期刊
CLASSICAL AND QUANTUM GRAVITY
卷 28, 期 21, 页码 -出版社
IOP PUBLISHING LTD
DOI: 10.1088/0264-9381/28/21/215018
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资金
- EPSRC [EP/H027106/1]
- EPSRC
- STFC
- Engineering and Physical Sciences Research Council [EP/H00355X/2, EP/H00355X/1, EP/H027106/1] Funding Source: researchfish
- Science and Technology Facilities Council [ST/G000743/1, PP/D003970/1] Funding Source: researchfish
- EPSRC [EP/H027106/1, EP/H00355X/2, EP/H00355X/1] Funding Source: UKRI
- STFC [ST/G000743/1, PP/D003970/1] Funding Source: UKRI
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)/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(N-c(2)) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.
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