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 $n\ge 3$), 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.

25 pages, The results become more rigorous, some physical examples and physical meaning of the geometric objects have been added, all results of Cheng-Shen Finslerian field equation for positive-definite generalized m-Kropina metrics have been deleted as this equation could not be applied for our singular metrics