Generic Off--Diagonal Solutions and Solitonic Hierarchies in Einstein and Modified Gravity

TL;DR

This paper presents a method to encode solutions of Einstein and modified gravity field equations into solitonic hierarchies, enabling the construction of general off-diagonal solutions and linking geometric data to integrable systems.

Contribution

It introduces a framework for representing Einstein and modified gravity solutions as solitonic hierarchies using nonholonomic curve flows and bi-Hamilton structures.

Findings

Solutions can be encoded into sine-Gordon and mKdV equations.

Off-diagonal solutions depend on functions of all spacetime coordinates.

The approach applies to both general relativity and modified gravity theories.

Abstract

We summarize some 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.