Einstein Gravity in Almost Kähler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics

Abstract

We argue that the Einstein gravity theory can be reformulated in almost Kähler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation on manifolds enabled with nonholonomic distributions is considered. We prove that, for certain types of nonholonomic constraints, there are modelled effective Lagrangians which do not develop instabilities. It is also elaborated a linearization formalism for anholonomic noncommutative gravity theories models and analyzed the stability of stationary ellipsoidal solutions defining some nonholonomic and/or nonsymmetric deformations of the Schwarzschild metric. We show how to construct nonholonomic distributions which remove instabilities in nonsymmetric gravity theories. It is concluded that instabilities do not consist a general feature of theories of gravity with nonsymmetric metrics but a particular property of some models and/or unconstrained solutions.

On General Solutions for Field Equations in Einstein and Higher Dimension Gravity

“…As we noted in Refs [4,37,38,39],. an equivalence of both types of filed equations would be possible, for instance, if we introduce a generalized source T k β k δ containing contributions of the distortion tensor(22).…”

“…In this paper, we do not analyze possible explicit symmetries and physical properties of such solutions. We consider such problems in our recent papers [20,21,22,23,24,38,41,42,43] and plan to provide further developments and applications in our future works. Acknowledgement: Author is grateful to M. Anastasiei for discussions and support.…”

Einstein Gravity, Lagrange-Finsler Geometry, and Nonsymmetric Metrics

“…We proved that the Einstein gravity can be re-written equivalently in so-called almost Kähler variables, and related Lagrange-Finsler variables, which is used for a study of stability of possible nonsymmetric metric generalizations [40]. But such stable configurations seem to exist only for some classes of nonholonomic constraints and under the condition of metric compatibility if we wont to preserve a physical limit to general relativity.…”

“…That stimulated certain interest of the scientific community (together with more or less motivated criticism) and a lot of discussions how to cure such theories [39,11,40]. More recently, new evidences coming from the Ricci flow theory, deformation quantization, geometric methods of constructing exact solutions of gravitational field and evolution equations, string gravity and nonholonomic spinor and Clifford-Lie algebroids etc has revealed a quite unexpected and additional theoretical support for NGT (for simplicity, in this work, we do not concern various speculations on some cosmological and astrophysical evidences, see details in [10,11,12]).…”

Nonholonomic relativistic diffusion and exact solutions for stochastic Einstein spaces

“…We have to solve certain systems of quadratic algebraic equations and define some e α ′ α (u β ), choosing a convenient system of coordinates u α ′ = u α ′ (u β ) 19. as a matter of principle, any solution of gravitational field equations with certain general matter fields sources can be represented in such a form by corresponding frame and coordinate transforms; we have to involve certain additional physical considerations, suppositions on symmetry of interactions and boundary conditions in order to model realistic gravitational interactions etc.…”