Lemma: Trace reversal under the declared transport

For (eq:htransport),

Hence

Proof. Raising both indices of $-{R_{\lambda }}^{♭}$ with ${\eta }_{-}=-{\eta }_{+}$ gives $-{R_{\lambda }}^{\mu \nu }$. Its minus-signature trace is ${\eta }_{+,\alpha \beta }{R_{\lambda }}^{\alpha \beta }$. Substitution in (eq:trace-reverse-cov) yields $-{\mathcal{P}}_{+}{R}_{\lambda }$. Equation (eq:V) and ${\mathcal{P}}_{+}^2=\mathrm{id}$ give $V_{\lambda }$. Spatial homogeneity then gives the divergence formula.