Solutions to the Ricci Flow via Einstein Field Equations

TL;DR

This paper establishes a novel connection between Ricci flow solutions on Lorentzian manifolds and Einstein's field equations, using matter deformations driven by quadratic stress-energy functionals, with applications to symmetric spacetimes and nonlinear electrodynamics.

Contribution

It introduces a new method linking Ricci flow to Einstein equations through matter deformations, providing explicit solutions in diverse physical scenarios.

Findings

Explicit solutions in maximally symmetric spacetimes

Application to Born-Infeld nonlinear electrodynamics

Analysis of topological monopole configurations

Abstract

We show how solutions to the Ricci flow on Lorentzian manifolds, along with its generalizations, can be linked to Einstein's field equations. The approach involves deformations of the matter sector that are generated by quadratic functionals of the stress-energy tensor. We provide illustrative examples by explicitly constructing analytical solutions within maximally symmetric spacetimes and in the context of Born-Infeld's nonlinear electrodynamics. Finally, we discuss configurations involving global topological monopoles, emphasizing the versatility of this approach across various geometric and physical settings.