Instabilities of Simpson-Visser black bounces sourced by a phantom scalar and nonlinear electrodynamics

Abstract

We study the linear stability of Simpson-Visser ``black-bounce'' space-times (regular black holes and wormholes) under spherically symmetric (monopole) perturbations, taking as a source a phantom scalar field with a self-interaction potential and a magnetic field within nonlinear electrodynamics (NED). The metric contains two parameters: the mass $M$ and the bounce parameter $a >0$, such that $a \leq 2M$ corresponds to black holes and $a > 2M$ to wormholes. The stability problem reduces to a boundary-value problem for a Schrödinger-like equation with a certain effective potential. All space-times under study are found to be unstable, and the corresponding exponential growth rates are calculated. For black holes the results are obtained without complications, while the wormhole stability problem faces difficulties that seem to be previously unnoticed. Thus, the Darboux transformation, used to remove the wall shaped $\approx 2/z^2$ in the effective potentials at wormhole throats ($z$ is the tortoise coordinate), creates nonphysical poles outside the throat for $a/(2M) \lesssim 4.98$. We circumvent this difficulty by using a gauge-invariant approach in the spirit of Cremona, Pizzocchero and Sarbach, leading to a master equation with a simple and regular potential. Its further analysis reveals unstable modes: the mode with a variable throat radius is always present; for $a/(2M) \gtrsim 4.98$ there are two real growing throat modes, which merge into a complex-conjugate pair (oscillatory instability) for smaller $a$. A further analysis reveals three exceptional points that separate ranges of $a/(2M)$ with different perturbation regimes. Furthermore, the leading unstable black-hole and wormhole modes approach the same limiting growth rate $Mκ\simeq 0.10121$ as $a/(2M)$ approaches unity from either side.