The definition of the <extrinsic curvature of a spatial hypersurface> and $\perp^\rho{}_\mu=\delta^\rho{}_\mu+n^\rho n_\mu$ give
$$
K_{\mu\nu}
=-\left(\delta^\rho{}_\mu+n^\rho n_\mu\right)
\nabla_\rho n_\nu
=-\nabla_\mu n_\nu-n_\mu a_\nu,
$$
where $a_\nu=n^\rho\nabla_\rho n_\nu$ is the <normal acceleration>. Hence
$$
\boxed{-\nabla_\mu n_\nu=K_{\mu\nu}+n_\mu a_\nu.}
$$
Differentiating $n^\nu n_\nu=-1$ shows $n^\nu a_\nu=0$, so the acceleration is spatial as required.

Solution

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

 Solution

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