We analyze the asymptotic symmetries and their associated charges at spatial infinity in $4$-dimensional asymptotically-flat spacetimes. We use the covariant formalism of Ashtekar and Hansen where the asymptotic fields and symmetries live on the $3$-manifold of spatial directions at spatial infinity, represented by a timelike unit-hyperboloid (or de Sitter space). Using the covariant phase space formalism, we derive formulae for the charges corresponding to asymptotic supertranslations and Lorentz symmetries at spatial infinity. With the motivation of, eventually, proving that these charges match with those defined on null infinity -- as has been conjectured by Strominger -- we do not impose any restrictions on the choice of conformal factor in contrast to previous work on this problem. Since we work with a general conformal factor we expect that our charge expressions will be more suitable to prove the matching of the Lorentz charges at spatial infinity to those defined on null infinity, as has been recently shown for the supertranslation charges.

中文翻译：

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广义相对论中空间无穷远的渐近对称性和电荷

我们分析了 $4$ 维渐近平坦时空中的渐近对称性及其在空间无穷远处的相关电荷。我们使用 Ashtekar 和 Hansen 的协变形式主义，其中渐近场和对称性存在于空间无限远的 $3$-空间方向流形上，由类时间单位双曲面（或 de Sitter 空间）表示。使用协变相空间形式主义，我们推导出对应于空间无穷大处的渐近超平移和洛伦兹对称性的电荷的公式。出于最终证明这些电荷与零无穷大上定义的电荷相匹配的动机——正如 Strominger 所推测的那样——与之前在这个问题上的工作相比，我们没有对保形因子的选择施加任何限制。