10 research outputs found
Nonlinear Connections and Spinor Geometry
We present an introduction to the geometry of higher order vector and
co-vector bundles (including higher order generalizations of the Finsler
geometry and Kaluza--Klein gravity) and review the basic results on Clifford
and spinor structures on spaces with generic local anisotropy modeled by
anholonomic frames with associated nonlinear connection structures. We
emphasize strong arguments for application of Finsler like geometries in modern
string and gravity theory and noncommutative geometry and noncommutative field
theory and gravity.Comment: It is a short submission variant of math.DG/0205190 reduced to 48
pages with modifications requested by the Editor of IJMM
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
Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes
We study the fractional gravity for spacetimes with non-integer dimensions.
Our constructions are based on a geometric formalism with the fractional Caputo
derivative and integral calculus adapted to nonolonomic distributions. This
allows us to define a fractional spacetime geometry with fundamental
geometric/physical objects and a generalized tensor calculus all being similar
to respective integer dimension constructions. Such models of fractional
gravity mimic the Einstein gravity theory and various Lagrange-Finsler and
Hamilton-Cartan generalizations in nonholonomic variables. The approach
suggests a number of new implications for gravity and matter field theories
with singular, stochastic, kinetic, fractal, memory etc processes. We prove
that the fractional gravitational field equations can be integrated in very
general forms following the anholonomic deformation method for constructing
exact solutions. Finally, we study some examples of fractional black hole
solutions, fractional ellipsoid gravitational configurations and imbedding of
such objects in fractional solitonic backgrounds.Comment: latex2e, 11pt, 40 pages with table of conten
Finsler and Lagrange Geometries in Einstein and String Gravity
We review the current status of Finsler-Lagrange geometry and
generalizations. The goal is to aid non-experts on Finsler spaces, but
physicists and geometers skilled in general relativity and particle theories,
to understand the crucial importance of such geometric methods for applications
in modern physics. We also would like to orient mathematicians working in
generalized Finsler and Kahler geometry and geometric mechanics how they could
perform their results in order to be accepted by the community of ''orthodox''
physicists.
Although the bulk of former models of Finsler-Lagrange spaces where
elaborated on tangent bundles, the surprising result advocated in our works is
that such locally anisotropic structures can be modelled equivalently on
Riemann-Cartan spaces, even as exact solutions in Einstein and/or string
gravity, if nonholonomic distributions and moving frames of references are
introduced into consideration.
We also propose a canonical scheme when geometrical objects on a (pseudo)
Riemannian space are nonholonomically deformed into generalized Lagrange, or
Finsler, configurations on the same manifold. Such canonical transforms are
defined by the coefficients of a prime metric and generate target spaces as
Lagrange structures, their models of almost Hermitian/ Kahler, or nonholonomic
Riemann spaces.
Finally, we consider some classes of exact solutions in string and Einstein
gravity modelling Lagrange-Finsler structures with solitonic pp-waves and
speculate on their physical meaning.Comment: latex 2e, 11pt, 44 pages; accepted to IJGMMP (2008) as a short
variant of arXiv:0707.1524v3, on 86 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