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

By Pau Figueras, James Lucietti, T. Wiseman. 2011

168 citations recorded by the source corpus.

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 AdS5/CFT4. 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(N2c) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/5ffcc479d6926545e236ef102f10b1801eeb2281

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Citations in the corpus

The thermoelectric properties of inhomogeneous holographic lattices (2014, background): “For the case of Einstein’s equations with a negative cosmological constant, it has been shown that there are no solutions of Eμν = ∇(μ ξν) with a non-trivial ξ, provided that ξ = 0 on the boundary of the given problem [40].”
Entanglement islands in higher dimensions (2019, background): “It can be shown that DeTurck solitons do not exist for static and certain stationary solutions of (11) with purely Dirichlet boundaries [36,37].”
Optical conductivity with holographic lattices (2012, background): “In certain circumstances one can show that solutions with ξ 6= 0, coined Ricci solitons, cannot exist [14].”