there is a natural vector space preserving isomorphism between the space of functions on I< and supertranslations on Spi, and that functions on l< which thus correspond to trans-lations are of the type (f(k))(1)) = ka1)a for some vector ka in the tangent space of iO. Consider the linear mapping f(k) -~ r 2 Eab(Dbf(k)kamndSW'n . s (23a) from the space of translations to the reals, where S2 is a 2-sphere cross section of the hyperboloid. Using the definition of f(k), it follows that DaD/Jf(k) '" -f(k)hab• Thus, D'lf(k) is a conformal Killing field on l<. Since Eab is both trace and divergence free, it follows that the integral in Eq. (22) is independent of the choice of the cross section. Thus, we have obtained a conserved quantity which takes values in the dual of the vector space of translations. This is the total 4-momentum. It is not difficult to show that this conserved quantity is essentially the same as the ADM 4-momentum. 6,3 (That is, the two agree when both are defined. )

the ADM pw can be written in the Ashtekar-Hansen form [lo]?: ppXp = lim (1 (-det g)l/2~wvapXpxYRnPpu dxP A dx" r+m r=constant d((-det g)'/2&,,,px"XpgaYT~p dxp) (32~)-' +2 I r=constant ) = lim (i (-det g)l/2R,,apXpx" dS"@)(16~)-'