Proposition: Coordinate coefficient equations

The problem (eq:coordproblem) has a unique formal solution. Its coefficients are the zero-data solutions of

that is,

Both coefficients and their first time derivatives vanish on $\Sigma$. The value ${{\mathsf{X}}_{\mathsf{j}}}^{\mu }(y)$ depends on $(h,k)$ only in $J^{-}(y)\cap M_T$. The coefficient ${\mathsf{X}}_2$ is affine in $k$:

Proof. Since $\square x^{\mu }=0$, inserting Lemma lem:Dexp into (eq:coordproblem) gives at order $\kappa$ the equation $\square {{\mathsf{X}}_1}^{\mu }+{\mathcal{V}}_{\epsilon }[h]x^{\mu }=0$ and at order ${\kappa }^2$ the equation $\square {{\mathsf{X}}_2}^{\mu }+{\mathcal{V}}_{\epsilon }[h]{{\mathsf{X}}_1}^{\mu }+{\mathcal{W}}_{\epsilon }[h,k]x^{\mu }=0$. The displayed right-hand sides follow from (eq:V), (eq:W), ${\partial }_{\beta }{x}^{\mu }={{\delta }^{\mu }}_{\beta }$, and (eq:C). Existence, uniqueness, and dependence are Lemma lem:cauchy. Equation (eq:X2affine) holds because the $k$-dependent part of the second source is $-{\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \mu }[k])$, the first source for $k$.