Spatial projection tensor (${\perp }^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu }+n^{\mu }{n}_{\nu }$)

For a unit timelike normal $n^{\mu }{n}_{\mu }=-1$, the spatial projection tensor annihilates $n^{\mu }$ and projects every spacetime tensor onto a spatial hypersurface.

The spatial covariant derivative projects the spacetime covariant derivative and every free tensor index onto a spatial hypersurface. It is the Levi-Civita connection of the induced spatial metric.

The normal acceleration measures failure of the hypersurface unit normals to be geodesics. It is spatial, $n^{\mu }{a}_{\mu }=0$, and in a 3+1 decomposition satisfies $a_i=D_i\log \alpha$.