Spacetime and free propagation

Let

and write

All coefficient fields are smooth up to ${\Sigma }_0$. Indices are raised and lowered with $\eta$ unless stated otherwise.

For $m\ge 0$ set

The operator ${\mathsf{E}}_m$ denotes the zero-Cauchy-data solution operator:

For sources supported away from ${\Sigma }_0$, it agrees on $M_{+}$ with the ordinary retarded Green operator applied to their extension by zero.

We use ${\mathsf{E}}_m$ on smooth sources. Sources need not have globally compact spatial support: on every compact observation region, finite propagation reduces the Cauchy problem to a compact portion of the source and initial data. The relevant existence, uniqueness, and finite-propagation statements are standard Cauchy theory for normally hyperbolic operators; see BGP (Chapter 3).