RG-2 flow, mass and entropy

TL;DR

This paper investigates the RG-2 flow, a two-loop approximation of the non-linear sigma model's renormalization group flow, in the context of general relativity, revealing properties like monotonic area evolution and connections to Hawking mass and entropy.

Contribution

It extends the understanding of RG-2 flow in general relativity, showing monotonicity of surface area and relations to Hawking mass, which were previously known only for Ricci flow.

Findings

Area of a closed surface is monotonous under RG-2 flow.

The inequality relating surface area and Hawking mass holds for RG-2 flow.

Discussion of Perelman's W-entropy and gradient formulation for RG-2 flow.

Abstract

The RG-2 flow is the two-loop approximation for the world-sheet non-linear sigma model renormalization group flow. The first truncation of the flow is the well known Ricci flow, at two loops higher order curvature terms appear, changing almost completely the behaviour of the evolution equation. In this article we study the RG-2 flow in the context of general relativity. Contrary to what happens with the Ricci flow the RG-2 flow theory has not been studied exhaustively, and from a mathematical point of view there are big differences between both flows. Considering a time symmetric foliation of an asymptotically flat spacetime, we show that the area $A$ of a closed two dimensional surface $S$ is monotonous under the RG-2 flow, refining and extending the previous results already known for the Ricci flow. We show that the inequality that relates the area of the surface $S$ and the Hawking mass, already found for the Ricci flow, is still satisfied when we make evolve the area under the new flow. Finally, we discuss about Perelman`s $W$-entropy for the RG-2 flow and the physically motivated path towards the gradient formulation of the RG-2 flow.