At fixed $N$ and $h_{ij}$, varying the shift gives
$$
\delta K_{ij}=-\frac1{2N}\left({}^{(3)}\nabla_i\delta N_j+{}^{(3)}\nabla_j\delta N_i\right)
$$
and
$$
\delta X=-\frac1{N^2}(\dot\phi-N^j\partial_j\phi)\partial_i\phi\,\delta N^i.
$$
Integration by parts in the gravitational term turns the variation of $K_{ij}K^{ij}-K^2$ into the spatial divergence of $K_i{}^j-\delta_i{}^jK$. Since the shift has no time derivative, its Euler-Lagrange equation is the <ADM momentum constraint for a P(X, phi) scalar field>
$$
\boxed{{}^{(3)}\nabla_j(K_i{}^j-\delta_i{}^jK)
+\frac{P_{,X}}{M_{\rm Pl}^2N}\partial_i\phi
(N^j\partial_j\phi-\dot\phi)=0}.
$$

Solution

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

{c}
{title2=ADM momentum constraint for a $P(X,\phi)$ scalar field}

Varying the shift in the <3+1 decomposition of spacetime> gives a nondynamical momentum constraint. For a $P(X,\phi)$ field it equates the spatial divergence of $K_i{}^j-\delta_i^jK$ to the scalar momentum density.

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