标题： 类局部质量的提议用于类时平均曲率的2-曲面 显示英文标题

标题： A proposal of quasi-local mass for 2-surfaces of timelike mean curvature

摘要： 一个准局部质量，通常定义为在类空$2$-曲面$\Sigma$上的积分，应包含关于由$\Sigma$限定的有限、扩展区域内的引力场的信息。因此，在尝试量子化引力时，可以考虑一个无限维的$2$-曲面空间，而不是一个无限维的$4$-维洛伦兹时空空间。然而，现有的准局部质量定义仅适用于平均曲率矢量为类空的视界外曲面。在本文中，我们提出将 Wang-Yau 准局部能量/质量扩展到平均曲率矢量为类时的曲面，特别是包括被俘曲面。我们采用与 Wang-Yau 准局部能量相同的规范，但允许回拉到物理时空的“守恒矢量”为类空。我们在 Wang-Yau 准局部能量的哈密顿表述中定义新的准局部能量。新定义产生了一个正定的表面能量密度和一个新的散度为零的电流。展示了克尔族时空中的坐标球面的计算结果。在球对称情况下，我们的定义简化为先前的一个定义\cite{lundgren2007self}。 显示英文摘要

摘要： A quasi-local mass, typically defined as an integral over a spacelike $2$-surface $\Sigma$, should encode information about the gravitational field within a finite, extended region bounded by $\Sigma$. Therefore, in attempts to quantize gravity, one may consider an infinite dimensional space of $2$-surfaces instead of an infinite dimensional space of $4$-dimensional Lorentzian spacetimes. However, existing definitions for quasilocal mass only applies to surfaces outside an horizon whose mean curvature vector is spacelike. In this paper, we propose an extension of the Wang-Yau quasi-local energy/mass to surfaces with timelike mean curvature vector, including in particular trapped surfaces. We adopt the same canonical gauge as in the Wang-Yau quasi-local energy but allow the pulled back "killing vector" to the physical spacetime to be spacelike. We define the new quasi-local energy along the Hamiltonian formulation of the Wang-Yau quasi-local energy. The new definition yields a positive definite surface energy density and a new divergence free current. Calculations for coordinate spheres in Kerr family spacetime are shown. In the spherical symmetric case, our definition reduces to a previous definition \cite{lundgren2007self}.