128 research outputs found
The Entropy of Lagrange-Finsler Spaces and Ricci Flows
We formulate a statistical analogy of regular Lagrange mechanics and Finsler
geometry derived from Grisha Perelman's functionals generalized for
nonholonomic Ricci flows. There are elaborated explicit constructions when
nonholonomically constrained flows of Riemann metrics result in Finsler like
configurations, and inversely, and geometric mechanics is modelled on Riemann
spaces with preferred nonholonomic frame structure.Comment: latex2e, 20 pages, v3, the variant accepted to Rep. Math. Phy
Off-Diagonal Deformations of Kerr Metrics and Black Ellipsoids in Heterotic Supergravity
Geometric methods for constructing exact solutions of motion equations with
first order corrections to the heterotic supergravity action
implying a non-trivial Yang-Mills sector and six dimensional, 6-d,
almost-K\"ahler internal spaces are studied. In 10-d spacetimes, general
parametrizations for generic off-diagonal metrics, nonlinear and linear
connections and matter sources, when the equations of motion decouple in very
general forms are considered. This allows us to construct a variety of exact
solutions when the coefficients of fundamental geometric/physical objects
depend on all higher dimensional spacetime coordinates via corresponding
classes of generating and integration functions, generalized effective sources
and integration constants. Such generalized solutions are determined by generic
off-diagonal metrics and nonlinear and/or linear connections. In particular, as
configurations which are warped/compactified to lower dimensions and for
Levi-Civita connections. The corresponding metrics can have (non) Killing
and/or Lie algebra symmetries and/or describe (1+2)-d and/or (1+3)-d domain
wall configurations, with possible warping nearly almost-K\"ahler manifolds,
with gravitational and gauge instantons for nonlinear vacuum configurations and
effective polarizations of cosmological and interaction constants encoding
string gravity effects. A series of examples of exact solutions describing
generic off-diagonal supergravity modifications to black hole/ ellipsoid and
solitonic configurations are provided and analyzed. We prove that it is
possible to reproduce the Kerr and other type black solutions in general
relativity (with certain types of string corrections) in 4-d and to generalize
the solutions to non-vacuum configurations in (super) gravity/ string theories.Comment: latex2e, 44 pages with table of content, v2 accepted to EJPC with
minor typos modifications requested by editor and referee and up-dated
reference
Locally Anisotropic Structures and Nonlinear Connections in Einstein and Gauge Gravity
We analyze local anisotropies induced by anholonomic frames and associated
nonlinear connections in general relativity and extensions to affine Poincare
and de Sitter gauge gravity and different types of Kaluza-Klein theories. We
construct some new classes of cosmological solutions of gravitational field
equations describing Friedmann-Robertson-Walker like universes with rotation
(ellongated and flattened) ellipsoidal or torus symmetry.Comment: 37 page
On General Solutions for Field Equations in Einstein and Higher Dimension Gravity
We prove that the Einstein equations can be solved in a very general form for
arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases
following a geometric method of anholonomic frame deformations for constructing
exact solutions in gravity. The main idea of this method is to introduce on
(pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection)
metric compatible linear connection which is also completely defined by the
same metric structure. Such a canonically distinguished connection is with
nontrivial torsion which is induced by some nonholonomy frame coefficients and
generic off-diagonal terms of metrics. It is possible to define certain classes
of adapted frames of reference when the Einstein equations for such an
alternative connection transform into a system of partial differential
equations which can be integrated in very general forms. Imposing nonholonomic
constraints on generalized metrics and connections and adapted frames
(selecting Levi-Civita configurations), we generate exact solutions in Einstein
gravity and extra dimension generalizations.Comment: latex 2e, 11pt, 40 pages; it is a generalizaton with modified title,
including proofs and additional results for higher dimensional gravity of the
letter v1, on 14 pages; v4, with new abstract, modified title and up-dated
references is accepted by Int. J. Theor. Phy
Finsler Branes and Quantum Gravity Phenomenology with Lorentz Symmetry Violations
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.Comment: latex2e, 11pt, 34 pages, the version accepted to Class. Quant. Gra
New Classes of Off-Diagonal Cosmological Solutions in Einstein Gravity
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.Comment: latex2e, 11pt, 33 pages with table of content, a variant accepted to
IJT
Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles
We provide a method of converting Lagrange and Finsler spaces and their
Legendre transforms to Hamilton and Cartan spaces into almost Kaehler
structures on tangent and cotangent bundles. In particular cases, the Hamilton
spaces contain nonholonomic lifts of (pseudo) Riemannian / Einstein metrics on
effective phase spaces. This allows us to define the corresponding Fedosov
operators and develop deformation quantization schemes for nonlinear mechanical
and gravity models on Lagrange- and Hamilton-Fedosov manifolds.Comment: latex2e, 11pt, 35 pages, v3, accepted to J. Math. Phys. (2009
Clifford-Finsler Algebroids and Nonholonomic Einstein-Dirac Structures
We propose a new framework for constructing geometric and physical models on
nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry
and nonlinear connection structure. Explicit parametrizations of generic
off-diagonal metrics and linear and nonlinear connections define different
types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to
spinor fields and Dirac operators on nonholonomic manifolds motivates the
theory of Clifford algebroids defined as Clifford bundles, in general, enabled
with nonintegrable distributions defining the nonlinear connection. In this
work, we elaborate the algebroid spinor differential geometry and formulate the
(scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids.
The paper communicates new developments in geometrical formulation of physical
theories and this approach is grounded on a number of previous examples when
exact solutions with generic off-diagonal metrics and generalized symmetries in
modern gravity define nonholonomic spacetime manifolds with uncompactified
extra dimensions.Comment: The manuscript was substantially modified following recommendations
of JMP referee. The former Chapter 2 and Appendix were elliminated. The
Introduction and Conclusion sections were modifie
Nonholonomic Ricci Flows: II. Evolution Equations and Dynamics
This is the second paper in a series of works devoted to nonholonomic Ricci
flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows
of Riemannian metrics we can model mutual transforms of generalized
Finsler-Lagrange and Riemann geometries. We verify some assertions made in the
first partner paper and develop a formal scheme in which the geometric
constructions with Ricci flow evolution are elaborated for canonical nonlinear
and linear connection structures. This scheme is applied to a study of
Hamilton's Ricci flows on nonholonomic manifolds and related Einstein spaces
and Ricci solitons. The nonholonomic evolution equations are derived from
Perelman's functionals which are redefined in such a form that can be adapted
to the nonlinear connection structure. Next, the statistical analogy for
nonholonomic Ricci flows is formulated and the corresponding thermodynamical
expressions are found for compact configurations. Finally, we analyze two
physical applications: the nonholonomic Ricci flows associated to evolution
models for solitonic pp-wave solutions of Einstein equations, and compute the
Perelman's entropy for regular Lagrange and analogous gravitational systems.Comment: v2 41 pages, latex2e, 11pt, the variant accepted by J. Math. Phys.
with former section 2 eliminated, a new section 5 with applications in
gravity and geometric mechanics, and modified introduction, conclusion and
new reference
On General Solutions of Einstein Equations
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.Comment: 15 pages, latex2e, submitted to arXiv.org on September 22, 2009,
equivalent to arXiv: 0909.3949v1 [gr-qc]; an extended/modified variant
published in IJTP 49 (2010) 884-913, equivalent to arXiv: 0909.3949v4 [gr-qc
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