arXiv subjects

Fidel F. Villaseñor

Publications and source records attributed to Fidel F. Villaseñor.

7 recordsLinked to original sources

arXiv2026

Reduction of the Finsler gravity vacuum equation and dynamics for the cosmological Landsberg spacetimes

When solving the Einstein vacuum equations, a very helpful feature is that they reduce simply to the vanishing of the Ricci tensor. In Finsler gravity, a promising extension of general relativity that can describe the gravitational field of kinetic gases from a phase space perspective in terms of Finsler geometry, such a reduction is not as straightforward. In this article, we identify precise conditions under which the scalar Finsler gravity vacuum equation (in either its purely metric or its Palatini formulation) reduces to the vanishing of the Finslerian Ricci curvature. Through analytic arguments, we find that this happens if there exists some power $F^n$ of the Finsler function $F$ that is sufficiently regular and whose associated Finsler metric is non-degenerate on the light cones. Moreover, the Landsberg term in the scalar equation must vanish. This result significantly generalizes the findings of [Villasenor2024], where a reduction theorem was established under the quite strong assumption that $F^2$ is regular, which is not satisfied by many examples currently under consideration. We demonstrate the impact of our findings by applying them to solve the Finsler gravity equations for homogeneous and isotropic Finsler spacetime functions of Landsberg type.

gr-qc↗

arXiv2026

Einstein-Kropina Metrics and Their Application in Finsler Gravity

We generalize the Einstein condition for Kropina metrics obtained in the positive definite setting by Zhang, Shen and others to all signatures. As examples of Einstein-Kropina metrics $L=L_{a,b}$, we construct new explicit positive definite ones and the first ones with Lorentzian signature. Next, we classify all Einstein-Kropina solutions to the vacuum equation of Finsler gravity by Pfeifer and Wohlfarth, in arbitrary dimension and including a possible cosmological constant $Λ$. For the Lorentzian and positive definite cases, the local picture is as follows. In dimension 4 or lower, only the trivial solution exists: a Minkowski or Euclidean space, essentially. In dimension 5 and higher, the solutions are those $L_{a,b}$ for which the metric $a$ is a product of the real line with a Ricci-flat metric (Lorentzian or Riemannian) and the vector field $b$ is the unique unit vector on the first factor. As a very surprising rigidity phenomenon, all Einstein-Kropina solutions to the $Λ$-vacuum equation are Berwald and Ricci-flat, and the cosmological constant necessarily vanishes.

math-ph↗

arXiv2025

The ladder of Finsler-type objects and their variational problems on spacetimes

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.

math.DG↗

arXiv2023

Schur theorem for the Ricci curvature of any weakly Landsberg Finsler metric

The Ricci version of the Schur theorem is shown to hold for a wide class of Finsler metrics. What is more, let $F$ be any (positive definite) Finsler metric such that $\text{Ric} =ρF^2$ with $ρ\colon M^n\rightarrow\mathbb{R}$ (i.e., $(M^n,F)$ is Einstein) and $n\geq 3$. For $x\in M$, we express $\text{d}ρ_x$ as an average over the indicatrix in $\text{T}_xM$ of the Hilbert $1$-form weighted by a combination of derivatives of the mean Landsberg tensor. As a consequence of this general expression, if the metric is weakly Landsberg, then $ρ$ must be constant. The proof is based on the invariance of natural functionals under $\text{Diff}(M)$. Furthermore, we revisit an independent argument which proves the Schur theorem for the class of pseudo-Finsler metrics with quadratic Ricci scalar, improving previous results on the topic.

math.DG↗

arXiv2023

The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry

A systematic development of the so-called Palatini formalism is carried out for pseudo-Finsler metrics $L$ of any signature. Substituting in the classical Einstein-Hilbert-Palatini functional the scalar curvature by the Finslerian Ricci scalar constructed with an independent nonlinear connection $\mathrm{N}$, the affine and metric equations for $(\mathrm{N},L)$ are obtained. In Lorentzian signature with vanishing mean Landsberg tensor Lan$_i$, both the Finslerian Hilbert metric equation and the classical Palatini conclusions are recovered by means of a combination of techniques involving the (Riemannian) maximum principle and an original argument about divisibility and fiberwise analyticity. Some of these findings are also extended to (positive definite) Riemannian metrics by using the eigenvalues of the Laplacian. When Lan$_i\neq 0$, the Palatini conclusions fail necessarily, however, a good number of properties of the solutions remain. The framework and proofs are built up in detail.

math.DG↗

arXiv2022

Anisotropic connections and parallel transport in Finsler spacetimes

The general notion of anisotropic connections $\nabla$ is revisited, including its precise relations with the standard setting of pseudo-Finsler metrics, i.e., the canonic nonlinear connection and the (linear) Finslerian connections. In particular, the vertically trivial Finsler connections are identified canonically with anisotropic connections. So, these connections provide a simple intrinsic interpretation of a part of any Finsler connection closer to the Koszul formulation in $M$. Moreover, a new covariant derivative and parallel transport along curves is introduced, taking first a self-propagated vector ({\em instantaneous observer}) so that it serves as a reference for the propagation of the others. The covariant derivative of any anisotropic tensor is given by the natural derivative of a curve of tensors obtained by parallel transport along a curve and, in the case of pseudo-Finsler metrics, this is used to characterize the Levi-Civita--Chern anisotropic connection as the one that preserves the length of parallely propagated vectors.

math.DG↗

arXiv2022

On the significance of the stress-energy tensor in Finsler spacetimes

We revisit the physical arguments which lead to the definition of the stress-energy tensor $T$ in the Lorentz-Finsler setting $(M,L)$ starting at classical Relativity. Both the standard heuristic approach using fluids and the Lagrangian one are taken into account. In particular, we argue that the Finslerian breaking of Lorentz symmetry makes $T$ an anisotropic 2-tensor (i. e., a tensor for each $L$-timelike direction), in contrast with the energy-momentum vectors defined on $M$. Such a tensor is compared with different ones obtained by using a Lagrangian approach. The notion of divergence is revised from a geometric viewpoint and, then, the conservation laws of $T$ for each observer field are revisited. We introduce a natural {\em anisotropic Lie bracket derivation}, which leads to a divergence obtained from the volume element and the non-linear connection associated with $L$ alone. The computation of this divergence selects the Chern anisotropic connection, thus giving a geometric interpretation to previous choices in the literature.

gr-qc↗