 Scalar-field matter projections

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.