标题： 带有宇宙常数的Bondi质量 显示英文标题

标题： Bondi mass with a cosmological constant

摘要： 由于宇宙常数 $\Lambda\in\R$ 的存在，孤立引力系统因引力波携带的能量而产生的质量损失最近被推导出来，使用的是 Newman-Penrose-Unti 方法。 在同一篇文章中，提出了孤立系统在 $M_\Lambda$ 情况下的 Bondi 质量表达式，适用于 $\Lambda>0$ 情况。 规定的质量 $M_\Lambda$ 将确保在没有来自其他地方的引力辐射输入时，发出的引力波必须带走一个正定能量。 然而，该建议的量引入了对 Bondi 质量 $M_B$ 的 $\Lambda$ 修正项（其中 $M_B$ 是渐近平直时空的通常 Bondi 质量），这不仅涉及系统在该时刻的状态信息，而且似乎还涉及其过去的历史。 在本文中，我们通过基于 Nester-Witten 恒等式的 Frauendiener 构造的超曲面上的积分公式推导出相同的质量损失方程，并认为可以采用 Bondi 质量的推广形式$\Lambda\in\R$\emph{无需任何修正}，即$M_\Lambda=M_B$对于任何$\Lambda\in\R$。此外，借助$M_\Lambda=M_B$，我们证明了对于由孤立系统发出的\emph{纯四极引力波}（即） 当“Bondi新闻” $\sigma^o$ 仅包含“带有自旋权重2的球谐函数”的 $l=2$ 成分时，带走的能量为 \emph{明显正定的} 对于 $\Lambda>0$ 情况而言。 对于具有更高多极矩的一般 $\sigma^o$ ，如果这些 $l>2$ 贡献较弱，那么在 $\Lambda>0$ 情况下的这一明显性质仍然成立——更准确地说，如果它们满足本文给出的任何不等式。 显示英文摘要

摘要： The mass loss of an isolated gravitating system due to energy carried away by gravitational waves with a cosmological constant $\Lambda\in\R$ was recently worked out, using the Newman-Penrose-Unti approach. In that same article, an expression for the Bondi mass of the isolated system, $M_\Lambda$, for the $\Lambda>0$ case was proposed. The stipulated mass $M_\Lambda$ would ensure that in the absence of any incoming gravitational radiation from elsewhere, the emitted gravitational waves must carry away a positive-definite energy. That suggested quantity however, introduced a $\Lambda$-correction term to the Bondi mass $M_B$ (where $M_B$ is the usual Bondi mass for asymptotically flat spacetimes) which would involve not just information on the state of the system at that moment, but ostensibly also its past history. In this paper, we derive the identical mass-loss equation using an integral formula on a hypersurface formulated by Frauendiener based on the Nester-Witten identity, and argue that one may adopt a generalisation of the Bondi mass with $\Lambda\in\R$ \emph{without any correction}, viz. $M_\Lambda=M_B$ for any $\Lambda\in\R$. Furthermore with $M_\Lambda=M_B$, we show that for \emph{purely quadrupole gravitational waves} given off by the isolated system (i.e. when the "Bondi news" $\sigma^o$ comprises only the $l=2$ components of the "spherical harmonics with spin-weight 2"), the energy carried away is \emph{manifestly positive-definite} for the $\Lambda>0$ case. For a general $\sigma^o$ having higher multipole moments, this perspicuous property in the $\Lambda>0$ case still holds if those $l>2$ contributions are weak --- more precisely, if they satisfy any of the inequalities given in this paper.