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A 3+1 decomposition writes $d{s}^2=-{\alpha }^{2}d{t}^2+{\gamma }_{ij}(d{x}^i+{\beta }^{i}dt)(d{x}^j+{\beta }^{j}dt)$ in terms of lapse, shift, and spatial metric.

A spacetime foliation separates a spatial metric, lapse and shift. The ADM formalism uses this decomposition to express Einstein evolution and constraint equations in Hamiltonian form.

The Arnowitt-Deser-Misner energy is the asymptotic surface charge generating time translations on an asymptotically flat spatial hypersurface. In asymptotically Cartesian coordinates,

The Arnowitt-Deser-Misner energy is the asymptotic Hamiltonian boundary charge in the ADM formalism for an asymptotically flat spacetime.