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

TL;DR

This paper extends the Ricci-DeTurck flow method to find Einstein metrics on manifolds with boundaries and specific asymptotics, proving the non-existence of Ricci solitons and providing numerical evidence for stable Einstein solutions relevant to AdS/CFT correspondence.

Contribution

It generalizes Ricci-DeTurck flow to new classes of manifolds with boundaries and asymptotics, and proves the absence of Ricci solitons in these cases, supporting numerical construction of Einstein metrics.

Findings

Ricci solitons do not exist in the considered cases.

Numerical evidence supports stable fixed points of Ricci-DeTurck flow.

Constructed Einstein metrics describe gravity duals for CFTs in Schwarzschild backgrounds.

Abstract

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