The ladder of Finsler-type objects and their variational problems on spacetimes
Miguel Sánchez, Fidel F. Villaseñor
TL;DR
The paper develops a unified ladder for anisotropic tensors on spacetimes, built from the functorial actions of the Liouville contraction $\imath_{\mathbb{C}}$ and the vertical derivative $\dot{\partial}$ on $\alpha$-homogeneous objects, connecting geodesic sprays, nonlinear connections, anisotropic connections, and linear connections, and extending this framework to metric-type objects via Legendre transforms. A central result is the relation $\overset{\alpha}{\imath_{\mathbb{C}}}\circ\underset{\alpha}{\dot{\partial}}=\alpha\,\mathrm{Id}$, which organizes transitions and defines residues that recover torsion and the Landsberg tensor; the paper also analyzes obstructions and signatures when moving metric-type objects on the ladder. The authors apply this structure to compare and relate variational formulations of generalized Einstein equations in semi-Finsler/Lorentz-Finsler gravity (Pfeifer-Wohlfarth-Hohmann-Voicu, Javaloyes-Sánchez-Villaseñor, García-Parrado-Minguzzi), providing a method to transport functionals across ladder levels and offering a geometric explanation for why certain non-quadratic Lorentz norms are not vacuum solutions. Overall, the ladder framework yields a robust, coordinate-invariant language to study the interplay between anisotropic geometry and variational gravity theories, clarifying how choices of objects and levels affect admissible actions, variations, and solutions.
Abstract
The space of anisotropic $r$-contravariant $s$-covariant $α$-homogeneous tensors on a manifold admits a functorial structure where vertical derivatives $\dot{\partial}$ and contractions $\imath_{\mathbb{C}}$ by the Liouville vector field $\mathbb{C}$ are operators which maintain $s+α$ constant. In (semi-)Finsler geometry, this structure is transmitted faithfully to connection-type elements yielding the following ladder: geodesic sprays / nonlinear connections / anisotropic connections / linear (Finslerian) connections. However, it is more loosely transmitted to metric-type ones: Finslerian Lagrangians / Legendre transformations / anisotropic metrics. We will study this structure in depth and apply it to discuss the recent variational proposals (Einstein-Hilbert, Einstein-Palatini, Einstein-Cartan) for generalizing Einstein equations to the Finsler setting.