Finsler Geometry Applications in Gravity Theories

Finsler geometry generalises the Riemannian framework by allowing the metric to depend not only on spacetime position but also on the tangent‐vector direction, thereby introducing intrinsic anisotropy. In gravity theories, this structure yields extensions of general relativity that naturally incorporate directional dependencies, modified dispersion relations and possible violations of local Lorentz invariance. The formalism employs nonlinear connections and generalised curvature tensors, leading to field equations that encompass extra degrees of freedom beyond those of the Einstein tensor. Such models have been explored in contexts ranging from anisotropic cosmologies and bounce scenarios to exotic compact objects and traversable wormholes. By capturing velocity-dependent effects, Finsler‐based gravitational theories open new pathways for addressing singularity resolution, dark sector phenomenology and potential observational signatures in high-energy astrophysics and cosmology.

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Technical terms

Finsler metric: A generalised distance function on the tangent bundle that depends on both position and direction, allowing anisotropic line elements.

Randers space: A class of Finsler manifold whose metric is the sum of a Riemannian component and a one-form, often employed to model anisotropic gravitational effects.

Berwald space: A Finsler manifold whose connection coefficients depend only on position, yielding parallel transport rules analogous to affine connections in Riemannian geometry.

Nonlinear connection: A decomposition of the tangent bundle into horizontal and vertical subspaces, essential for defining curvature, torsion and geodesics in Finsler geometry.

Anisotropy: Directional dependence of geometric or physical quantities, central to Finsler extensions of isotropic spacetime models and key to novel gravitational phenomena.

References

1. Possible existence of traversable wormhole in Finsler–Randers geometry. European Physical Journal C (2023).
2. Cosmological Finsler Spacetimes. Universe (2020).
3. Bounce Cosmology in Generalized Modified Gravities. Universe (2019).
4. Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry. European Physical Journal C (2018).