Spherically symmetric, asymptotically flat Berwald vacuum solutions in Finsler gravity

Abstract: So-called Berwald-Finsler spacetimes are Finsler spacetimes that are closest to pseudo-Riemannian geometry, as their canonical nonlinear connection defines an affine connection on spacetime. In spherical symmetry, these geometries can be used to describe the gravitational field outside of compact objects. We solve the Finsler gravity vacuum equation for $SO(3)$-symmetric Berwald spacetimes that are asyptotically flat, but not Ricci flat. We find that among all spherically symmetric Berwald spacetimes, only one class is compatible with asymptotic flatness and a well defined causal structure. For this class, we completely solve the Finsler gravity vacuum equation and find three families of non-Ricci flat solutions -- which represent the first non-trivial, exact spherically symmetric vacuum solutions. They are so-called $(α,β)$-Finsler spacetimes that are constructed from a pseudo-Riemannnian metric and a 1-form. In particular, we show, by providing a concrete example, that in Finsler geometry there exist $SO(3)$-symmetric, asymptotically flat vacuum solutions that are not Ricci flat; these solutions are promising candidates to model the gravitational field around compact objects, beyond their Riemannian description.

• The paper introduces explicit non-Ricci-flat, spherically symmetric vacuum solutions in Berwald-Finsler gravity, breaking GR’s uniqueness theorem.
• It employs an (α,β)-type Finsler Lagrangian with a parallel 1-form to derive well-defined causal and asymptotically flat spacetimes.
• Key families of solutions reveal observable deviations in test particle dynamics, offering new insights for compact object exteriors.

Spherically Symmetric, Asymptotically Flat Berwald Vacuum Solutions in Finsler Gravity

Introduction and Theoretical Context

This work presents an exhaustive analysis of spherically symmetric, asymptotically flat vacuum solutions within the framework of Berwald-type Finsler gravity. In contrast to General Relativity (GR), where the uniqueness of the Schwarzschild solution in the spherically symmetric vacuum is guaranteed by the Jebsen-Birkhoff theorem, Finslerian generalizations admit new geometric structures where the standard correspondence between vacuum solutions and Ricci flatness is broken. The study leverages recent developments in Finsler geometry, especially the action-based scalar field equations that provide physically motivated dynamics tied to pseudo-Finsler spacetimes. Of particular importance are Berwald spaces, in which the canonical nonlinear Finsler connection is equivalent to an affine spacetime connection, making them the closest Finslerian analogues to (pseudo-)Riemannian geometry.

Berwald-Finsler Geometry and Field Equations

The analysis is restricted to Finsler spacetimes of Berwald type, characterized by the existence of a symmetric affine connection $\Gamma$ so that autoparallel and geodesic curves coincide. While any Ricci-flat Berwald metric solves the Finslerian vacuum equation, the converse does not hold—there exist non-Ricci-flat Berwald metrics that satisfy the Finsler gravity field equations.

The vacuum field equation for Berwald-Finsler spacetimes simplifies to

$$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$

where $R_{ab}(x)$ is the Ricci tensor of the Berwald connection, and $g^{ab}$ is the inverse Finsler metric. This points to a dramatic extension beyond Einstein's vacuum equations, wherein spatial symmetry does not constrain the spacetime to be either Minkowski or Schwarzschild.

Classification of Spherically Symmetric Berwald-Finsler Spacetimes

Utilizing the complete local classification of $SO(3)$-symmetric Berwald structures, the authors prove that demanding asymptotic flatness and a non-degenerate causal structure restricts admissible solutions to a unique subclass among five possible ones: so-called "Class 3" metrics. These are precisely $(\alpha,\beta)$-type Finsler Lagrangians, expressible in the form

$$L = \mathbf{B}^2 \Theta\left( \frac{\mathbf{A}}{\mathbf{B}^2} \right) = \mathbf{A} \Psi(s), \qquad s = \frac{\mathbf{B}^2}{\mathbf{A}}$$

with $\mathbf{A}$ a pseudo-Riemannian metric, $\mathbf{B}$ an absolutely parallel 1-form, and $\Theta$, $$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$0 arbitrary functions subject to non-degeneracy constraints. This class exhibits a canonical affine connection that is Levi-Civita for $$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$1 and for which the 1-form $$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$2 is covariantly constant.

