Nonholonomic relativistic diffusion and exact solutions for stochastic Einstein spaces

Abstract

We develop an approach to the theory nonholonomic relativistic stochastic processes in curved spaces. The Itô 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 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.

Deforming black hole and cosmological solutions by quasiperiodic and/or pattern forming structures in modified and Einstein gravity

“…Such constructions will be applied in order to extend the approach for solutions encoding quasiperiodic and/or pattern forming, solitonic distributions and nonlinear wave gravitational and matter field structures. Details, examples with small parametric decompositions and rigorous mathematical proofs are contained in [8,9,10,11,12,13,14,20,21,22,23,24,25,26], see also a summary of necessary N-adapted formulas in Appendix A.…”

“…One shall not be considered summation on repeating "low-low", or "up-up" indices. In our works , we elaborated a system of "N-adapted notations" with boldface symbols for manifolds and fiber bundles enables with a nontrivial N-connection structure N. Details on nonholonomic differential geometry and N-connections are explained in [11,12,13,14,20,21,22,23,24,25,26] and references therein. If the anholonomy coefficients C γ αβ in (5) are nontrivial, a metric g αβ (1) can not be diagonalized in a local finite, or infinite, spacetime region with respect to coordinate frames.…”

“…In references [27,28,29,30,31], there are provided details and main examples for constructing physically important solutions by using diagonal ansatz, for instance, with spherical/cylinder symmetries reducing the (generalized) Einstein equations to certain systems of nonlinear ODEs. Proofs of results and a number of examples how the AFDM should be applied in order to generate exact solutions of gravitational, matter field and evolution nonlinear systems of PDEs are considered in [8,9,10,11,12,13,14,20,21,22,23,24,25,26].…”

“…One can be found a STC pattern with µ = ξ = 0.000707, when the correlation length is about 1-2 wave-lengths. We can ad new terms with fractional chaos and diffusion [21,22].…”

Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity

“…It is interesting to understand more deeply the "discrete gravitational coupling constant" and its relation to the quantum world in addition to its role in determining atomic and nuclear physical constants. Nevertheless, the occurrence of a discrete gravitational coupling constant in our scenario can have important impacts on quantum gravity and some basic concept of bound states of Dirac particles in a Schwarzschild field, in addition to some concepts of the standard cosmological model related to the growth of the universe, interaction of collapsing stars with nearby matter [75], fractional dynamics from Einstein gravity, and diffusion in Einstein spaces [76]. A more exhaustive analysis of the discrete gravitational coupling constant will be presented in a subsequent paper.…”