Geometric Flows and Perelman's Thermodynamics for Black Ellipsoids in $R^2$ and Einstein Gravity Theories

TL;DR

This paper explores geometric flows and thermodynamics in modified gravity theories, constructing exact solutions like black ellipsoids and analyzing their properties through Perelman's functionals, linking geometric evolution with gravitational phenomena.

Contribution

It generalizes Perelman's thermodynamic functionals to modified gravity theories, enabling the study of geometric flow evolution and exact solutions such as black ellipsoids in these frameworks.

Findings

Constructed off-diagonal black ellipsoid and solitonic solutions in MGTs and GR.

Analyzed geometric flow evolution scenarios and nonlinear interactions.

Computed Perelman's energy and entropy for black ellipsoids in $R^2$ gravity.

Abstract

We study geometric relativistic flow and Ricci soliton equations which (for respective nonholonomic constraints and self-similarity conditions) are equivalent to the gravitational field equations of $R^2$ gravity and/or to the Einstein equations with scalar field in general relativity, GR. Perelman's functionals are generalized for modified gravity theories, MGTs, which allows to formulate an analogous statistical thermodynamics for geometric flows and Ricci solitons. There are constructed and analyzed generic off-diagonal black ellipsoid, black hole and solitonic exact solutions in MGTs and GR encoding geometric flow evolution scenarios and nonlinear parametric interactions. Such new classes of solutions in MGTs can be with polarized and/or running constants, nonholonomically deformed horizons and/or imbedded self-consistently into solitonic backgrounds. They exist also in GR as generic off-diagonal vacuum configurations with effective cosmological constant and/or mimicking effective scalar field interactions. Finally, we compute Perelman's energy and entropy for black ellipsoids and evolution solitons in $R^2$ gravity.