proposes a new critical baryonic acceleration related to observed acceleration laws.

## Abstract

Pandres has developed a theory which extends the geometrical structure of a real four-dimensional space-time via a field of orthonormal tetrads with an enlarged covariance group. This new group, called the conservation group, contains the group of diffeomorphisms as a proper subgroup and we hypothesize that it is the foundational group for quantum geometry. Using the curvature vector, $C_\mu$, we find a free-field Lagrangian density $C^\mu C_\mu \sqrt{-g}\,$. When massive objects are present a source term is added to this Lagrangian density. Spherically symmetric solutions for both the free field and the field with sources have been derived. The field equations require nonzero stress-energy tensors in regions where no source is present and thus may bring in dark matter and dark energy in a natural way. A simple model for a galaxy is given which satisfies our field equations. This model includes flat rotation curves. In this paper we compare our results with recently reported results of McGaugh, Lelli and Schombert which exhibit a new law between the observed radial acceleration and the baryonic radial acceleration. We find a slightly different model which relates these accelerations. In conjunction with our model, the McGaugh, Lelli and Schombert relation imply a new critical baryonic acceleration. When applied to bulge-dominated galaxies, this critical baryonic acceleration may be used to predict the radial velocity curve value by using the radius of the bulge.