时间对称3切片中小度规球的Bartnik-Bray外质量
The Bartnik-Bray outer mass of small metric spheres in time-symmetric 3-slices
摘要：
给定一个球，其Bartnik数据接近于欧氏空间中的圆球的Bartnik数据，我们计算其Bartnik-Bray外质量到数据与标准球的偏差的一阶。霍金质量给出了一个众所周知的下界，而上界是通过估计静态真空扩展的质量得到的。作为一个应用，我们证实了在时间对称切片中，收缩到一点的同心测地线球的质量体积比收敛到其中心的能量密度，这与物理预期和其它准局域质量的行为一致.对于收缩到一个平坦点的球，我们也可以计算半径的五阶外质量-该项与中心处的数量曲率的拉普拉斯算子成比例-但我们的估计不够精确，无法在一个仅仅是标量平坦的点处识别该项。特别是，它无法分辨引力对质量的贡献。
英文摘要：
Given a sphere with Bartnik data close to that of a round sphere in Euclidean 3-space, we compute its Bartnik-Bray outer mass to first order in the data's deviation from the standard sphere. The Hawking mass gives a well-known lower bound, and an upper bound is obtained by estimating the mass of a static vacuum extension. As an application we confirm that in a time-symmetric slice concentric geodesic balls shrinking to a point have mass-to-volume ratio converging to the energy density at their center, in accord with physical expectation and the behavior of other quasilocal masses. For balls shrinking to a flat point we can also compute the outer mass to fifth order in the radius---the term is proportional to the Laplacian of the scalar curvature at the center---but our estimate is not refined enough to identify this term at a point which is merely scalar flat. In particular it cannot discern gravitational contributions to the mass.
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