arXiv subjects
Publications and source records attributed to Sergiu I. Vacaru.
The aims of this work are 1) to sketch a proof that there are such parameterizations of the local frame and canonical connection structures when the gravitational field equations in f(R,T)-modified gravity, MG, can be integrated in generic off-diagonal forms with metrics depending on all spacetime coordinates and 2) to provide some examples of exact solutions.
Using Finsler brane solutions [see details and methods in: S. Vacaru, Class. Quant. Grav. 28 (2011) 215001], we show that neutrinos may surpass the speed of light in vacuum which can be explained by trapping effects from gravity theories on eight dimensional (co) tangent bundles on Lorentzian manifolds to spacetimes in general and special relativity. In nonholonomic variables, the bulk gravity is described by Finsler modifications depending on velocity/ momentum coordinates. Possible super-luminal phenomena are determined by the width of locally anisotropic brane (spacetime) and induced by generating functions and integration functions and constants in coefficients of metrics and nonlinear connections. We conclude that Finsler brane gravity trapping mechanism may explain neutrino super-luminal effects and almost preserve the paradigm of Einstein relativity as the standard one for particle physics and gravity.
In this paper, nonholonomic gerbes will be naturally derived for manifolds and vector bundle spaces provided with nonintegrable distributions (in brief, nonholonomic spaces). An important example of such gerbes is related to distributions defining nonlinear connection (N-connection) structures. They geometrically unify and develop the concepts of Riemann-Cartan manifolds and Lagrange-Finsler spaces. The obstruction to the existence of a spin structure on nonholonomic spaces is just the second Stiefel-Whitney class, defined by the cocycle associated to a $\mathbb{Z} /2$ gerbe, which is called the nonholonomic spin gerbe. The nonholonomic gerbes are canonically endowed with N-connection, Sasaki type metric, canonical linear connection connection and (for odd dimension spaces) almost complex structures. The study of nonholonomic spin structures and gerbes have both geometric and physical applications. Our aim is to prove the Atiyah--Singer theorems for such nonholonomic spaces.
We model pseudo-Finsler geometries, with pseudo-Euclidean signatures of metrics, for two classes of four dimensional nonholonomic manifolds: a) tangent bundles with two dimensional base manifolds and b) pseudo-Riemannian/ Einstein manifolds. Such spacetimes are enabled with nonholonomic distributions and associated nonlinear connection structures and theirs metrics are solutions of the field equations in general relativity or in generalized gravity theories with nonholonomic variables. We rewrite the Schwarzschild metric in Finsler variables and use it for generating new classes of locally anisotropic black holes and (or) stationary deformations to ellipsoidal configurations. There are analyzed the conditions when such metrics describe imbedding of black hole solutions into nontrivial solitonic backgrounds.
We consider models of accelerating Universe elaborated for Finsler like gravity theories constructed on tangent bundles to Lorentz manifolds. In the osculating approximation, certain locally anisotropic configurations are similar to those for f(R) gravity. This allows us to generalize a proposal (by Nojiri, Odintsov and Saez-Gomez, arXiv: 1108.0767) in order to reconstruct and compare two classes of Einstein-Finsler gravity, EFG, and f(R) gravity theories using modern cosmological data and realistic physical scenarios. We conclude that EFG provides inflation, acceleration and little rip evolution scenarios with realistic alternatives to standard Lambda CDM cosmology. The approach is based on a proof that there is a general decoupling property of gravitational field equations in EFG and modified theories which allows us to generate off-diagonal cosmological solutions.
We prove the existence of hidden symmetries in the general relativity theory defined by exact solutions with generic off-diagonal metrics, nonholonomic (non-integrable) constraints, and deformations of the frame and linear connection structure. A special role in characterization of such spacetimes is played by the corresponding nonholonomic generalizations of Stackel-Killing and Killing-Yano tensors. There are constructed new classes of black hole solutions and studied hidden symmetries for ellipsoidal and/or solitonic deformations of "prime" Kerr-Sen black holes into "target" off-diagonal metrics. In general, the classical conserved quantities (integrable and not-integrable) do not transfer to the quantized systems and produce quantum gravitational anomalies. We prove that such anomalies can be eliminated via corresponding nonholonomic deformations of fundamental geometric objects (connections and corresponding Riemannian and Ricci tensors) and by frame transforms.
