Threshold asymptotics and decay for massive Maxwell on subextremal Reissner--Nordstr\"om

TL;DR

This paper analyzes the late-time behavior and decay properties of massive Maxwell (Proca) fields on subextremal Reissner--Nordström black hole exteriors, revealing universal decay laws and detailed asymptotics through spectral and polarization analysis.

Contribution

It develops a threshold spectral theory for the Proca equation, proving meromorphic continuation, ruling out resonances, and deriving explicit asymptotics and decay laws for the field.

Findings

Universal late-time decay of $t^{-5/6}$ for the Proca field.

Explicit asymptotic expansions with quantitative remainders.

Decay estimates for the full field and its radiative component.

Abstract

We study the neutral massive Maxwell (Proca) equation on subextremal Reissner--Nordstr\"om exteriors. After spherical-harmonic decomposition, the odd sector is scalar, while the even sector remains a genuinely coupled $2\times2$ system. Our starting point is that this even system admits an exact asymptotic polarization splitting at spatial infinity. The three resulting channels carry effective angular momenta $\ell-1$, $\ell$, and $\ell+1$, and these are precisely the indices that govern the late-time thresholds. % For each fixed angular momentum we develop a threshold spectral theory for the cut-off resolvent. We prove meromorphic continuation across the massive branch cut, rule out upper-half-plane modes and threshold resonances, and obtain explicit small- and large-Coulomb expansions for the branch-cut jump. Inverting this jump yields polarization-resolved intermediate tails together with the universal very-late $t^{-5/6}$ branch-cut law. % At the full-field level, high-order angular regularity allows us to sum the modewise leading terms on compact radial sets and obtain a two-regime asymptotic expansion for the radiative branch-cut component of the Proca field, with explicit coefficient fields and quantitative remainders. We also analyze the quasibound resonance branches created by stable timelike trapping, prove residue and reconstruction bounds, and derive a fully self-contained dyadic packet estimate. As a result, the unsplit full Proca field obeys logarithmic compact-region decay, while the radiative branch-cut contribution retains explicit polynomial asymptotics and explicit leading coefficients.