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, ℏ, and string, := 3 s ∕6ℏ, 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 t but with possible dependencies on momentum like coordinates on star deformed phase spaces. This geometric techniques of decoupling and integrating nonlinear systems of physically important partial differential equations (the anholonomic frame and connection deformation method, AFCDM) is applied in our partner works for constructing nonassociative and locally anisotropic generalizations of black hole and cosmological solutions and elaborating geometric flow evolution and classical and quantum information theories.

We elaborate on nonassociative differential geometry of phase spaces endowed with nonholonomic (non-integrable) distributions and frames, nonlinear and linear connections, symmetric and nonsymmetric metrics, and correspondingly adapted quasi-Hopf algebra structures. The approach is based on the concept of nonassociative star product introduced for describing closed strings moving in a constant R-flux background. Generalized Moyal-Weyl deformations are considered when, for nonassociative and noncommutative terms of star deformations, there are used nonholonomic frames (bases) instead of local partial derivatives. In such modified nonassociative and nonholonomic spacetimes and associated complex/real phase spaces, the coefficients of geometric and physical objects depend both on base spacetime coordinates and conventional (co) fiber velocity/momentum variables like in (non) commutative Finsler-Lagrange-Hamilton geometry. For nonassociative and (non) commutative phase spaces modelled as total spaces of (co) tangent bundles on Lorentz manifolds enabled with star products and nonholonomic frames, we consider associated nonlinear connection, N-connection, structures determining conventional horizontal and (co) vertical (for instance, 4+4) splitting of dimensions and N-adapted decompositions of fundamental geometric objects. There are defined and computed in abstract geometric and N-adapted coefficient forms the torsion, curvature and Ricci tensors. We extend certain methods of nonholonomic geometry in order to construct R-flux deformations of vacuum Einstein equations for the case of N-adapted linear connections and symmetric and nonsymmetric metric structures. Introduction, Motivations and GoalsWe initialize a research on nonassociative nonholonomic geometry induced by R-flux backgrounds in string gravity and classical and quantum models on phase spaces modelled as (co) tangent Lorentz bundles endowed with nonholonomic (non-integrable, equivalently, anholonomic) structures and nonsymmetric metrics. Such a nonholonomic geometric formalism was elaborated for various types of (non) commutative gravity and matter field theories, modified (relativistic) geometric flows, and classical and quantum information theories, see details in a series of our previous works [1][2][3][4][5][6][7][8][9][10][11] and references therein. In this article, nonassociative geometric and physical models are formulated in a form which is accessible to researchers in particle physics and modern cosmology. There are provided nonholonomic generalizations of two recent models of nonassociative geometry and vacuum gravity due to [12] and, with

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We show that a geometric techniques can be elaborated and applied for constructing generic off-diagonal exact solutions in f (R, T )-modified gravity for systems of gravitational-Yang-Mills-Higgs equations. The corresponding classes of metrics and generalized connections are determined by generating and integration functions which depend, in general, on all space and time coordinates and may possess, or not, Killing symmetries. For nonholonomic constraints resulting in Levi-Civita configurations, we can extract solutions of the Einstein-Yang-Mills-Higgs equations. We show that the constructions simplify substantially for metrics with at least one Killing vector. There are provided and analyzed some examples of exact solutions describing generic off-diagonal modifications to black hole/ellipsoid and solitonic configurations.MSC 2010: 83T13, 83C15 (primary); 53C07, 70S15 (secondary)

We construct nonassociative quasi-stationary solutions describing deformations of Schwarzschild black holes, BHs, to ellipsoid configurations, which can be black ellipsoids, BEs, and/or BHs with ellipsoidal accretion disks. Such solutions are defined by generic off-diagonal symmetric metrics and nonsymmetric components of metrics (which are zero on base four dimensional, 4-d, Lorentz manifold spacetimes but nontrivial in respective 8-d total (co) tangent bundles). Distorted nonassociative BH and BE solutions are found for effective real sources with terms proportional to $$\hbar \kappa $$ ħ κ (for respective Planck and string constants). These sources and related effective nontrivial cosmological constants are determined by nonlinear symmetries and deformations of the Ricci tensor by nonholonomic star products encoding R-flux contributions from string theory. To generate various classes of (non) associative /commutative distorted solutions we generalize and apply the anholonomic frame and connection deformation method for constructing exact and parametric solutions in modified gravity and/or general relativity theories. We study properties of locally anisotropic relativistic, optically thick, could and thin accretion disks around nonassociative distorted BHs, or BEs, when the effects due to the rotation are negligible. Such configurations describe angular anisotropic deformations of axially symmetric astrophysical models when the nonassociative distortions are related to the outer parts of the accretion disks.