Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

Abstract

We apply the method of moving anholonomic frames, with associated nonlinear connections, in (pseudo) Riemannian spaces and examine the conditions when various types of locally anisotropic (la) structures (Lagrange, Finsler like and more general ones) could be modeled in general relativity. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps and introduce the tensorial na-integration as the inverse operation to both covariant derivation and deformation of connections by na-maps. The problem of redefinition of the Einstein gravity theory on na-backgrounds, provided with a set of na-map invariant conditions and local conservation laws, is analyzed. There are illustrated some examples of generation of vacuum Einstein fields by Finsler like metrics and chains of na-maps.Comment: 41 page

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

“…The equation (33) contains only derivatives on y 4 = v and allows us to define h 4 (x i , v) for a given h 5 (x i , v), or inversely, for h * 4,5 = 0. Having defined h 4 and h 5 , we can compute the coefficients (36), which allows us to find w i from algebraic equations (34) and to compute n i by integrating two times on v as follow from equations (35). Similar properties hold true for equations on higher order shells.…”

“…In explicit form, the (super) gravitational gauge field equations and conservations laws were analyzed in Refs. [33,34,35,36].…”

“…• Having defined h 4 and h 5 and computed γ from (36), we can solve the equation (35) by integrating on variable "v" the equation n * * i + γn * i = 0. The exact solution is…”

“…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.…”

On a covariant version of Caianiello’s model

“…An extensive and general treatment of the notion of non-linear connection, allowing for connections in associated bundles that are g-compatible, can be found in the work of Vacaru et al (for example [12] and [14]) in the contest of Lagrange spaces and other generalizations. In this general framework, the coefficients of an alternative non-linear connection are given by:…”