Scalar Perturbations of Black Holes in the <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="bold-italic">f</mi><mrow><mo mathvariant="bold">(</mo><mi mathvariant="bold-italic">R</mi><mo mathvariant="bold">)</mo></mrow><mo mathvariant="bold">=</mo><mi mathvariant="bold-italic">R</mi><mo mathvariant="bold">−</mo><mn mathvariant="bold">2</mn><mi mathvariant="bold-italic">α</mi><msqrt><mi mathvariant="bold-italic">R</mi></msqrt></mrow></semantics></math></inline-formula> Model

Abstract

In this paper, we study the perturbations of the charged static spherically symmetric black holes in the f(R)=R−2αR model by a scalar field. We analyze the quasinormal modes spectrum, superradiant modes, and superradiant instability of the black holes. The frequency of the quasinormal modes is calculated in the frequency domain by the third-order WKB method, and in the time domain by the finite difference method. The results by the two methods are consistent and show that the black hole stabilizes quicker for larger α satisfying the horizon condition. We then analyze the superradiant modes when the massive charged scalar field is scattered by the black hole. The frequency of the superradiant wave satisfies ω∈(μ2,ωc), where μ is the mass of the scalar field, and ωc is the critical frequency of the superradiance. The amplification factor is also calculated by numerical method. Furthermore, the superradiant instability of the black hole is studied analytically, and the results show that there is no superradiant instability for such a system.

Keywords

• f(R) theory
• black hole perturbation
• quasinormal modes
• superradiant scattering
• superradiant instability