In this paper we establish two results concerning four-dimensional
asymptotically flat spacetimes at spatial infinity. First, we show that the six
conserved Lorentz charges are encoded in two unique, distinct, but mutually
dual symmetric divergence free tensors that we construct from the equations of
motion. Second, we show that integrability of Einstein's equations in the
asymptotic expansion is sufficient to establish the equivalence between
counter-term charges defined from the variational principle and charges defined
by Ashtekar and Hansen. These results clarify earlier constructions of
conserved charges in the hyperboloid representation of spatial infinity. In
showing this, parity condition on the mass aspect is not needed. Along the way
in establishing these results, we prove two lemmae on tensor fields on three
dimensional de Sitter spacetime stated by Ashtekar-Hansen and Beig-Schmidt and
state and prove three additional lemmae.Comment: 26 pages; no figures; v2: minor changes; v3: clarifications +
references + a new lemma added, results unaffecte