Section current and its boundary pairing

Set

Then

For a compact metric test $h$,

For $h\in {\mathcal{D}}_{\mathrm{ret}}$, the unambiguous identity instead is

The second pairing has compact support intersection: $D(h)$ vanishes sufficiently far in the past, while $A$ has advanced support from a compact source.

These assertions follow from

and $K^{\ast }{D}^{\ast }A=\square A=\mathbf{H}$. For noncompact $D(h)$, one inserts a compact cutoff on the causal region relevant to the compact source $\mathbf{H}$.

A temporal cutoff makes the boundary issue explicit. Choose $\theta =0$ sufficiently far in the past and $\theta =1$ by the time either $J_{\mathrm{raw}}$ or $D(h)$ can be nonzero. Then

Indeed, $D(\theta h{)}^{\nu }=\theta D(h{)}^{\nu }+({\partial }_{\mu }\theta ){\bar{h}}^{\mu \nu }$. Every displayed pairing has compact support intersection. Conservation of $J_{\sec }$ alone does not authorize deletion of the second term.

Figure fig:harmonic-boundary shows why harmonic prescriptions retain residual gauges and why the boundary pairing must be kept.