Coordinate domain versus solution-jet domain

The necessary first metric condition in VI's named ${\mathcal{J}}_T$ is

where its fixed linearized Einstein tensor is

Here lowering in the latter formula uses ${\eta }_{-}$, and parentheses denote symmetrization with factor $1/2$. This normalization is sufficient for our domain test.

The higher metric coefficient, scalar coefficients, and their equations impose further conditions in ${\mathcal{J}}_T$. We do not assert that an arbitrary smooth metric argument extends to that solution-jet space CheungVI (:jet-space).

consequence. Failure of (eq:VI-vacuum) prevents applying VI's solution-jet descent theorem to that input. It does not prevent solving the independent smooth coordinate Cauchy problem (eq:VI-data).