Nonholonomic Black Ring and Solitonic Solutions in Finsler and Extra Dimension Gravity Theories

We study stationary configurations mimicking nonholonomic locally anisotropic black rings (for instance, with ellipsoidal polarizations and/or imbedded into solitonic backgrounds) in three/six dimensional pseudo-Finsler/ Riemannian spacetimes. In the asymptotically flat limit, for holonomic configurations, a subclass of such spacetimes contains the set of five dimensional black ring solutions with regular rotating event horizon. For corresponding parameterizations, the metrics and connections define Finsler-Einstein geometries modeled on tangent bundles, or on nonholonomic (pseudo) Riemannian manifolds. In general, there are vacuum nonholonomic gravitational configurations which can not be generated in the limit of zero cosmological constant.

💡 Research Summary

The paper investigates stationary, non‑holonomic configurations that emulate locally anisotropic black rings—such as those with ellipsoidal polarizations or embedded in solitonic backgrounds—within three‑ and six‑dimensional pseudo‑Finsler or pseudo‑Riemannian spacetimes. The authors begin by introducing a non‑integrable (non‑holonomic) splitting of the tangent bundle into horizontal (base) and vertical (fiber) subspaces, characterized by a nonlinear connection (N‑connection). This structure allows the definition of adapted frames in which the metric acquires off‑diagonal components that encode anisotropic deformations.

Using a generalized Levi‑Civita connection compatible with the N‑connection, they derive a set of field equations that extend the usual Einstein equations by additional anisotropic stress‑energy terms. The metric ansatz is written in a block‑diagonal form with respect to the adapted frames: \