Insert $Z_\mu=\Theta_\mu+n_\mu\Theta$ from part (i) into the projected divergence:
$$
D^\mu Z_\mu
=D^\mu\Theta_\mu
+\perp^{\mu\nu}\nabla_\mu(n_\nu\Theta).
$$
The term containing $\nabla_\mu\Theta$ vanishes because $\perp^{\mu\nu}n_\nu=0$. The remaining contraction is
$$
\Theta\perp^{\mu\nu}\nabla_\mu n_\nu=-K\Theta
$$
by the definition of the <extrinsic curvature of a spatial hypersurface>. Therefore
$$
\boxed{D^\mu Z_\mu=D^\mu\Theta_\mu-K\Theta.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-357/1/a/iv/solution