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

Abstract: 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.

• The paper establishes analytic conditions to reduce Finsler gravity's vacuum equation to a Ricci-flat form in homogeneous and isotropic Landsberg spacetimes.
• It details how the regularity of Finsler function powers ensures non-degeneracy on light cones, linking both metric and Palatini formulations.
• The findings facilitate explicit cosmological models, exemplified by the cosmological unicorn spacetime, with implications for alternative dark energy theories.

Reduction of the Finsler Gravity Vacuum Equation for Cosmological Landsberg Spacetimes

Overview

The paper systematically develops new analytic methods to reduce the scalar Finsler gravity vacuum field equations to the vanishing of the Finslerian Ricci curvature for a broad class of Finsler spacetimes, with special emphasis on homogeneous and isotropic Finsler geometries of Landsberg type. The authors rigorously establish sufficient conditions on the regularity of certain powers of the Finsler function to enable this reduction. Their analysis covers both purely metric and Palatini formulations, providing precise mathematical criteria for when the simplification to Ricci-flatness is justified, and they demonstrate implications for the construction of explicit solutions—such as the cosmological unicorn geometry.

Mathematical Formulation and Reduction Criteria

The central technical achievement is the proof that the vacuum Finsler gravity equation—typically a complicated scalar partial differential equation involving curvature and Landsberg terms—reduces to the Ricci-flat Finslerian equation $R=0$ (or $R=0$ in Palatini formalism) under specific regularity conditions. The authors dissect the cases where the Landsberg term vanishes, primarily focusing on Landsberg and Berwald geometries, and elucidate two scenarios:

1. Regularity of Finsler Function's Powers: If there exists an integer $n \geq 2$ such that $F^n$ is sufficiently smooth (at least $C^8$) and its associated $n$-Finsler metric is non-degenerate on the light cones $\partial T$, then the reduction to $R=0$ is valid.
2. Analyticity and Partial Regularity: For certain analytic and non-degenerate powers of $F$ (with technical restrictions on $n$), even if the regularity holds only on subsets of the light cone, the reduction is still possible. This accommodates broader classes of Finsler spacetimes, including those with metric singularities or weaker differentiability.

Formal proofs leverage maximum principles for elliptic PDEs and division-by-zero arguments to show that the variable $R=0$0 vanishes identically in the relevant domains, provided these regularity criteria.

Structure of the Vacuum Equation

The Finsler gravity field equations, derived from the action principle, are presented in both metric and Palatini formulations. Their generic complexity rests with the interplay between curvature, metric, and the Landsberg tensor:

• Metric Equation: $R=0$1
• Palatini Equation: Involves an independent nonlinear connection and yields additional tensorial structure.

The central result is that in Landsberg and Berwald spacetimes (for which the Landsberg tensor trace vanishes), and under the stated regularity conditions, these equations reduce to $R=0$2, analogous to the Ricci-flat vacuum form in general relativity.

Application: Cosmological Unicorn Geometry

The practical utility of the reduction criteria is exemplified through the cosmological unicorn Finsler spacetime—a homogeneous, isotropic geometry of Landsberg type found in previous literature. The authors analyze the Finsler function and demonstrate powers of $R=0$3 on the relevant causal cones (timelike and lightlike directions) satisfy the non-degeneracy and analyticity conditions necessary for their reduction theorems. Thus, the field equation for the scale factor $R=0$4, governing cosmological expansion, simplifies drastically:

• The explicit Ricci scalar is algebraically solvable, yielding a scale factor linear in $R=0$5 and requiring vanishing or negative spatial curvature for physical solutions.

This reduction enables analytic cosmological modeling within Finsler gravity frameworks and facilitates comparative studies with standard cosmological models.

Implications and Future Directions

The results clarify the physical and mathematical motivation for working with arbitrary powers of the Finsler function, as opposed to restricting analysis to $R=0$6 as commonly done. This broadens the landscape of possible Finsler geometry models and ensures the equivalence between action-based Finsler gravity equations and the Ricci-flatness condition, foundational for constructing tractable solutions.

Practically, these reduction techniques accelerate the search for exact vacuum solutions in Finsler gravity, underpinning investigations into alternatives to dark energy via kinetic gas matter couplings. Theoretically, the precise regularity criteria and analytic framework open paths for extending the methodology to matter-coupled Finsler spacetimes and for systematically generalizing energy conditions (weak, strong, null) in the Finslerian context.

Anticipated future work includes analysis of nonzero Landsberg terms in matter-dominated universes, and reformulation of convergence and energy conditions paralleling those in Lorentzian relativity. The approach detailed herein stands to impact both the foundational theory and practical solution space of Finsler gravity.

Conclusion

The paper rigorously establishes analytic criteria for reducing the Finsler gravity vacuum equations to Ricci-flatness under Landsberg-type regularity, both expanding and simplifying the solution space for action-based Finsler gravity. The methods bridge previously unresolved gaps in the reduction of scalar equations for Finsler spacetimes, facilitate tractable construction of explicit cosmological solutions, and lay mathematical groundwork for future advances in Finsler gravity and its potential applications to cosmology and gravitational theory (2606.24698).