Normal scalar-field momentum
= Normal scalar-field momentum
{title2=$\Pi=n^\mu\partial_\mu\phi$}
For a <minimally coupled scalar field> with spatial <induced metric> $h_{ij}$, the field's derivative along the future <unit normal> is $\Pi$. If the <shift vector> enters as $dx^i-N^idt$, then $\Pi=(\dot\phi+N^i\partial_i\phi)/N$. The canonical momentum of the density is $\pi_\phi=\sqrt h\,\Pi$. Confusing this density with the normal derivative loses a volume factor.