= Spatial metric sign for a unit timelike normal
{title2=$h_{\mu\nu}=n_\mu n_\nu-g_{\mu\nu}$}
With signature $+---$ and $n^\mu n_\mu=1$, the <spatial projection tensor> is $P^\mu{}_\nu=\delta^\mu{}_\nu-n^\mu n_\nu$. Projecting the spacetime <metric tensor> gives $q_{\mu\nu}=g_{\mu\nu}-n_\mu n_\nu$, which is negative definite on spatial vectors. The positive spatial <induced metric> is $h_{\mu\nu}=-q_{\mu\nu}$. A plus sign in $g_{\mu\nu}+n_\mu n_\nu$ belongs to a different normal/signature convention and is not transverse in this one. Fully projected <covariant derivatives> of either spatial metric vanish by <metric compatibility>.
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