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.

The lapse function measures proper-time separation between neighboring spatial hypersurfaces along their unit normal.

The shift vector describes how spatial coordinates move tangentially between neighboring hypersurfaces.

With future unit normal $n^{\mu }$, the convention $K_{\mu \nu }=-{\perp }^{\rho }{}_{\mu }{\nabla }_{\rho }{n}_{\nu }$ measures how the hypersurface is embedded in spacetime.

For $J^{\mu }=Q{n}^{\mu }+Q^{\mu }$ with spatial $Q^{\mu }$, current conservation gives $n^{\mu }{\nabla }_{\mu }Q=KQ-D_{\mu }{Q}^{\mu }-Q^{\mu }{D}_{\mu }\log \alpha$ under the stated extrinsic-curvature convention.