Quasinormal modes of three $(2+1)$-dimensional black holes in string theory, conformal gravity, and Hu-Sawicki $F(R)$ theory via the Heun function

TL;DR

This paper analyzes quasinormal modes of various three-dimensional black holes in different theories using Heun functions, revealing new frequency families and confirming stability of fermionic perturbations.

Contribution

It introduces a novel method employing Heun functions to compute quasinormal modes in three-dimensional black holes, extending previous results and discovering new frequency families.

Findings

Reproduced known quasinormal modes for BTZ and conformal gravity black holes.

Discovered new families of quasinormal frequencies, including purely imaginary modes.

Confirmed stability of fermionic perturbations with negative imaginary frequencies.

Abstract

We study the propagation of massless fermionic fields, implementing a family of special functions: Heun functions, in solving the wave equation in three three-dimensional backgrounds, including the BTZ black hole in string theory and Lifshitz black hole solutions in conformal gravity and Hu-Sawicki $F(R)$ theory. The main properties of the selected black hole solutions is that their line elements are Weyl related to that of a homogeneous spacetime, whose spatial part possesses Lie symmetry, described by Lobachevsky-type geometry with arbitrary negative Gaussian curvature. Using the Weyl symmetry of massless Dirac action, we consider the perturbation equations of fermionic fields in relation to those of the homogeneous background, which having definite singularities, are transformed into Heun equation. We point out the existence of quasinormal modes labeled by the accessory parameter of the Heun function. The distribution of the quasinormal modes has been clarified to satisfy the boundary conditions that require ingoing and decaying waves at the event horizon and conformal infinity, respectively. It turned out that the procedure based on the Heun function, beside reproducing the previously known results obtained via hypergemetric function for the BTZ and Lifshitz black hole solution in conformal gravity, brings up new families of quasinormal frequencies, which can also contain purely imaginary modes. Also, the analysis of the quasinormal modes shows that with the negative imaginary part of complex frequencies $\omega=\omega_{Re}+i\omega_{Im}$, the fermionic perturbations are stable in this background.