Perturbation domain

For the calculations through second order, take smooth, spatially compact perturbations with a common compact support:

The tensors $h,k$ are arbitrary symmetric covariant tensors. No scalar, vector, or tensor metric component is discarded.

The formulas are formal Taylor coefficients. For an actual nonperturbative coordinate map, one must additionally restrict to configurations for which the scalar level sets are spacelike, the harmonic-coordinate problem has the stated boundary solution, and the resulting coordinate map is invertible on the region used by $f$. We do not assert such global properties for every smooth configuration.

For the formal coefficients considered here, all elliptic sources are smooth and spatially compact, and the required decaying solutions are constructed explicitly.