Charged scalar fields on Reissner--Nordstr\"om spacetimes II: late-time tails and instabilities

TL;DR

This paper analyzes the late-time decay and stability of charged scalar fields on Reissner--Nordstr"om black holes using energy estimates, establishing decay rates and instabilities without small charge assumptions.

Contribution

It provides the first pointwise decay estimates for charged scalar fields on black holes without assuming small charge, and demonstrates asymptotic instabilities along null infinity and horizons.

Findings

Established inverse-power decay tails for charged scalar fields.

Proved existence of asymptotic instabilities at null infinity and horizons.

Derived decay estimates without small charge assumptions.

Abstract

This is the second part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordstr\"om spacetimes via energy estimates. In this paper, we use purely physical-space based methods to establish the precise late-time behaviour of solutions to the charged scalar field equation in the form of oscillating and decaying late-time tails that satisfy inverse-power laws, assuming global integrated energy decay estimates, which are proved in the companion paper [Gaj26]. This paper provides the first pointwise decay estimates for charged scalar fields on black hole backgrounds without an assumption of smallness of the scalar field charge. We also prove the existence of asymptotic instabilities for the radiation field along future null infinity and, in the extremal case, also along the future event horizon. Both the energy methods and the precise late-time asymptotics derived in this paper are expected to play an important role in future nonlinear studies of black hole dynamics in the context of the spherically symmetric (Einstein--)Maxwell--charged scalar field equations, as well as in the context of extremal Kerr spacetimes.