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

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 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. We consider possible applications of the AFCDM with explicit examples of off-diagonal deformations of black holes, cylindrical metrics and wormholes, black ellipsoids and torus configurations. In general, such solutions encode nonassociative and/or with geometric flow variables. For another types of generic off-diagonal (nonassociative) solutions, we study models with nonholonomic cosmological solitonic and spheroid deformations involving vertices and solitonic vacua for voids. We emphasize that such new classes of generic off-diagonal solutions can not be considered, in general, in the framework of the Bekenstein–Hawking entropy paradigm. This motivates relativistic/nonassociative phase space extensions of the G. Perelman thermodynamic approach to geometric flows and MGTs defined by nonholonomic Ricci solitons. In Appendix, Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, summarize the AFCDM for various classes of quasi-stationary and cosmological solutions in MGTs with 4-d and 10-d spacetimes and (nonassociative) phase space variables on (co) tangent bundles.