We study the geometric and physical foundations of Finsler gravity theories with metric compatible connections defined on tangent bundles, or (pseudo) Riemannian manifolds). There are analyzed alternatives to Einstein gravity (including theories with broken local Lorentz invariance) and shown how general relativity and modifications can be equivalently re-formulated in Finsler like variables. We focus on prospects in modern cosmology and Finsler acceleration of Universe. All known formalisms are outlined - anholonomic frames with associated nonlinear connection structure, the geometry of the Levi-Civita and Finsler type connections, all defined by the same metric structure, Einstein equations in standard form and/or with nonholonomic/ Finsler variables - and the following topics are discussed: motivation for Finsler gravity; generalized principles of equivalence and covariance; fundamental geometric/ physical structures; field equations and nonholonomic constraints; equivalence with other models of gravity and viability criteria. Einstein-Finsler gravity theories are elaborated following almost the same principles as in the general relativity theory but extended to Finsler metrics and connections. Gravity models with anisotropy can be defined on (co) tangent bundles or on nonholonomic pseudo-Riemannian manifolds. In the second case, Finsler geometries can be modelled as exact solutions in Einstein gravity. Finally, some examples of generic off-diagonal metrics and generalized connections, defining anisotropic cosmological Einstein-Finsler spaces are analyzed; certain criteria for Finsler accelerating evolution are analyzed.
In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative geometry, almost Kaehler and Clifford structures, Dirac operators and effective Lagrange-Hamilton and Riemann-Finsler spaces.
There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric non-compatible) connections, the Ricci flag curvature etc. In a series of works, we studied (non)commutative metric compatible Finsler and nonholonomic generalizations of the Ricci flow theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and references therein]. The goal of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived nonholonomic Hamilton evolution equations, when metric noncompatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric noncompatible connections.
We develop an approach to the theory nonholonomic relativistic stochastic processes on curved spaces. The Ito and Stratonovich calculus are formulated for spaces with conventional horizontal (holonomic) and vertical (nonholonomic) splitting defined by nonlinear connection structures. Geometric models of relativistic diffusion theory are elaborated for nonholonomic (pseudo) Riemannian manifolds and phase velocity spaces. Applying the anholonomic deformation method, the field equations in Einstein gravity and various modifications are formally integrated in general forms, with generic off-diagonal metrics depending on some classes of generating and integration functions. Choosing random generating functions we can construct various classes of stochastic Einstein manifolds. We show how various types of stochastic gravitational interactions with mixed holonomic/ nonholonomic and random variables can be modelled in explicit form and study their main geometric and stochastic properties. Finally, there are analyzed the conditions when non-random classical gravitational processes transform into stochastic ones and inversely.
We study possible links between quantum gravity phenomenology encoding Lorentz violations as nonlinear dispersions, the Einstein-Finsler gravity models, EFG, and nonholonomic (non-integrable) deformations to Hořava-Lifshitz, HL, and/or Einstein's general relativity, GR, theories. EFG and its scaling anisotropic versions formulated as Hořava-Finsler models, HF, are constructed as covariant metric compatible theories on (co) tangent bundle to Lorentz manifolds and respective anisotropic deformations. Such theories are integrable in general form and can be quantized following standard methods of deformation quantization, A-brane formalism and/or (perturbatively) as a nonholonomic gauge like model with bi-connection structure. There are natural warping/trapping mechanisms, defined by the maximal velocity of light and locally anisotropic gravitational interactions in a (pseudo) Finsler bulk spacetime, to four dimensional (pseudo) Riemannian spacetimes. In this approach, the HL theory and scenarios of recovering GR at large distances are generated by imposing nonholonomic constraints on the dynamics of HF, or EFG, fields.
