Spatially project <stress-energy conservation>,
$$
\perp^\nu{}_\alpha\nabla_\mu T^\mu{}_\nu=0,
$$
and substitute the decomposition from part a. The $\rho n^\mu n_\nu$ term contributes $\rho a_\alpha$. The two momentum terms combine into
$$
\mathcal L_nj_\alpha-Kj_\alpha,
$$
because the two contractions with the full extrinsic curvature cancel and
$$
\nabla_\mu n^\mu=-K.
$$
Finally, part b gives
$$
\perp^\nu{}_\alpha\nabla_\mu S^\mu{}_\nu
=D_\mu S^\mu{}_\alpha+S^\mu{}_\alpha a_\mu.
$$
Thus
$$
\rho a_\alpha+\mathcal L_nj_\alpha-Kj_\alpha
+D_\mu S^\mu{}_\alpha+S^\mu{}_\alpha a_\mu=0.
$$
The <momentum equation in a 3+1 decomposition> is
$$
\boxed{
\mathcal L_nj_\alpha
=-D_\mu S^\mu{}_\alpha
-S^\mu{}_\alpha a_\mu
+Kj_\alpha-\rho a_\alpha}.
$$
Therefore
$$
\boxed{c_1=-1,\qquad c_2=-1,\qquad c_3=1,\qquad c_4=-1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 357 / 2 / d / Solution

With $K_{\mu\nu}=-\perp^\rho{}_\mu\nabla_\rho n_\nu$ and $a_\mu=n^\rho\nabla_\rho n_\mu$, spatial projection of $\nabla_\mu T^{\mu\nu}=0$ gives
$$
\mathcal L_nj_\alpha=-D_\mu S^\mu{}_\alpha-S^\mu{}_\alpha a_\mu+Kj_\alpha-\rho a_\alpha.
$$


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