  The Kerr metric outside the ergosphere is transformed into ADM coordinates up
to the orders $1/r^4$ and $a^2$, respectively in radial coordinate $r$ and
reduced angular momentum variable $a$, starting from the Kerr solution in
quasi-isotropic as well as harmonic coordinates. The distributional source
terms for the approximate solution are calculated. To leading order in linear
momenta, higher-order-in-spin interaction Hamiltonians for black-hole binaries
are derived.
