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At linear order a coordinate displacement ${\xi }^{\mu }$ changes a metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. Work in conformal time with ${\bar{g}}_{\mu \nu }=a^2\mathrm{diag}(-1,1,1,1)$, write ${\xi }^0=\alpha$ and lower spatial indices with ${\delta }_{ij}$. Directly evaluating the Lie derivative of a covariant tensor field givesHere primes on $\alpha ,{\xi }_i$ mean conformal-time derivatives, while primes on $q_{0\mu }^{\prime }$ label transformed components. Setting the transformed lapse and shift perturbations to zero requiresThese are first-order time equations. Their explicit local solutions areThus four coordinate functions can impose the four synchronous conditions in any regular perturbative patch. This is synchronous gauge fixing by coordinate displacement. The arbitrary spatial functions $C,D_i$ remain as residual gauge freedom; the conditions alone do not fix the coordinate system completely.

Geometrically, synchronous coordinates follow a congruence of freely falling observers launched normally from a spatial slice, using their proper time. This gives unit lapse and zero shift in cosmic time; conversion to background conformal time gives the displayed convention. “Always possible” is local: geodesic caustics or singularities can prevent a global synchronous chart. Such global obstructions do not invalidate local linear cosmological perturbation theory.