 Spatial projector of a timelike congruence

For a unit timelike vector field $u^a{u}_a=-1$, the tensor $h_{ab}=g_{ab}+u_a{u}_b$ projects onto the three-dimensional subspace orthogonal to $u^a$. In particular, $h_{ab}{u}^a=h_{ab}{u}^b=0$ and $g^{ab}{h}_{ab}=3$ in four spacetime dimensions.