Support classes

Fix a reference time $t_0$. For this paper, a smooth field is called spacelike compact if, for some $R_0$,

This concrete cone condition is sufficient for every use below. It follows from compact Cauchy data for homogeneous wave and Klein–Gordon solutions.

Let ${\mathcal{E}}_{+}$ be the smooth fields supported in $J_{\eta }^{+}(K_0)$ for some compact spacetime set $K_0$. Use this notation componentwise for vectors and tensors. Such fields have zero data on a sufficiently early Cauchy slice and bounded spatial support on each finite time interval.

Compact spacetime support, compact Cauchy data, and future support from a compact set are different conditions.