Covariant counter-terms and conserved charges in asymptotically flat spacetimes

Recent work has shown that the addition of an appropriate covariant boundary term to the gravitational action yields a well-defined variational principle for asymptotically flat spacetimes and thus leads to a natural definition of conserved quantities at spatial infinity. Here we connect such results to other formalisms by showing explicitly (i) that for spacetime dimension d ⩾ 4 the canonical form of the above-mentioned covariant action is precisely the ADM action, with the familiar ADM boundary terms and (ii) that for d = 4 the conserved quantities defined by counter-term methods agree precisely with the Ashtekar–Hansen conserved charges at spatial infinity.