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.

Differentiating the spatial projection tensor and projecting its derivative and covariant indices leaves $-n^{\lambda }{K}_{\sigma \rho }$, where $K$ is the extrinsic curvature of a spatial hypersurface. Contracting $\lambda$ with $\rho$ gives zero because $K$ is transverse. This distinction separates the informative tensor identity, with a free normal index, from its vanishing trace. Hypersurface orthogonality implies symmetric extrinsic curvature.

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.

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.

For a unit timelike normal $n$, the induced metric is $h_{\mu \nu }=g_{\mu \nu }+n_{\mu }{n}_{\nu }$. The spatial covariant derivative projects the derivative and all free indices with the spatial projection tensor $P^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu }+n^{\mu }{n}_{\nu }$. Metric compatibility gives ${\nabla }_{\lambda }{h}_{\mu \nu }=({\nabla }_{\lambda }{n}_{\mu })n_{\nu }+n_{\mu }{\nabla }_{\lambda }{n}_{\nu }$; projecting kills the normal factor in each term. Thus the spatial covariant derivative is compatible with the induced 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 $a_i=D_i\log \alpha$.