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

At fixed $N$ and $h_{ij}$, varying the shift givesandIntegration by parts in the gravitational term turns the variation of $K_{ij}{K}^{ij}-K^2$ into the spatial divergence of $K_i{}^j-{\delta }_i{}^{j}K$. Since the shift has no time derivative, its Euler-Lagrange equation is the ADM momentum constraint for a P(X, phi) scalar field