A novel approach to autoparallels for the theories of symmetric teleparallel gravity

TL;DR

This paper introduces a new method to distinguish autoparallel curves from geodesics in symmetric teleparallel gravity, with applications to Schwarzschild metrics and implications for dark matter and orbital dynamics.

Contribution

It presents a novel approach to autoparallel curves and geodesics in symmetric teleparallel gravity within the coincident gauge, highlighting differences from Riemannian geometry.

Findings

Autoparallel curves differ from geodesics in non-Riemannian geometries.

Application to Schwarzschild-type metrics demonstrates practical implications.

Remarks on dark matter and orbit equations suggest potential physical relevance.

Abstract

Although the autoparallel curves and the geodesics coincide in the Riemannian geometry in which only the curvature is nonzero among the nonmetricity, the torsion and the curvature, they define different curves in the non-Riemannian ones. We give a novel approach to autoparallel curves and geodesics for theories of the symmetric teleparallel gravity written in the coincident gauge. Then we apply our autoparallel equation to a Schwarzschild-type metric and give remarks about dark matter and orbit equation.