We use an important decoupling property of gravitational field equations in the general relativity theory and modifications, written with respect to nonholonomic frames with 2+2 spacetime decomposition. This allows us to integrate the Einstein equations (eqs) in very general forms with generic off--diagonal metrics depending on all spacetime coordinates via generating and integration functions containing (un--)broken symmetry parameters. We associate families of off-diagonal Einstein manifolds to certain classes of covariant gravity theories which have a nice ultraviolet behavior and seem to be (super) renormalizable in a sense of covariant modifications of Horava-Lifshits gravity. The apparent breaking of Lorentz invariance is present in some "partner" anisotropically induced theories due to nonlinear coupling with effective parametric interactions determined by nonholonomic constraints and generic off-diagonal gravitational and matter fields configurations. Finally, we show how the constructions can be extended to include exact solutions for conjectured covariant reonormalizable models with Einstein-Yang-Mills-Higgs fields.
A consistent theory of quantum gravity (QG) at Planck scale almost sure contains manifestations of Lorentz local symmetry violations (LV) which may be detected at observable scales. This can be effectively described and classified by models with nonlinear dispersions and related Finsler metrics and fundamental geometric objects (nonlinear and linear connections) depending on velocity/ momentum variables. We prove that the trapping brane mechanism provides an accurate description of gravitational and matter field phenomena with LV over a wide range of distance scales and recovering in a systematic way the general relativity (GR) and local Lorentz symmetries. In contrast to the models with extra spacetime dimensions, the Einstein-Finsler type gravity theories are positively with nontrivial nonlinear connection structure, nonholonomic constraints and torsion induced by generic off-diagonal coefficients of metrics, and determined by fundamental QG and/or LV effects.
We show how the Einstein equations with cosmological constant (and/or various types of matter field sources) can be integrated in a very general form following the anholonomic deformation method for constructing exact solutions in four and five dimensional gravity (S. Vacaru, IJGMMP 4 (2007) 1285). In this letter, we prove that such a geometric method can be used for constructing general non-Killing solutions. The key idea is to introduce an auxiliary linear connection which is also metric compatible and completely defined by the metric structure but contains some torsion terms induced nonholonomically by generic off-diagonal coefficients of metric. There are some classes of nonholonomic frames with respect to which the Einstein equations (for such an auxiliary connection) split into an integrable system of partial differential equations. We have to impose additional constraints on generating and integration functions in order to transform the auxiliary connection into the Levi-Civita one. This way, we extract general exact solutions (parametrized by generic off-diagonal metrics and depending on all coordinates) in Einstein gravity and five dimensional extensions.
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and associated solitonic hierarchies. The constructions yield horizontal/vertical pairs of fractional vector sine-Gordon equations and fractional vector mKdV equations when the hierarchies for corresponding curve fractional flows are described in explicit forms by fractional wave maps and analogs of Schrodinger maps.
We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists in a proof that for corresponding classes of nonholonomic distributions a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.
We analyze the foundations of Finsler gravity theories with metric compatible connections constructed on nonholonomic tangent bundles, or (pseudo) Riemannian manifolds. There are considered "minimal" modifications of Einstein gravity (including theories with violation of local Lorentz invariance) and shown how the general relativity theory and generalizations can be equivalently re--formulated in Finsler like variables. We focus on Finsler branes solutions and perspectives in modern cosmology.
In this work, we apply the anholonomic deformation method for constructing new classes of anisotropic cosmological solutions in Einstein gravity and/or generalizations with nonholonomic variables. There are analyzed four types of, in general, inhomogeneous metrics, defined with respect to anholonomic frames and their main geometric properties. Such spacetimes contain as particular cases certain conformal and/or frame transforms of the well known Friedman-Robertson-Walker, Bianchi, Kasner and Godel universes and define a great variety of cosmological models with generic off-diagonal metrics, local anisotropy and inhomogeneity. It is shown that certain nonholonomic gravitational configurations may mimic de Sitter like inflation scenaria and different anisotropic modifications without satisfying any classical false-vacuum equation of state. Finally, we speculate on perspectives when such off-diagonal solutions can be related to dark energy and dark matter problems in modern cosmology.