We use the formalism developed by Wald and Zoupas to derive explicit covariant expressions for the charges and fluxes associated with the Bondi–Metzner–Sachs symmetries at null infinity in asymptotically flat spacetimes in vacuum general relativity. Our expressions hold in non-stationary regions of null infinity, are local and covariant, conformally-invariant, and are independent of the choice of foliation of null infinity and of the chosen extension of the symmetries away from null infinity. While similar expressions have appeared previously in the literature in Bondi–Sachs coordinates (to which we compare our own), such a choice of coordinates obscures these properties. Our covariant expressions can be used to obtain charge formulae in any choice of coordinates at null infinity. We also include detailed comparisons with other expressions for the charges and fluxes that have appeared in the literature: the Ashtekar–Streubel flux formula, the Komar formulae, and the linkage and twistor charge formulae. Such comparisons are easier to perform using our explicit expressions, instead of those which appear in the original work by Wald and Zoupas.

中文翻译：

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广义相对论中零无穷处渐近电荷的 Wald-Zoupas 处方

我们使用 Wald 和 Zoupas 开发的形式主义来推导出与真空广义相对论中渐近平坦时空的零无穷处 Bondi-Metzner-Sachs 对称性相关的电荷和通量的显式协变表达式。我们的表达式在零无穷大的非平稳区域中成立，是局部和协变的，共形不变的，并且独立于零无穷大的叶状结构选择和远离零无穷大的对称性扩展的选择。虽然类似的表达式以前出现在 Bondi-Sachs 坐标的文献中（我们将自己的坐标与之进行比较），但这种坐标选择掩盖了这些属性。我们的协变表达式可用于在零无穷远的任何坐标选择中获得电荷公式。我们还详细比较了文献中出现的电荷和通量的其他表达式：Ashtekar–Streubel 通量公式、Komar 公式以及连接和扭量电荷公式。使用我们的显式表达式更容易执行此类比较，而不是 Wald 和 Zoupas 的原始作品中出现的那些。