Definition: Declared flat completion

The analytical completion is the separate Minkowski space

All coordinate and scalar Cauchy problems are solved on this completion. Smooth fields are defined on a neighbourhood of $M_{T_{\ast }}$, without decay assumptions. A compact laboratory test is transported to $\hat{M}$ and extended by zero. The initial surface is the complete flat $\Sigma$, not a claimed Cauchy surface of ${\mathcal{M}}^{\circ }$.