Oscar Lasso Andino

The target space of the non-linear $\sigma$-model is a Riemannian manifold. Although it can be any Riemannian metric, there are certain physically interesting geometries which are worth to study. Here, we numerically evolve the time-symmetric foliations of a family of spherically symmetric asymptotically flat wormholes under the $1$-loop renormalization group flow of the non-linear $\sigma$-model, the Ricci flow, and under the $2$-loop approximation, RG-2 flow. We rely over some theorems adapted from the compact case for studying the evolution of different wormhole types, specially those with high curvature zones. Some metrics expand and others contract at the beginning of the flow, however, all metrics pinch-off at a certain time. This is related with the fact that the flow does not converge to a fixed point when its starting geometry is the spatial sections of a Morris-Thorne wormhole, and therefore the corresponding non-linear $\sigma$-model is non-integrable/renormalizable. We present a numerical study of the evolution of wormhole singularities in three dimensions extending the theoretical estimations. Finally, we compute the evolution of the Hamilton's entropy and the Brown-York energy.

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.