On the generalized $m$-Kropina metrics

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 $m \in 2 \mathbb{Z}$, the arithmetic nature of $m$ imposes strong rigidity constraints: 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. Furthermore, we show that if $F$ has almost isotropic flag curvature ($m \in 2 \mathbb{Z}$ and $n \geq 3$), then the flag curvature is constant. We, hence, deduce under what conditions a generalized $m$-Kropina metric $F$ becomes an exact solution to "Pfeifer and Wohlfarth's vacuum field equation". Finally, we provide several four-dimensional examples arising in modified gravity and cosmology.