Dynamical spontaneous scalarization in Einstein-Maxwell-scalar theory

We study the linear instability and nonlinear dynamical evolution of the Reissner-Nordstr?m(RN) black hole in the Einstein-Maxwell-scalar theory in asymptotic flat spacetime. We focus on the coupling function f(?) = e-b?~2, which facilitates both scalar-free RN and scalarized black hole solutions. We first present the evolution of system parameters during dynamic scalarization. For parameter regions in which spontaneous scalarization occurs, we observe that the evolution of the scalar field at the horizon is dominated by the fundamental unstable mode from linear analysis at early times. At late times, the nonlinear evolution can be considered to be the perturbation of scalarized black holes.