Collective onset of matter-induced scalarization around a black hole with two thin shells

Abstract

We study the linear onset of matter-induced scalarization for a static black hole surrounded by two spatially separated thin matter shells. We assume that the conformal matter coupling is unity on the scalar-free background and that its logarithmic derivative vanishes there. The scalar perturbation then decouples from the metric and matter perturbations at first order, while the invariant surface traces of the shells give singular terms in the scalar equation. For a static spherical black-hole exterior with no bulk scalar effective-mass term between the shells, the onset reduces to the finite-rank condition

$$ 1-a_1-a_2+(1-\chi )a_1a_2=0, \qquad \chi =\frac{\mathcal{S}(R_2)}{\mathcal{S}(R_1)}, $$

where \(a_j\) is the attractive strength of shell j normalized by its one-shell threshold on the same background and \(\mathcal{S}\) is the static radial resistance. This relation shows that two individually subcritical shells can collectively destabilize the scalar-free black hole. We derive the critical scalar cloud in closed form and verify numerically, in the Schwarzschild probe problem, that a growing mode with a finite growth rate appears beyond the threshold. For an exact scalar-free background consisting of three Schwarzschild regions joined by two Israel shells, we separately derive the static onset condition and the corresponding critical cloud. The exact Israel trace also exhibits a sign transition whose light-shell limit occurs at the Schwarzschild photon sphere.