Zero-data identities

For $\xi \in {\mathfrak{g}}_U$, both $\xi$ and $X_{\xi }{\varphi }_0$ vanish on a neighbourhood of ${\Sigma }_0$. Accordingly,

The relevant fact is the vanishing Cauchy data. Compact support is a sufficient condition here because the admitted support is strictly to the future of ${\Sigma }_0$.