Dynamical evolution of a scalar field coupling to Einstein's tensor in the Reissner-Nordstr\"{o}m black hole spacetime

TL;DR

This paper investigates how a scalar field coupled to Einstein's tensor evolves dynamically around Reissner-Nordström black holes, revealing effects of the coupling on quasinormal modes and stability, including conditions for instability.

Contribution

It provides a detailed analysis of the scalar field's wave dynamics and stability criteria in the black hole background, highlighting the impact of the coupling constant on quasinormal frequencies and instabilities.

Findings

Increasing coupling constant decreases real parts of frequencies.

Strong coupling can induce instability for non-zero angular momentum modes.

Zero angular momentum mode always decays regardless of coupling.

Abstract

We study the dynamical evolution of a scalar field coupling to Einstein's tensor in the background of Reissner-Nordstr\"{o}m black hole. Our results show that the the coupling constant $\eta$ imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant $\eta$ the real parts of the fundamental quasinormal frequencies decrease and the absolute values of imaginary parts increase for fixed charge $q$ and multipole number $l$. In the strong coupling, we find that for $l\neq0$ the instability occurs when $\eta$ is larger than a certain threshold value $\eta_c$ which deceases with the multipole number $l$ and charge $q$. However, for the lowest $l=0$, we find that there does not exist such a threshold value and the scalar field always decays for arbitrary coupling constant.