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.

For signature $(+---)$ and future normal $n$, write $\Pi =n^{\mu }{\partial }_{\mu }\varphi$ and use the positive induced metric ${\gamma }_{ij}$ for $s^2={\gamma }^{ij}{D}_i\varphi D_j\varphi$. The Klein-Gordon scalar stress-energy tensor has normal-frame density $\rho =({\Pi }^2+s^2)/2+V$, momentum $J_i=-\Pi D_i\varphi$ and spatial stress $S_{ij}=D_i\varphi D_j\varphi +{\gamma }_{ij}[({\Pi }^2-s^2)/2-V]$. The spatial index convention and normal orientation must be specified.

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