Exact Families of Non-Ricci-flat, Asymptotically Flat Solutions

Through a detailed and explicit integration of the Berwald field equation, three two-parameter families of solutions are obtained. These solutions are not Ricci flat yet possess asymptotic flatness and well-defined causal structures, thereby providing the first such exact families in Finsler gravity:

1. Branch 1 ($$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$3): Solutions where the Ricci scalar of $$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$4 is a nonzero constant; the general form of the Finslerian factor $$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$5 is

$$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$6

with $$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$7 and $$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$8 for a non-degenerate cone.

2. Branch 2 ($$(g^{ab}(x,\dot{x}) - 3 \frac{\dot{x}^a \dot{x}^b}{L(x,\dot{x})}) R_{ab}(x) = 0,$$9): Solutions for vanishing Ricci scalar for $R_{ab}(x)$0, split into two subfamilies defined by the sign in front of a square root. The solutions are

$R_{ab}(x)$1

with integration constants $R_{ab}(x)$2, $R_{ab}(x)$3, and $R_{ab}(x)$4.

The solutions recover a $R_{ab}(x)$5-type Finsler geometry at the level of the fundamental function and always admit well-defined lightcone structures at spatial infinity. The construction includes a fully explicit example with all relevant coordinate dependencies specified, including a pseudo-Riemannian sector with constant scalar curvature.

Key Properties and Physical Implications

Contradiction with GR expectations: The explicit non-Ricci-flat Berwald vacuum solutions break the uniqueness result familiar from general relativity, as Ricci flatness is no longer necessary for validity of the vacuum equation.

Geodesic Equivalence: For all Class 3 solutions, Finslerian geodesics coincide with those of the Riemannian metric $R_{ab}(x)$6, but proper time measurements are given by the full Finslerian length, implying observable deviations for test particle dynamics, potentially with relevance for compact object exteriors and their observable properties.

$R_{ab}(x)$7 Structure and Physical Candidates: The solutions are of $R_{ab}(x)$8-type, the 1-form $R_{ab}(x)$9 is absolutely parallel, and the underlying Riemannian metric has either constant nonzero or vanishing Ricci scalar, depending on the family. These models thus offer concrete, physically viable candidates for describing the static exteriors of compact objects beyond the Riemannian regime.

Causality and Asymptotic Behavior: Causality is preserved (non-degenerate lightcones exist where required), and spatial infinity is flat (all curvature components vanish as $g^{ab}$0). The vacua are shown to have consistent asymptotic structures compatible with physical expectations.

Future Developments

The explicit identification of non-Ricci-flat, exact vacuum solutions broadens the landscape of physically meaningful Finslerian geometries. Immediate directions include:

• Investigation of phenomenological implications for light propagation, stability, and observables in compact object exteriors
• Generalization to axially symmetric rotating solutions, potentially relevant for Finslerian analogs of Kerr metrics
• Exploration of the impact on the global properties of spacetimes (horizons, causal boundaries)
• Extension to the inclusion of matter and coupling to kinetic gases, leveraging the kinetic foundations of the employed field equation

Conclusion

This work achieves a complete classification and explicit construction of spherically symmetric, asymptotically flat, non-Ricci-flat vacuum solutions in Berwald-Finsler gravity. The solutions are necessarily of $g^{ab}$1-type with an absolutely parallel $g^{ab}$2-form and either constant or vanishing underlying Riemannian Ricci scalar. The results establish the existence of physically viable, nontrivial Finslerian generalizations of Schwarzschild-like exteriors, opening new directions for both the mathematical structure and physical phenomenology of Finsler gravity (2606.05427).