Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 1 / a

The normalized timelike normal has $n_{\alpha }{n}^{\alpha }=-1$. Multiplying the linear projections givesAlso $P^{\mu }{}_{\alpha }{n}_{\mu }=n_{\alpha }+(n_{\mu }{n}^{\mu })n_{\alpha }=0$. ThusThe operator fixes every vector tangent to the spacelike hypersurface and annihilates its normal. The repeated-index typo in the TeX is resolved by the actual PDF's matrix-composition indices.