 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 ${\tilde{x}}^i=x^i+{\xi }^i(\tau ,\mathbf{x})$ and unchanged time, the inverse differentials are, to first order,Inserting these into the background line element gives the transformed componentsThe spatial components can now depend on position, although no physical density inhomogeneity or spacetime curvature perturbation has been introduced. For a scalar displacement ${\xi }^i={\partial }^{i}L$, these are pure coordinate-generated scalar cosmological perturbations: $E=-L$, $C=-{\nabla }^{2}L/3$, $B=-L^{\prime }$ and $A=0$ in the transformed coordinates.

The 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.