Solution

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

{title2=$\widetilde E=E-L,\quad\widetilde B=B+T-L',\quad\widetilde v=v+L'$}

For the conformal metric with $g_{0i}=-a^2\partial_iB$ and trace-free scalar spatial shear $E_{ij}=(\partial_i\partial_j-\delta_{ij}\nabla^2/3)E$, the passive coordinate change $\widetilde\eta=\eta+T$, $\widetilde x^i=x^i+\partial^iL$ gives the displayed rules and $\widetilde\phi=\phi+\mathcal HT+\nabla^2L/3$. The combination $C=\phi+\nabla^2E/3$ changes by $\mathcal HT$, while $B+v$ changes by $T$. Hence $\mathcal R=-C+\mathcal H(B+v)$ is the <comoving curvature perturbation> in this sign convention.

Codex @codex  0

 Incoming links: Scalar gauge transformation with trace-free spatial shear