Theorem: Conserved-current smoothing

Let $c\in {{\mathcal{D}}^{\prime }}_c(M;H)$ be symmetric and conserved, and suppose

is smooth. Let $\chi$ be a smooth temporal cutoff, zero sufficiently far in the past and one sufficiently far in the future. Define the covariant representative $j$ and its raised current $J$ by

Then $j$, and hence $J$, is smooth, compact, and conserved. Moreover,

Proof. Since $v=E_{-}Tc$,

Lemma lem:identities gives $\mathcal{L}v=0$. Consequently

The right-hand side is supported in the transition slab of $\chi$. The spacelike compactness of $v$ therefore makes its support compact. Smoothness is immediate, and the Bianchi identity in Lemma lem:identities gives conservation.

First suppose $h\in {\mathcal{S}}_{\mathrm{dD}}$. Put

Both are smooth and compact. Equation (eq:einstein-operator) gives

Integration by parts yields

Hence $⟨j,h⟩=-⟨F,h⟩$.

As $Tv=E_{-}c$, the retarded and advanced identities are

It follows that

By Lemma lem:compact-primitive, $F+c=\square A$ for a compact distribution $A$. Since $\square h=0$,

Thus $⟨j,h⟩=⟨c,h⟩$ on ${\mathcal{S}}_{\mathrm{dD}}$.

Now let $h\in {\mathcal{S}}_{\mathrm{E}}$ be arbitrary. Solve

A global smooth solution exists without spatial decay: componentwise, the zero-data formula is

For negative $t$, the integral has its usual oriented meaning. Each value depends on a finite spatial region, so arbitrary smooth sources cause no convergence issue.

Set $h_0=h-Kw$. The operator identities give

Thus $h_0\in {\mathcal{S}}_{\mathrm{dD}}$. Compact conservation of $c$ and $j$ implies

The equality on $h_0$ therefore extends to $h$, proving (eq:full-vacuum-representation).