The Anholonomic Frame and Connection Deformation Method for constructing off-diagonal solutions in (modified) Einstein gravity and nonassociative geometric flows and Finsler-Lagrange-Hamilton theories

TL;DR

The paper presents the Anholonomic Frame and Connection Deformation Method (AFCDM) as a comprehensive framework to construct generic off-diagonal solutions in general relativity, modified gravity theories, and nonassociative phase-space geometries. By employing nonholonomic (2+2) (and higher shell) splittings, N-connections, and a canonical d-connection  hat{D}, the method decouples complex nonlinear PDEs and yields exact and parametric solutions, including black holes, wormholes, and cosmological models with local anisotropy. It extends these constructions to 8-d phase spaces with star-product Deformations, enabling nonassociative Finsler-Lagrange-Hamilton geometries and nonholonomic geometric flows guided by Perelman-type functionals. The work also develops a detailed formalism for nonassociative geometric flows, supports concrete off-diagonal solutions (Kerr-KdS, cylindrical, toroidal horizons, BE configurations), and provides a roadmap to connect these classical-geometric structures with quantum gravity and deformation quantization scenarios. Overall, AFCDM offers a versatile, systematically decoupled approach to generate and analyze physically rich off-diagonal solutions and their geometric-flow thermodynamics in both commutative and nonassociative settings, with extensive examples and tables in the Appendix to guide construction and interpretation.

Abstract

This article is a status report on the Anholonomic Frame and Connection Deformation Method, AFCDM, for constructing generic off-diagonal exact and parametric solutions in general relativity, GR, relativistic geometric flows, and modified gravity theories, MGTs. Such models can be generalized to nonassociative and noncommutative star products on phase spaces and modelled equivalently as nonassociative Finsler-Lagrange-Hamilton geometries. Our approach involves a nonholonomic geometric reformulation of classical models of gravitational and matter fields described by Lagrange and Hamilton densities on relativistic phase spaces. Using nonholonomic dyadic variables, the Einstein equations in GR and MGTs can be formulated as systems of nonlinear partial differential equations(PDEs), which can be decoupled and integrated in some general off-diagonal forms. In this approach, the Lagrange and Hamilton dynamics and related models of classical and quantum evolution are equivalently described in terms of generalized Finsler-like or canonical metrics and (nonlinear) connection structures on deformed phase spaces defined by solutions of modified Einstein equations. New classes of exact and parametric solutions in (nonassociative) MGTs are formulated in terms of generating and integration functions and generating effective/ matter sources. The physical interpretation of respective classes of solutions depends on the type of (non) linear symmetries, prescribed boundary/ asymptotic conditions, or posed Cauchy problems.