= 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 variation
$$
\delta H_{\rm bulk}=\text{bulk functional derivatives}-\frac1{16\pi G}\lim_{r\to\infty}\int_{S_r}n^i(\partial_j\delta h_{ij}-\partial_i\delta h_{jj})\,dA.
$$
The 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.
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