Aretakis Hair for Extreme Kerr Black Holes with Axisymmetric Scalar Perturbations

TL;DR

This paper investigates scalar perturbations on extreme Kerr black holes, revealing a measurable relationship between horizon charges and finite-distance quantities, extending the understanding of black hole hair beyond previous modes.

Contribution

It introduces the analysis of mode coupling effects on horizon charges and demonstrates the potential to measure Aretakis charges at finite distances in extreme Kerr spacetimes.

Findings

Aretakis constant and Ori-Sela prefactor are linearly related for near-field initial data.

Mode coupling excites higher multipole modes and horizon charges.

Linear relationship breaks down for initial data farther from the horizon.

Abstract

We study the evolution of axially-symmetric scalar field perturbations on an extreme Kerr spacetime for initial data with multipole moments $\ell^{\prime}$ higher than the least radiative mode, and we measure modes $\ell$ -- and for the first time also horizon charges -- that are excited by mode coupling interactions. We then find the Ori-Sela prefactors, a certain quantity that can be evaluated at finite distances and the Aretakis constant along the event horizon of the extreme Kerr black hole for a sequence of initial data preparations that differ only by their distance from the event horizon. We find that for initial data in the near field there is a linear relationship of the Aretakis constant and the Ori-Sela prefactor. For initial data farther than these the linear relationship is not universal, and we propose that stronger numerical simulations would be needed to regain linearity. The linear relationship suggests that the Aretakis charge along the event horizon can be measured at a finite distance, thereby extending this type of violation of the no-hair theorems from the least radiative axisymmetric mode also to situations that involve mode coupling.