The asymptotic symmetry of an isolated gravitating system, or the Bondi–Metzner–Sachs (BMS) group, contains an infinite-dimensional subgroup of supertranslations. Despite decades of study, the difficulties with the “supertranslation ambiguity” persisted in making sense of fundamental notions such as the angular momentum carried away by gravitational radiation. The issues of angular momentum and center of mass were resolved by the authors recently. In this paper, we address the issues for conserved charges with respect to both the classical BMS algebra and the extended BMS algebra. In particular, supertranslation ambiguity of the classical charge for the BMS algebra, as well as the extended BMS algebra, is completely identified. We then propose a new invariant charge by adding correction terms to the classical charge. With the presence of these correction terms, the new invariant charge is then shown to be free from any supertranslation ambiguity. Finally, we prove that both the classical and invariant charges for the extended BMS algebra are invariant under the boost transformations.
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