3+1 decomposition of the stress-energy tensor

Relative to a future unit normal $n^{\mu }$, the stress-energy tensor decomposes aswhere $j_{\mu }$ and both indices of $S_{\mu \nu }$ are spatial.

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