Kinematics in Matrix Gravity

TL;DR

This paper introduces Matrix Gravity, a non-commutative deformation of General Relativity, leading to a Finsler geometric framework that predicts potential violations of the equivalence principle and corrections to Newtonian gravity.

Contribution

It develops the kinematics in Matrix Gravity, replacing Riemannian geometry with a collection of Finsler geometries, and analyzes the resulting physical implications.

Findings

First and second order corrections to geodesic flow.

Evaluation of anomalous nongeodesic acceleration.

Potential violation of the equivalence principle.

Abstract

We develop the kinematics in Matrix Gravity, which is a modified theory of gravity obtained by a non-commutative deformation of General Relativity. In this model the usual interpretation of gravity as Riemannian geometry is replaced by a new kind of geometry, which is equivalent to a collection of Finsler geometries with several Finsler metrics depending both on the position and on the velocity. As a result the Riemannian geodesic flow is replaced by a collection of Finsler flows. This naturally leads to a model in which a particle is described by several mass parameters. If these mass parameters are different then the equivalence principle is violated. In the non-relativistic limit this also leads to corrections to the Newton's gravitational potential. We find the first and second order corrections to the usual Riemannian geodesic flow and evaluate the anomalous nongeodesic acceleration in a particular case of static spherically symmetric background.