RG-2 flow, mass and entropy

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 the 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 also discuss about the possibility of extending the result to all loops. We show that the inequality relating 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 the Perelman`s $W$-entropy for the RG-2 flow and the physically motivated path towards the gradient formulation of the RG-2 flow.