 3+1 decomposition of spacetime

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.