Scalarizations of qOS-extremal black hole and Aretakis instability

We study scalarization of quantum Oppenheimer–Snyder (qOS)-extremal black hole in the Einstein–Gauss–Bonnet-scalar theory. This black hole is described by mass ( M=4√α /3√3 M = 4 α / 3 3 ) with α α quantum parameter. Investigating the onset of scalarization, we find the appearance of the single branch of scalarized qOS-extremal black holes. To obtain the tachyonic scalar cloud being a seed to generate the single branch of scalarized qOS-extremal black holes, we consider the near-horizon geometry of the Bertotti–Bobinson (BR) spacetime. In this case, it is shown that the appearance of a large scalar cloud at the horizon ( ρ =0 ρ = 0 ) is a new feature to represent onset scalarization of extremal black holes for tachyon with negative mass. However, it is not related to the Aretakis instability of a propagating scalar with standard mass around the BR spacetime, showing polynomial instability of the ingoing time v . The Aretakis instability is connected to the scalar cloud with standard mass, indicating the blow-up at the horizon.