On the generalized $m$-Kropina metrics
Ebtsam H. Taha
TL;DR
The paper analyzes generalized $m$-Kropina metrics in pseudo-Finsler geometry, proving they are almost rational (AR-Finsler) in the directional variable $y$ and extracting the rationality properties of their Finslerian objects, including the geodesic spray. It establishes rigorous rigidity results: for $m otinbZ$, Einstein implies Ricci-flat; for $m otin 2bZ$, isotropic/vanishing-curvature conditions yield strong reductions to Berwald/Landsberg types or $ oldsymbol{H}=0$, and for $m otin 2bZ$ almost isotropic flag curvature leads to constant flag curvature under suitable dimensions. The work further connects these metrics to Finsler gravity, identifying conditions under which they solve Chen–Shen or Pfeifer–Wohlfarth field equations, and provides explicit 4D examples with cosmological and gravity applications. Overall, it clarifies structural and physical implications of generalized $m$-Kropina metrics and offers practical criteria for their use as exact solutions in Finslerian gravity.
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.