ADM momentum ($P_i=(8\pi G{)}^{-1}\lim \int (k_{ij}-k{h}_{ij})s^{j}dS$)

For an asymptotically flat spatial metric $h$ and $k_{ij}=-g({\nabla }_{i}n,e_j)$, the displayed surface integral gives ADM spatial momentum. Together with ADM energy it forms the asymptotic energy-momentum four-vector. Reversing the sign convention for extrinsic curvature reverses the corresponding surface-integral sign.