Lemma: Agreement with the regular invariant equations

For the regular tensors (eq:regularstress),

For every smooth radial scalar $G(r)$, the mixed improvement $(g{\square }_g-{\nabla }^g{\nabla }^g{)}^{\mu }{}_{\nu }G$ equals $\mathcal{I}[G{]}^{\mu }{}_{\nu }$. Consequently,

Proof. In polar coordinates the only nonzero difference between the transverse Christoffel symbols of $g$ and $\bar{g}$ is

Its contraction with the diagonal mixed tensor in (eq:regularstress) vanishes in the mixed divergence. The two volume densities differ by the constant $\alpha$, so vector divergences agree as well.

For a radial scalar, both mixed Hessians have radial entry $G^{\prime \prime }$ and angular entry $G^{\prime }/r$; both wave operators give $G^{\prime \prime }+G^{\prime }/r$. This proves the improvement assertion. Finally,

on $r>0$, which gives the trace and improvement identities.