Polynomial time decay for solutions of the Klein--Gordon equation on a subextremal Reissner--Nordstr\"{o}m black hole

Abstract

We consider the massive scalar field equation $\Box_{g_{RN}} \phi = m^2 \phi$ on any subextremal Reissner--Nordstr\"{o}m exterior metric $g_{RN}$. We prove that solutions with localized initial data decay pointwise-in-time at the polynomial rate $t^{-\frac{5}{6}+\delta}$ in any spatially compact region (including the event horizon), for some small $ \delta\leq \frac{1}{23} $. Moreover, assuming the validity of the Exponent Pair Conjecture on exponential sums in Number Theory, our result implies that decay upper bounds hold at the rate $t^{-\frac{5}{6}+\epsilon}$, for any arbitrarily small $\epsilon>0$. In our previous work, we proved that each fixed angular mode decays at the exact rate $t^{-\frac{5}{6}}$, thus the upper bound $t^{-\frac{5}{6}+\epsilon}$ is sharp, up to a $t^{\epsilon}$ loss. Without the restriction to a fixed angular mode, the solution turns out to have an unbounded Fourier transform due to discrete frequencies associated to quasimodes, and caused by the occurrence of stable timelike trapping. Our analysis nonetheless shows that inverse-polynomial asymptotics in $t$ still hold after summing over all angular modes.