Hod–Piran asymptotic behaviour conjecture for weakly charged scalar fields

Let $(\phi,F)$ be a solution of the Maxwell-Charged-Scalar field system with no particular symmetry assumption on a Reissner--Nordström space-time. Define the Maxwell charge $Q$ by

$$F_{uv}=\frac{2Q\Omega^2}{r^2}.$$

Suppose that the data for $|\phi|$ is sufficiently decaying towards spatial infinity, and denote the asymptotic charge by

$$e=\lim_{v\rightarrow +\infty}Q_{|\mathcal{H}^+}(v).$$

For $\epsilon>0$, there should exist $\delta>0$ such that, if $q_0|e|<\delta$, then on the event horizon, null infinity, and a far-away curve $\gamma$ with $t\sim u\sim v\sim r$, there are constants $\Gamma_0$, $\Gamma'_0$, and $\Gamma”_0$, and a function satisfying $0<\eta(q_0e)<\epsilon$, such that

$$\phi_{|\mathcal{H}^+}(v)\sim \Gamma_0 e^{iq_0e\frac{r^*}{r_+}}v^{-2+\eta(q_0e)},$$ $$\psi_{|\mathcal{I}^+}(u)\sim \Gamma'_0\left(\frac{u}{v}\right)^{iq_0e}u^{-1+\eta(q_0e)},$$ $$\phi_{|\gamma}(t)\sim \Gamma”_0t^{iq_0e}v^{-2+\eta(q_0e)},$$

as $v\rightarrow+\infty$. Consequently, the energy on the specified $V$ foliation should satisfy

$$E(u)\sim E_0u^{-3+2\eta(q_0e)}$$

for a constant $E_0$. This conjecture predicts that weakly charged scalar fields decay more slowly than uncharged perturbations, with a rate depending on the asymptotic Maxwell charge; the asserted energy asymptotic is an implication rather than an explicitly stated part of the cited work.