A linear stability analysis shows that all Simpson-Visser space-times, both regular black holes and wormholes, are unstable to spherically symmetric perturbations. The metric depends on two parameters: the mass M and the bounce parameter a > 0, with a ≤ 2M describing black holes and a > 2M wormholes. The problem reduces to a Schrödinger-like equation with an effective potential, and the authors compute the exponential growth rates of the unstable modes.

The matter source considered combines a phantom scalar field with a self-interaction potential and a magnetic field described by nonlinear electrodynamics. For black holes, the analysis is resolved without complications. Wormholes, however, present a technical difficulty that the authors say had gone unnoticed: the Darboux transformation, commonly used to remove the wall-shaped barrier of approximately 2/z² in the effective potential at the throat (where z is the tortoise coordinate), generates nonphysical poles outside the throat when a/(2M) ≲ 4.98.

To circumvent this obstacle, the team uses a gauge-invariant approach inspired by the work of Cremona, Pizzocchero and Sarbach, leading to a master equation with a simple and regular potential. Further analysis reveals unstable modes: the mode associated with a variable throat radius is always present; for a/(2M) ≳ 4.98 there are two real growing throat modes that merge into a complex-conjugate pair, giving rise to an oscillatory instability for smaller values of a. The study also identifies three exceptional points separating ranges of a/(2M) with different perturbation regimes.

One notable result is that the leading unstable black-hole and wormhole modes converge to the same limiting growth rate, Mκ ≈ 0.10121, as a/(2M) approaches unity from either side. This continuity suggests that the transition between the two geometries does not remove the instability, but makes it evolve smoothly. Whether the result holds outside the spherically symmetric sector, and what role other forms of matter play, remains to be seen.