Proposition: Smooth compact does not imply conserved representability

There exists a smooth compact symmetric tensor $c$ such that:

1. $c-c_{\mathrm{sm}}=\square r$ with $c_{\mathrm{sm}}=c$ and $r=0$;

2. $E_{-}Tc$ is smooth;

3. no smooth compact conserved tensor represents $h\mapsto ⟨c,h⟩$ on ${\mathcal{S}}_{\mathrm{E}}$, or even on ${\mathcal{S}}_{\mathrm{dD}}$.

Proof. Choose $g\in {C_c}^{\infty }({\mathbb{R}}^3)$ with $g(x_0)>0$. Let $\zeta$ solve

It is smooth and spacelike compact. For example,

By continuity, ${\partial }_t\zeta >0$ on some nonempty spacetime neighborhood, which may be chosen with $t>0$.

Choose a nonzero nonnegative $\psi \in {C_c}^{\infty }(M)$ supported in that neighborhood. Set

and let $c$ be its lowered tensor. Define

Then

Thus $h\in {\mathcal{S}}_{\mathrm{dD}}$. Nevertheless,

Every compact conserved tensor $j$ annihilates this smooth pure gauge:

Consequently no such $j$ represents $c$ on ${\mathcal{S}}_{\mathrm{dD}}$ or ${\mathcal{S}}_{\mathrm{E}}$. Assertions (i) and (ii) hold because $c$ itself is smooth and compact.