On static Poincare-Einstein metrics
On static Poincare-Einstein metrics
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DOI：
10.1007/jhep06(2015)051
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发表时间：
2015-06-09
影响因子：
5.4
通讯作者：
Woolgar, Eric
中科院分区：
文献类型：
--
作者：
Galloway, Gregory J.;Woolgar, Eric
静态真空Einstein方程在给定闭流形或渐近平坦流形上的解的分类是一个长期存在且研究较多的问题。解的特征在于一个完整的黎曼n-流形（M，g）和一个正函数N，称为lapse。本文研究了渐近Poincare-Einstein n-流形上的这一问题，其中n >= 3，当无穷远共形边界为圆球、平坦环面或其光滑商，或紧双曲流形时.这样的流形有明确的Wang质量，是静态、真空、渐近反德西特时空的时间对称切片。通过对Lindblom、Shen、Wang等人所用恒等式的一个温和的推广形式进行积分，我们给出了这种流形的质量公式。没有正质量的解。因此，我们观察到，要么是平凡的流逝和（M，g）是庞加莱-爱因斯坦或王质量是负的，在时间对称切片的AdS孤子的情况下。作为应用，我们利用质量公式计算了近似或等于（M-3，g）x（N2）（S-1，dt（2））的翘曲积（X，gamma）的重整化体积，并给出了在视界外的时间推移函数为零的区域内度规为静态时的质量公式.然后流形（X，gamma）被称为有一个“螺栓”，其中S-1因子收缩到零长度。在物理基础上，（X，γ）的重整化体积被期望具有黑洞在固定温度下与辐射浴平衡时的每单位温度自由能的形式。当M是3维的且存在视界时，我们应用这个质量公式计算了（X，gamma）的重整化体积，证明了它确实具有期望的量子激发形式，并讨论了静态真空渐近Poincare-Einstein流形的几个公开问题.
The classification of solutions of the static vacuum Einstein equations, on a given closed manifold or an asymptotically flat one, is a long-standing and much-studied problem. Solutions are characterized by a complete Riemannian n-manifold (M, g) and a positive function N, called the lapse. We study this problem on Asymptotically Poincare-Einstein n-manifolds, n >= 3, when the conformal boundary-at-infinity is either a round sphere, a flat torus or smooth quotient thereof, or a compact hyperbolic manifold. Such manifolds have well-defined Wang mass, and are time-symmetric slices of static, vacuum, asymptotically anti-de Sitter spacetimes. By integrating a mildly generalized form of an identity used by Lindblom, Shen, Wang, and others, we give a mass formula for such manifolds. There are no solutions with positive mass. In consequence, we observe that either the lapse is trivial and (M, g) is Poincare-Einstein or the Wang mass is negative, as in the case of time symmetric slices of the AdS soliton. As an application, we use the mass formula to compute the renormalized volume of the warped product (X, gamma) similar or equal to (M-3, g) x (N2) (S-1, dt(2)).We also give a mass formula for the case of a metric that is static in the region exterior to a horizon on which the lapse function is zero. Then the manifold (X, gamma) is said to have a "bolt" where the S-1 factor shrinks to zero length. The renormalized volume of (X, gamma) is expected on physical grounds to have the form of the free energy per unit temperature for a black hole in equilibrium with a radiation bath at fixed temperature. When M is 3-dimensional and admits a horizon, we apply this mass formula to compute the renormalized volume of (X, gamma) and show that it indeed has the expected thermodynamically motivated form.We also discuss several open questions concerning static vacuum asymptotically Poincare-Einstein manifolds.