Scalar identification and formal coefficients

Let ${\varphi }_0$ be a free scalar with smooth compact initial data. Lemma lem:wave defines $S_{t_0}(h){\varphi }_0$, with support in the causal evolution of those data on each side of $t_0$.

If $H_{t_0}(u)$ denotes the homogeneous scalar solution with the Cauchy data of $u$ at $t_0$, then

Both sides have zero data and solve $P\psi =P(X_{\xi }{\varphi }_0)$, by (eq:gauge-identities); uniqueness proves the equality.