Critical Remarks on Finsler Modifications of Gravity and Cosmology by Zhe Chang and Xin Li

TL;DR

This paper critiques recent claims about Finsler geometry in gravity and cosmology, emphasizing the existence of multiple connections and the importance of metric compatibility for realistic physical models.

Contribution

It clarifies the misconceptions about the uniqueness and properties of Finsler connections, advocating for the use of metric-compatible connections in physical theories.

Findings

Finsler geometry admits multiple linear connections with the same metric structure.

The Chern connection is not generally metric compatible, contrary to some claims.

Using metric-compatible connections like Cartan's can lead to more realistic Finsler gravity models.

Abstract

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in Phys. Lett., respectively, B668 (2008) 453 and B676 (2009) 173). They consider that \textit{"In Finsler manifold, there exists a unique linear connection - the Chern connection ... It is torsion freeness and metric compatibility ... "}. There are well known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections \textbf{are not metric compatible.} For instance, the Chern's one (being with zero torsion and "weak" compatibility on the base manifold of tangent bundle) is not generally compatible with the metric structure on total space. This results in a number of additional difficulties and sophistication in definition of Finsler spinors and Dirac operators and in additional problems with further generalizations for quantum gravity and noncommutative/string/brane/gauge theories. I conclude that standard physics theories can be generalized naturally by gravitational and matter field equations for the Cartan and/or any other Finsler metric compatible connections. This allows us to construct more realistic models of Finsler spacetimes, anisotropic field interactions and cosmology.