Locally Anisotropic Structures and Nonlinear Connections in Einstein and Gauge Gravity

Vacaru, Sergiu I.; Dehnen, H.

doi:10.1023/a:1022388909622

Cited by 26 publications

(83 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper we shall prove the existence for noncommutative spaces of gauge models of gravity which agrees with usual gauge gravity theories being equivalent or extending the general relativity theory (see works [11,12] for locally isotropic spaces and corresponding reformulations and generalizations respectively for anholonomic frames [13] and locally anisotropic (super) spaces [14]) in the limit of commuting spaces.…”

Section: Introductionmentioning

confidence: 81%

Gauge and Einstein gravity from non-abelian gauge models on noncommutative spaces

Vacaru

2001

Physics Letters B

Self Cite

140

View full text Add to dashboard Cite

Following the formalism of enveloping algebras and star product calculus we formulate and analyze a model of gauge gravity on noncommutative spaces and examine the conditions of its equivalence to general relativity. The corresponding Seiber-Witten maps are established which allow the definition of respective dynamics for a finite number of gravitational gauge field components on noncommutative spaces.

show abstract

Section: Introductionmentioning

confidence: 81%

Gauge and Einstein gravity from non-abelian gauge models on noncommutative spaces

Vacaru

2001

Physics Letters B

Self Cite

140

View full text Add to dashboard Cite

show abstract

“…Such constructions are similar to those presented in above Theorems and in Refs. [1,2,3,4,30,31,32,6,7,8,9,34,35,36]. Some additional necessary formulas are given in Appendices.…”

Section: Introducing Valuesmentioning

confidence: 99%

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

Vacaru

2010

Int J Theor Phys

Self Cite

237

View full text Add to dashboard Cite

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.

show abstract

“…We emphasize that the theory of anisotropic spinors may be related not only to generalized Finsler, Lagrange, Cartan, and Hamilton spaces or their higher-order generalizations, but also to anholonomic frames with associated nonlinear connections which appear naturally even in (pseudo-) Riemannian and Riemann-Cartan geometries if off-diagonal metrics are considered [94,96,97,98,102,103,104,105,110]. In order to construct exact solutions of the Einstein equations in general relativity and extradimensional gravity (for lower dimensions see [85,96,107,108]), it is more convenient to diagonalize space-time metrics by using some anholonomic transforms.…”

Section: Nonlinear Connections and Spinor Geometry 1191mentioning

confidence: 99%

“…Various types of Finsler-like structures can be parametrized by generic off-diagonal metrics, which cannot be diagonalized by coordinate transforms but only by anholonomic maps with associated nonlinear connection (in brief, N-connection). Such structures may be defined as exact solutions of gravitational field equations in the Einstein gravity and its generalizations [75,79,80,94,95,96,97,98,99,100,102,103,104,105,109,110,111], for instance, in the metric-affine [19,23,56] Riemann-Cartan gravity [24,25]. Finsler-like configurations are considered in locally anisotropic thermodynamics, kinetics, related stochastic processes [85,96,107,108], and (super-) string theory [84,87,90,91,92].…”

mentioning

confidence: 99%

Nonlinear connections and spinor geometry

Vacaru¹,

Vicol

2004

International Journal of Mathematics and Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

We present an introduction to the geometry of higher-order vector and covector 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, noncommutative geometry and noncommutative field theory, and gravity.2000 Mathematics Subject Classification: 15A66, 58B20, 53C60, 83C60, 83E15.1. Introduction. Nowadays, interest has been established in non-Riemannian geometries derived in the low-energy string theory [18,64,65], noncommutative geometry [1,3,8,12,15,22,32,34,53,55,67,109,111,112], and quantum groups [33,35,36,37]. Various types of Finsler-like structures can be parametrized by generic off-diagonal metrics, which cannot be diagonalized by coordinate transforms but only by anholonomic maps with associated nonlinear connection (in brief, N-connection). Such structures may be defined as exact solutions of gravitational field equations in the Einstein gravity and its generalizations [75,79,80,94,95,96,97,98,99,100,102,103,104,105,109,110,111], for instance, in the metric-affine [19,23,56] Riemann-Cartan gravity [24,25]. Finsler-like configurations are considered in locally anisotropic thermodynamics, kinetics, related stochastic processes [85,96,107,108], and (super-) string theory [84,87,90,91,92].The following natural step in these lines is to elucidate the theory of spinors in effectively derived Finsler geometries and to relate this formalism of Clifford structures to noncommutative Finsler geometry. It should be noted that the rigorous definition of spinors for Finsler spaces and generalizations was not a trivial task because (on such spaces) there are no defined even local groups of automorphisms. The problem was solved in [82,83,88,89,93] by adapting the geometric constructions with respect to anholonomic frames with associated N-connection structure. The aim of this work is to outline the geometry of generalized Finsler spinors in a form more oriented to applications in modern mathematical physics.We start with some historical remarks: the spinors studied by mathematicians and physicists are connected with the general theory of Clifford spaces introduced in 1876 [14]. The theory of spinors and Clifford algebras play a major role in contemporary physics and mathematics. The spinors were discovered by Èlie Cartan in 1913 in 1190 S. I. VACARU AND N. A. VICOL mathematical form in his researches on representation group theory [10,11]; he showed that spinors furnish a linear representation of the groups of rotations of a space of arbitrary dimensions. The physicists Pauli [60] and Dirac [20] (in 1927, resp., for the three-dimensional and four-dimensional space-times) introduced spinors for the representation of the wave functions. In general relativity theory spinors and the Dirac equat...

show abstract

Locally Anisotropic Structures and Nonlinear Connections in Einstein and Gauge Gravity

Cited by 26 publications

References 31 publications

Gauge and Einstein gravity from non-abelian gauge models on noncommutative spaces

Gauge and Einstein gravity from non-abelian gauge models on noncommutative spaces

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

Nonlinear connections and spinor geometry

Contact Info

Product

Resources

About