Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua. (English) Zbl 1230.83015
Summary: 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. The 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 the Ricci-DeTurck flow preserves these classes of manifolds. As an example, we simulate the Ricci-DeTurck flow for a manifold with asymptotics relevant for AdS\(_{5}/\)CFT\(_{4}\). Our maximum principle dictates that 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 \(O(N^{2}_{c})\) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.
MSC:
83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
83E05 | Geometrodynamics and the holographic principle |
83-08 | Computational methods for problems pertaining to relativity and gravitational theory |
81T20 | Quantum field theory on curved space or space-time backgrounds |
83C75 | Space-time singularities, cosmic censorship, etc. |
81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
83C15 | Exact solutions to problems in general relativity and gravitational theory |
83C57 | Black holes |
83E15 | Kaluza-Klein and other higher-dimensional theories |
53Z05 | Applications of differential geometry to physics |
35Q51 | Soliton equations |