Magnetic-induced Spontaneous Scalarization in Dynamcial Chern-Simons Gravity

In the framework of the dynamical Chern-Simons gravity, we study the scalar field perturbations of the Reissner-Nordstr\"{o}m-Melvin spacetime, which describes a charged black hole permeated by a uniform magnetic field. In the presence of the magnetic field, the scalar field acquires an effective mass whose square takes negative value in the half domain of the angular direction. This inevitably introduces the tachyonic instability and associated spontaneous scalarization as long as the coupling constant between the scalar field and the Chern-Simons invariant exceeds a threshold value. We study the object pictures of the time evolutions of the scalar field perturbations at the linear level, and find that the presence of the magnetic field will dramatically change the waveforms and associated ringdown modes. Nonlinear evolutions for the unstable perturbations are also performed in the decoupling limit, which demonstrate the scalar cloud as the final fate. Influences of the coupling constant and the black hole charge on the wave dynamics are also studied.