Robustness of the $S$-deformation method for black hole stability analysis

Abstract

The $S$-deformation method is a useful way to show the linear mode stability of a black hole when the perturbed field equation takes the form of the Schrödinger equation. While previous works where many explicit examples are studied suggest that this method works well, general discussion is not given yet explicitly. In this paper, we show the existence of a regular $S$-deformation when a black hole spacetime is stable under some reasonable assumptions. This $S$-deformation is constructed from a solution of a differential equation. We also show that the boundary condition for the differential equation which corresponds to a regular $S$-deformation has a one-parameter degree of freedom with a finite range. This is the reason why any fine-tune technique is not needed to find $S$-deformation numerically.

arXiv comment14 pages, 1 figure, v2: minor revisions, v3: minor revisions

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• Spontaneous scalarization of black holes in the Horndeski theory
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• Shape dependence of spontaneous scalarization
• Mixed scalarization of charged black holes: from spontaneous to non-linear scalarization
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