Differentiability of the gravitational Hamiltonian at spatial infinity

For asymptotically Cartesian spatial metrics $h_{ij}={\delta }_{ij}+O(r^{-1})$, unit asymptotic lapse and vanishing asymptotic shift, the curvature term in the bulk gravitational Hamiltonian has variationThe surface variation is finite under the usual differentiable asymptotic falloff and is not removed by simply taking the boundary to infinity. Adding Arnowitt-Deser-Misner energy cancels it and makes Hamilton's equations well defined for allowed variations that need not fix the energy. On the constraint surface, the total Hamiltonian for that time translation equals the boundary energy.