The asymptotics of massive fields on stationary spherically symmetric black holes for all angular momenta

TL;DR

This paper establishes the first rigorous decay asymptotics for massive scalar fields on stationary spherically symmetric black holes, revealing precise tail behavior crucial for understanding black hole interiors and cosmic censorship.

Contribution

It provides the first rigorous proof of decay tails for massive scalar fields on black holes, including Coulomb-like long-range potentials, for all angular momenta.

Findings

Decay tail of order t^{-5/6} with explicit oscillating profile

Applicable to Schwarzschild and Reissner–Nordström black holes in sub-extremal range

Advances understanding of scalar field behavior relevant to black hole interior studies

Abstract

We study the massive scalar field equation $\Box_g \phi = m^2 \phi$ on a stationary and spherically symmetric black hole $g$ (including in particular the Schwarzschild and Reissner--Nordstr\"om black holes in the full sub-extremal range) for solutions $\phi$ projected on a fixed spherical harmonic. Our problem involves the scattering of an attractive long-range potential (Coulomb-like) and thus cannot be treated perturbatively. We prove precise (point-wise) asymptotic tails of the form $t^{-5/6} f(t)+ O(t^{-1+\delta})$, where $f(t)$ is an explicit oscillating profile. Our asymptotics appear to be the first rigorous decay result for a massive scalar field on a black hole. Establishing these asymptotics is also an important step in retrieving the assumptions used in work of the third author regarding the interior of dynamical black holes and Strong Cosmic Censorship.