Comment on "Superradiant stability of the Kerr black holes" (arXiv:1907.09118)

TL;DR

This paper refines the stability analysis of Kerr black holes with massive scalar fields by simplifying conditions, enlarging the stability region, and providing tighter bounds for rapidly spinning black holes, enhancing previous results.

Contribution

It offers a simplified and more comprehensive analytic stability criterion for Kerr black holes, improving upon prior conditions by analyzing polynomial coefficients.

Findings

Expanded the stability region for Kerr black holes

Provided tighter bounds for near-extremal black holes

Simplified the stability conditions by polynomial coefficient analysis

Abstract

We revisit the recent work of Huang on the superradiant stability of Kerr black holes coupled to massive scalar fields. While their analysis provides sufficient conditions for stability, it imposes an unnecessarily strong requirement by demanding that two roots of the relevant quartic equation be explicitly negative. By instead analyzing the polynomial's coefficients, we show that simpler constraints already exclude additional positive turning points, thereby slightly enlarging the region of guaranteed stability. We further present a near-extremal estimate that tightens the stability bound for rapidly spinning black holes. These refinements sharpen the analytic stability limits without introducing extra assumptions.