Abstract
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 (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.
Highlights
1.1 Motivations for nonassociative geometry, physics and gravityNonassociative algebras and nonassociative and noncommutative theories have a long and diverse history in mathematics and physics
Star products are defined in terms of coordinate bases ∂ when nonassociative generalizations of Riemann geometry are constructed with symmetric, g, and nonsymmetric, G, star-metric structures and a related nonassociative variant of LC-connection ∇
Our geometric method of constructing solutions in modified gravity theories and GR was extended to a level bearing direct relevance to observable nonassociative contributions using for relativistic thin disk models around such compact objects
Summary
Nonassociative algebras and nonassociative and noncommutative theories have a long and diverse history in mathematics and physics. There were studied certain cases with solitonic gravitational metrics for various classes of nonlinear waves when PDEs are solved using other geometric and analytic techniques Such methods can not be applied if our aim is to find exact solutions with generic off-diagonal metrics of type gkj (xi ) and/or gαβ (xi , pa) depending on some spacetime/phase space coordinates, generalized connections and/or LC-connections, even we project all geometric objects and equations on a GR background. There is a rigorous proof in [26] that nonassociative vacuum Einstein equations (1) can be decoupled in general form for quasi-stationary phase spaces and a nontrivial cosmological constant Using such a decoupling property and respective classes of nonlinear symmetries, we can encode the cofiber dynamics into real effective R-flux sources as in (2) and/or cosmological constants. Those formulas will be used in our further partner works in order to construct, for instance, nonassociative BH and solitonic solutions on 8-d phase spaces generalizing some respective classes of solutions from [37,38]
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