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 $ deltaleq 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

Future Roadmap: Challenges and Opportunities


  • The first challenge is to further investigate the decay of solutions to the massive scalar field equation on the subextremal Reissner-Nordström exterior metric. Currently, the decay rate of $t^{-frac{5}{6}+delta}$ is known for localized initial data in spatially compact regions, including the event horizon.
  • A potential challenge would be to find a tighter upper bound for the decay rate. The current upper bound is $t^{-frac{5}{6}+epsilon}$, where $epsilon$ can be arbitrarily small. However, it is unclear whether a sharper upper bound could be achieved.
  • The validity of the Exponent Pair Conjecture in Number Theory is assumed to imply decay upper bounds at a faster rate. Further research is needed to explore this assumption and determine its validity.
  • An important challenge is to understand the behavior of solutions when not restricted to a fixed angular mode. In previous work, it was found that the solution has an unbounded Fourier transform due to discrete frequencies associated with quasimodes. Stable timelike trapping is the cause of these frequencies. Investigating these frequencies and their impact on the solution is an area that requires further study.


  • The potential for discovering sharper decay rates for solutions to the massive scalar field equation opens up opportunities for improving our understanding of decay properties in this context.
  • If the Exponent Pair Conjecture in Number Theory is valid and can be applied to this problem, it would offer a powerful tool for deriving more precise decay upper bounds.
  • The unbounded Fourier transform and the presence of discrete frequencies associated with quasimodes present opportunities for exploring new mathematical techniques to analyze and understand the behavior of the solution. This could potentially lead to new insights into stable timelike trapping and its effects on the system.
  • Summing over all angular modes allows the observation of inverse-polynomial asymptotics in $t$. Further investigation into the summation process and its implications could uncover valuable information about the overall behavior of solutions.

Overall, the future roadmap for readers and researchers involves addressing several challenges, such as improving decay rates, validating conjectures, and understanding the impact of discrete frequencies on the solution. These challenges present exciting opportunities for advancing knowledge in this field and uncovering new mathematical techniques.

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