= Synchronous gauge fixing by coordinate displacement
{title2=$\alpha'+\mathcal H\alpha=-q_{00}/(2a^2),\quad\xi_i'=q_{0i}/a^2+\partial_i\alpha$}
For the <FRW metric> in <conformal time>, a displacement $\xi^\mu=(\alpha,\xi^i)$ changes a raw <metric perturbation> by $q_{\mu\nu}\mapsto q_{\mu\nu}-\mathcal L_\xi\bar g_{\mu\nu}$. The displayed first-order time equations set the lapse and shift perturbations to zero. They can be integrated locally for smooth perturbations, demonstrating accessibility of <synchronous gauge in cosmology>. Their <homogeneous solutions> leave residual spatial functions, so the gauge is not fully fixed. A globally regular synchronous chart need not survive caustics of its freely falling congruence.
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