= Scalar-field matter projections
{title2=$\rho=(\Pi^2+s^2)/2+V$}
For signature $(+---)$ and future normal $n$, write $\Pi=n^\mu\partial_\mu\phi$ and use the positive induced metric $\gamma_{ij}$ for $s^2=\gamma^{ij}D_i\phi D_j\phi$. The <Klein-Gordon scalar stress-energy tensor> has normal-frame density $\rho=(\Pi^2+s^2)/2+V$, momentum $J_i=-\Pi D_i\phi$ and spatial stress $S_{ij}=D_i\phi D_j\phi+\gamma_{ij}[(\Pi^2-s^2)/2-V]$. The spatial index convention and normal orientation must be specified.
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