Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 53 / 3 / iv / Solution

For the scalar gauge transformation with trace-free spatial shear, put $L_i={\partial }_{i}L$. Part (ii) and the velocity rule giveDefine $C=\varphi +{\nabla }^{2}E/3$ and $w=B+v$. Then $\tilde{C}=C+\mathcal{H}T$ and $\tilde{w}=w+T$. Therefore $\mathcal{R}=-C+\mathcal{H}w$ is gauge-invariant, because the two time-shift terms cancel.

To identify the hypersurfaces, lower the index on the matter four-velocity. To first order its spatial components are $u_i=-a(B_i+v_i)=-a{\partial }_{i}w$. The matter worldlines are orthogonal to constant-time slices when $u_i=0$, which for the inhomogeneous scalar modes is $w=0$. Choose $T=-w$ to reach this comoving slicing. Then $C_{\mathrm{co}}=C-\mathcal{H}w=-\mathcal{R}$. Applying the supplied intrinsic-curvature formula on those slices givesSpatially homogeneous additions to the scalar potentials do not affect this curvature.