Decoupling and Integrability of Nonassociative Vacuum Phase Space Gravitational Equations With Star and R‐flux Parametric Deformations

We prove that nonassociative star deformed vacuum Einstein equations can be decoupled and integrated in certain general forms on phase spaces involving real R-flux terms induced as parametric corrections on base Lorentz manifold spacetimes. The geometric constructions are elaborated with parametric (on respective Planck, $\hbar $, and string, $\kappa :=\mathit{\ell }_{s}^{3}/6\hbar $, constants) and nonholonomic dyadic decompositions of fundamental geometric and physical objects. This is our second partner work on elaborating nonassociative geometric and gravity theories with symmetric and nonsymmetric metrics, (non) linear connections, star deformations defined by generalized Moyal-Weyl products, endowed with quasi-Hopf algebra, or other type algebraic and geometric structures, and all adapted to nonholonomic distributions and frames. We construct exact and parametric solutions for nonassociative vacuum configurations (with nontrivial or effective cosmological constants) defined by star deformed generic off-diagonal (non) symmetric metrics and (generalized) nonlinear and linear connections. The coefficients of geometric objects defining such solutions are determined by respective classes of generating and integration functions and constants and may depend on all phase space coordinates [spacetime ones, $(x^{i},t)$; and momentum like variables, $(p_{a},E$)]. Quasi-stationary configurations are stated by solutions with spacetime Killing symmetry on a time like vector $\partial _{t}$ but with possible dependencies on momentum like coordinates on star deformed phase spaces.