Lemma: Compactness conditions

The following are equivalent:

Proof. The first two are (eq:compactprimitive); compactness of $A$ implies compactness of $J_{\sec }$.

Conversely suppose $J_{\sec }$ is compact. In an early half-space $t<t_{\ast }$, both $J_{\sec }$ and $J_{\mathrm{raw}}$ vanish, so $D^{\ast }A=0$. There $A$ is smooth by (eq:Aexplicit). Trace reversal of (eq:Dstar) shows that $A^{♯}$ satisfies the flat Killing equation. Differentiating that equation and cyclically permuting indices proves that all second derivatives vanish. Hence $A$ is affine in this connected half-space.

Its advanced support confines every spatial slice to a bounded set. An affine field with that property is zero. Thus $A$ vanishes sufficiently far in the past. Its advanced support also bounds it above in time; in that bounded slab, the causal past of its compact source has a uniform spatial bound. Hence $A$ is compact.