Momentum constraint (${\mathcal{H}}_i=-2D_j({\pi }^j{}_i/\sqrt{h})=0$)

The momentum constraint is the equation obtained by varying the shift vector in the Hamiltonian formulation of general relativity. It expresses the tangential-normal projection of the Einstein field equations and generates spatial coordinate transformations. Writing $p^{ij}={\pi }^{ij}/\sqrt{h}$ avoids ambiguities about covariant derivatives of tensor densities.

For scalar metric potentials $d{s}^2=a^2[-(1+2A)d{\tau }^2+2B_{,i}d\tau d{x}^i+((1-2\psi ){\delta }_{ij}+2E_{,ij})d{x}^{i}d{x}^j]$ and positive extrinsic curvature, $\delta K^i{}_j=a^{-1}[-({\psi }^{\prime }+\mathcal{H}A){\delta }^i{}_j+{\partial }^i{\partial }_j(E^{\prime }-B)]$. The scalar shear terms cancel between its divergence and trace gradient. With $j_i=-P^{\mu }{}_i{T}_{\mu \nu }{n}^{\nu }$, the momentum constraint becomes $2{\partial }_i({\psi }^{\prime }+\mathcal{H}A)/a=-8\pi G{j}_i$. Reversing the convention for $K$ reverses the geometric left side.