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

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.

Summary of a talk at MG13 (Stockholm, 2012), GT1, Exact Solutions in Four and Higher Dimensions: Mathematical Aspects; the results for modified gravity elaborated during a visit at CERN-TH; version accepted to Mod. Phys. Lett. A

References in corpus (6)

• Off-Diagonal Deformations of Kerr Black Holes in Einstein and Modified Massive Gravity and Higher Dimensions
• Fractional Nonholonomic Ricci Flows
• On General Solutions of Einstein Equations
• Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons
• Hidden Symmetries for Ellipsoid-Solitonic Deformations of Kerr-Sen Black Holes and Quantum Anomalies
• Modified Dynamical Supergravity Breaking and Off-Diagonal Super-Higgs Effects

Cited by in corpus (8)

• On Relativistic Generalization of Perelman's W-entropy and Statistical Thermodynamic Description of Gravitational Fields
• Geometric Flows and Perelman's Thermodynamics for Black Ellipsoids in $R^2$ and Einstein Gravity Theories
• Modified Dynamical Supergravity Breaking and Off-Diagonal Super-Higgs Effects
• Dark energy and dark matter configurations for wormholes and solitionic hierarchies of nonmetric Ricci flows and $F(R,T,Q,T_m)$ gravity
• Quantum geometric information flows and relativistic generalizations of G. Perelman thermodynamics for nonholonomic Einstein systems with black holes and stationary solitonic hierarchies
• Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions
• Bianchi IX geometry and the Einstein-Maxwell theory
• The Anholonomic Frame and Connection Deformation Method for constructing off-diagonal solutions in (modified) Einstein gravity and nonassociative geometric flows and Finsler-Lagrange-Hamilton theories