Generic off-diagonal solutions and solitonic hierarchies in (modified) gravity

Abstract

There are summarized our recent results on encoding exact solutions of field equations in Einstein and modified gravity theories into solitonic hierarchies derived for nonholonomic curve flows with associated bi-Hamilton structure. We argue that there is a canonical distinguished connection for which the fundamental geometric/ physical equations decouple in general form. This allows us to construct very general classes of generic off-diagonal solutions determined by corresponding types of generating and integration functions depending on all (spacetime) coordinates. If the integral varieties are constrained to zero torsion configurations, we can extract solutions for the general relativity theory. We conclude that the geometric and physical data for various classes of effective/modified Einstein spaces can be encoded into multi-component versions of the sine-Gordon, or modified Korteweg -de Vries equations.Keywords: solitonic hierarchies, off-diagonal exact solutions, (modified) gravity. PACS: 04.20.Jb, 04.50.Kd, 04.30.Nk, 02.30.lk Preliminaries: Let us consider a pseudo-Riemannian manifold V , dim V = n + m, (n, m ≥ 2), endowed with a conventional nonholonomic (equivalently, non-integrable, or anholonomic) horizontal (h) and vertical (v)

On relativistic generalization of Perelman’s W-entropy and thermodynamic description of gravitational fields and cosmology

“…A metric (62) There are possible more general 3-d solitonic gravitational waves which can propagate self-consistently in Minkowski spacetime or in a FLRW background. To generate such solutions we need to take a generating function s Ψ(x 1 , x 2 , t) which is a solution of the Kadomtsev-Petviashvili, KdP, equation [60,61,62]. If we consider that in the ansatz of type (58) y 3 = t (time like coordinate) and y 4 = ϕ (an angular space coordinate), we construct stationary solutions with Killing symmetry on ∂ 3 , see details in [41,42].…”

“…There are possible more general 3-d solitonic gravitational waves which can propagate self-consistently in Minkowski spacetime or in a FLRW background. To generate such solutions we need to take a generating function s Ψ(x 1 , x 2 , t) which is a solution of the Kadomtsev-Petviashvili, KdP, equation [60,61,62].…”

Off-diagonal wormhole and black hole deformations and dark energy in two measure/bi-metric modified gravity theories

“…Such classes of "generalized solutions" are very different from the well known and physically important black hole, wormhole and/or cosmological solutions with particular type (for instance, spherical, or cylindrical) symmetry determined by integration constants. Mathematically, the new classes of solutions can be encoded and parameterized as solitonic wave hierarchies 6 . For physicists, it is very important to study how such generalized solutions are related, for instance, to nonholonomic deformations of the Kerr and wormhole metrics 1,3 and to homogeneous and inhomogeneous cosmological solutions in various MGTs 2,5 .…”