The inverse <3+1 decomposition of spacetime> metric gives
$$
\boxed{X=\frac1{2N^2}(\dot\phi-N^i\partial_i\phi)^2
-\frac12h^{ij}\partial_i\phi\partial_j\phi}.
$$
The first term is the squared derivative along the hypersurface normal and the second is the spatial-gradient contribution.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 2 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-312/2/i/solution