Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 1 / a / iv / Solution

Insert $Z_{\mu }={\Theta }_{\mu }+n_{\mu }\Theta$ from part (i) into the projected divergence:The term containing ${\nabla }_{\mu }\Theta$ vanishes because ${\perp }^{\mu \nu }{n}_{\nu }=0$. The remaining contraction isby the definition of the extrinsic curvature of a spatial hypersurface. Therefore