Some Applications of Ricci Flow in Physics

I discuss certain applications of the Ricci flow in physics. I first review how it arises in the renormalization group (RG) flow of a nonlinear sigma model. I then review the concept of a Ricci soliton and recall how a soliton was used to discuss the RG flow of mass in 2-dimensions. I then present recent results obtained with Oliynyk on the flow of mass in higher dimensions. The final section discusses one way in which Ricci flow may arise in general relativity, particularly for static metrics.Comment: Minor corrections in Sections IV and VI. Invited talk at Theory Canada III meeting, June 2007; submitted to Proceedings. Dedicated to Rafael D Sorkin on the occasion of his 60th birthda

Bakry-\'Emery black holes

Scalar-tensor gravitation theories, such as the Brans-Dicke family of theories, are commonly partly described by a modified Einstein equation in which the Ricci tensor is replaced by the Bakry-\'Emery-Ricci tensor of a Lorentzian metric and scalar field. In physics this formulation is sometimes referred to as the "Jordan frame". Just as, in General Relativity, natural energy conditions on the stress-energy tensor become conditions on the Ricci tensor, in scalar-tensor theories expressed in the Jordan frame natural energy conditions become conditions on the Bakry-\'Emery-Ricci tensor. We show that, if the Bakry-\'Emery tensor obeys the null energy condition with an upper bound on the Bakry-\'Emery scalar function, there is a modified notion of apparent horizon which obeys analogues of familiar theorems from General Relativity. The Bakry-\'Emery modified apparent horizon always lies behind an event horizon and the event horizon obeys a modified area theorem. Under more restrictive conditions, the modified apparent horizon obeys an analogue of the Hawking topology theorem in 4 spacetime dimensions. Since topological censorship is known to yield a horizon topology theorem independent of the Hawking theorem, in an appendix we obtain a Bakry-\'Emery version of the topological censorship theorem. We apply our results to the Brans-Dicke theory, and obtain an area theorem for horizons in that theory. Our theorems can be used to understand behaviour observed in numerical simulations by Scheel, Shapiro, and Teukolsky of dust collapse in Brans-Dicke theory.Comment: 17 page

A causal order for spacetimes with C0^{0} lorentzian metrics: proof of compactness of the space of causal curves

Sometimes one wants to study the causal properties of metrics which are not differentiable, and may not even be invertible everywhere. With this purpose in mind, we extend the tools of ``global causal analysis'' to cover general C^0 metrics, working with a new causal relation which we call K^+, and formulating extended definitions of concepts like causal curve and global hyperbolicity based on K^+. We first establish some fundamental definitions and lemmas of this broader framework. We then prove for general C^0 metrics the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. A unique feature of our approach is its thoroughgoing reliance on order-theoretic arguments, coupled with a utilization of the Vietoris topology for the space of closed subsets of a compact set. We feel that this approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. (For example, in a spacetime free of causal loops, we characterize a causal curve simply as a compact connected subset which is linearly ordered by K^+.) Our results have application in a recent positive energy theorem [1], and may also prove useful in the study of topology change [2]. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry

A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order

Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explicitly known only to leading order in perturbation theory, but at that order is positive semi-definite, as follows from Perelman's work on the Ricci flow. This gives rise to a monotonicity formula for the flow which is expected to fail only if the beta function perturbation series fails to converge, which can happen if curvatures or their derivatives grow large. We test the validity of the monotonicity formula at next-to-leading order in perturbation theory by explicitly computing the second-order terms in the metric on the space of coupling constants. At this order, this metric is found not to be positive semi-definite. In situations where this might spoil monotonicity, derivatives of curvature become large enough for higher order perturbative corrections to be significant.Comment: 15 pages; Erroneous sentence in footnote 14 removed; this version therefore supersedes the published version (our thanks to Dezhong Chen for the correction