Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua
Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua
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DOI：
10.1088/0264-9381/28/21/215018
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发表时间：
2011-04
影响因子：
3.5
通讯作者：
P. Figueras;James Lucietti;T. Wiseman
中科院分区：
文献类型：
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作者：
P. Figueras;James Lucietti;T. Wiseman
椭圆爱因斯坦-德图尔克方程可以用来数值地寻找黎曼流形上的爱因斯坦度量。静态洛伦兹爱因斯坦度规被认为是解析继续到欧几里德时间。Ricci-DeTurck流是求解该方程的构造性算法，并且当解是稳定的不动点时易于实现，唯一的复杂性是可能存在不是爱因斯坦的Ricci孤子。在这里，我们扩展以前的工作考虑的爱因斯坦-DeTurck方程的黎曼流形的边界，和那些继续静态洛伦兹时空是渐近平坦的，卡鲁扎-克莱因，局部AdS或极值视野。利用极大值原理，我们证明了在这些情况下，里奇孤子不存在，所以任何解决方案是爱因斯坦。我们还认为，Ricci-DeTurck流保持这些类的流形。作为一个例子，我们模拟Ricci-DeTurck流的流形与渐近AdS 5/CFT 4相关。我们的最大值原理表明，没有孤子的解决方案，我们给出了强有力的数值证据表明，存在一个稳定的不动点的流量继续一个光滑的静态洛伦兹爱因斯坦度量。我们的渐近性是这样的，这描述了经典的重力对偶相关的CFT在史瓦西背景下的Unruh或Boulware真空。它决定了CFT应力张量的O（N2 c）部分，有趣的是，它在未来和过去的史瓦西视界上都是规则的。
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.