On the generalized $m$-Kropina metrics

TL;DR

This paper studies generalized m-Kropina metrics, proving their rationality properties, conditions for Einstein and special curvature types, and providing examples relevant to modified gravity and cosmology.

Contribution

It establishes the rationality of geometric objects of generalized m-Kropina metrics and characterizes their curvature properties and solutions to modified gravity field equations.

Findings

Geodesic spray coefficients are rational in y.

If Einstein with m not integer, then Ricci-flat.

Conditions under which metrics solve modified gravity field equations.

Abstract

Generalized $m$-Kropina metrics appear naturally as a spacetime geometry compatible with Lorentz symmetry breaking, leading to useful applications in modified gravity and cosmology. We prove that a generalized $m$-Kropina metric $F$ is an almost rational Finsler metric. Thereby, we study the rationality of its Finslerian geometric objects in the directional variable $y$. For example, its geodesic spray coefficients are rational in $y$. Consequently, we prove that if $F$ is an Einstein metric with $m \notin \mathbb{Z}$, then it is Ricci-flat. Moreover, for $m \in 2 \mathbb{Z}$, if $F$ has isotropic mean Berwald curvature, or has relatively isotropic Landsberg curvature, or has almost vanishing $\mathbf{H}$-curvature, then $F$ is weakly Berwaldian, or $F$ is Landsbergian, or $\mathbf{H}=0$, respectively. We, hence, deduce under what conditions a generalized $m$-Kropina metric $F$ becomes an exact solution to either "Chen and Shen's Finslerian nonvcuum field equations"or "Pfeifer and Wohlfath's vacuum field equation". Finally, some examples of generalized $m$-Kropina metrics in dimension $4$, which has significant applications in modified gravity and cosmology, are provided.