MG14 - Talk detail

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A brief summary of the anholonomic frame deformation method, AFDM, for generating exact solutions with generic off-diagonal metrics and generalized nonlinear connections in modified gravity theories, MGTs, is presented. We generalize the method to systems of nonlinear partial differential equations, PDEs, for constructing solutions describing generalized (effective) Einstein-Yang-Mills-Higgs, EYMH, interactions in two measure, f(R) and massive gravity theories. Finally, we speculate on possible applications of such generalized anisotropic and inhomogeneous solutions in modern acceleration cosmology.

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Let $\mathbf{g}$ be a pseudo-Riemanian metric on a manifold $\mathbf{V}$ with conventional $n+n$ dimensional splitting, $\ n\geq 2,$ for a nonholonomic (non-integrable) distribution $\mathcal{N}$ and consider a correspondingly adapted linear metric compatible connection $\widehat{\mathbf{D}}$ and its torsion $\widehat{\mathcal{T}}$, both completely determined by $\mathbf{g}$. We prove that there are certain generalized frame and/or jet transforms and prolongations with $(\mathbf{g},\mathbf{V})\rightarrow (\widehat{\mathbf{g}},\widehat{\mathbf{V}})$ into explicit classes of solutions of some generalized Einstein equations $\widehat{\mathbf{R}}\mathit{ic}=\Lambda \widehat{\mathbf{g}}$, $\Lambda =const$, encoding various types of (nonholonomic) Ricci soliton configurations and/or jet variables and symmetries, in particular, subject to the condition $\widehat{\mathcal{T}}=0$. This allows us to construct in general form generic off-diagonal exact solutions depending on all space time coordinates on $\mathbf{V}$ and its jet prolongations, via generating and integration functions and various classes of constants and associated symmetries. We consider an example when exact solutions are constructed as nonholonomic jet prolongations of the Kerr metrics, with possible Ricci soliton deformations, and characterized by generalized connections.

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We review the theory of geometric flows on nonholonomic manifolds and tangent bundles and self-similar configurations resulting in generalized Ricci solitons and Einstein-Finsler equations. There are provided new classes of exact solutions on Finsler-Lagrange f(R,F,L)-modifications of general relativity and discussed possible implications in acceleration cosmology.

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We find general parameterizations for generic off-diagonal spacetime metrics and matter sources in general relativity (GR) and two measure/ bi-metric modified gravity theories, MGTs, when the field equations decouple with respect to nonholonomic frames of reference. This allows us to construct various classes of exact solutions when the coefficients of the fundamental geometric/physical objects depend on all spacetime coordinates via corresponding classes of generating and integration functions and/or constants. Such (modified) spacetimes display Killing and non-Killing symmetries, describe nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants. Certain examples of exact locally anisotropic wormhole and generic off-diagonal cosmological solutions are analysed for MGTs of massive and/or modified f(R,T) gravity, two measure theory etc. We conclude that considering generic off-diagonal nonlinear parametric interactions in GR it is possible to mimic various effects in massive and/or modified gravity, or to distinguish certain classes of “generic” modified gravity solutions which cannot be encoded in GR.

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We outline a geometric method of constructing generic off--diagonal and diagonal cosmological solutions of effective Einstein equations modeling modified gravity theories with two non-Riemannian volume forms and associated bimetric and/or biconnection geometric structures. Such solutions are determined by generating functions, effective sources and integration constants and characterized by nonholonomic frame torsion effects. In the physical Einstein frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions which allows unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are "emergent universes" described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce in two measure theories massive gravity effects.

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