Momentum equation in a 3+1 decomposition
= Momentum equation in a 3+1 decomposition
With $K_{\mu\nu}=-\perp^\rho{}_\mu\nabla_\rho n_\nu$ and $a_\mu=n^\rho\nabla_\rho n_\mu$, spatial projection of $\nabla_\mu T^{\mu\nu}=0$ gives
$$
\mathcal L_nj_\alpha=-D_\mu S^\mu{}_\alpha-S^\mu{}_\alpha a_\mu+Kj_\alpha-\rho a_\alpha.
$$