= Solution
A <cosmological gauge transformation> changes the coordinates used to identify a perturbed spacetime with its homogeneous background. It can produce apparent <metric perturbations> even in a physically unperturbed <FRW metric>. For $\widetilde x^i=x^i+\xi^i(\tau,\mathbf x)$ and unchanged time, the inverse differentials are, to first order,
$$
dx^i=d\widetilde x^i-\partial_j\xi^i\,d\widetilde x^j
-\xi^{i\prime}\,d\tau.
$$
Inserting these into the background line element gives the transformed components
$$
\widetilde g_{ij}=a^2(\delta_{ij}-\partial_i\xi_j-\partial_j\xi_i),
\qquad \widetilde g_{0i}=-a^2\xi_i',\qquad
\widetilde g_{00}=-a^2.
$$
The spatial components can now depend on position, although no physical density inhomogeneity or spacetime curvature perturbation has been introduced. For a scalar displacement $\xi^i=\partial^iL$, these are pure coordinate-generated <scalar cosmological perturbations>: $E=-L$, $C=-\nabla^2L/3$, $B=-L'$ and $A=0$ in the transformed coordinates.
The \b[gauge problem] is that individual perturbation variables mix this coordinate freedom with physical fluctuations. A nonzero coordinate-dependent field therefore need not represent a physical effect. One must either choose a definite gauge, such as <Newtonian gauge in cosmology>, or construct invariant combinations, while ensuring that observables do not depend on that choice.
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