Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua

Abstract

The elliptic Einstein-DeTurck equation may be used to numerically find\nEinstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics\nare considered by analytically continuing to Euclidean time. Ricci-DeTurck flow\nis a constructive algorithm to solve this equation, and is simple to implement\nwhen the solution is a stable fixed point, the only complication being that\nRicci solitons may exist which are not Einstein. Here we extend previous work\nto consider the Einstein-DeTurck equation for Riemannian manifolds with\nboundaries, and those that continue to static Lorentzian spacetimes which are\nasymptotically flat, Kaluza-Klein, locally AdS or have extremal horizons. Using\na maximum principle we prove that Ricci solitons do not exist in these cases\nand so any solution is Einstein. We also argue that Ricci-DeTurck flow\npreserves these classes of manifolds. As an example we simulate Ricci-DeTurck\nflow for a manifold with asymptotics relevant for AdS_5/CFT_4. Our maximum\nprinciple dictates there are no soliton solutions, and we give strong numerical\nevidence that there exists a stable fixed point of the flow which continues to\na smooth static Lorentzian Einstein metric. Our asymptotics are such that this\ndescribes the classical gravity dual relevant for the CFT on a Schwarzschild\nbackground in either the Unruh or Boulware vacua. It determines the leading\nO(N^2) part of the CFT stress tensor, which interestingly is regular on both\nthe future and past Schwarzschild horizons.\n