Quasinormal modes and hidden conformal symmetry in the Reissner–Nordström black hole

Abstract

It is shown that the scalar wave equation in the near-horizon limit respects a hidden SL(2, R) invariance in the Reissner–Nordström (RN) black hole spacetimes. We use the SL(2, R) symmetry to determine algebraically the purely imaginary quasinormal frequencies of the RN black hole. We confirm that these are exactly quasinormal modes of scalar perturbation around the near-horizon region of a near-extremal black hole.

Appendix: A computation of QNFs around the near-extremal RN black hole

In this appendix, we show that a scalar propagating around the near-horizon region of a near-extremal RN black hole has purely imaginary QNFs. Let us start with the Schrödinger equation (59) with the near-horizon and near-extremal RN potential (62). Introducing a new variable [20]

$$ x=\frac{1}{\cosh^2(\tilde{\kappa}\rho_*)},\quad x\in[0,1], $$

(A.1)

Eq. (59) can be written as

(A.2)

By changing to a new wave function y through

$$ R=x^{-\frac{i\omega}{2\tilde{\kappa}}}(1-x)^{-\frac{l}{2}}y, $$

(A.3)

Eq. (A.2) can be transformed into a standard hypergeometric equation [21]

$$ x(1-x)y''+\bigl[c-(a+b+1)x \bigr]y'-aby=0 $$

(A.4)

with

(A.5)

In order to obtain the quasinormal modes, we have to check whether or not solutions of Eq. (A.2) satisfy the boundary conditions: ingoing waves near the horizon (x=0) and zero (Dirichlet condition) at infinity of x=1. Taking into account e ^−iωt, such a solution takes the form of

$$ R=x^{-\frac{i\omega}{2\tilde{\kappa}}}(1-x)^{-\frac{l}{2}}{}_2F_1[a,b,c;x], $$

(A.6)

where ₂ F ₁[a,b,c;x] is a standard hypergeometric function. Imposing the boundary condition at infinity (x→1, ρ _∗→0, $\tilde{\rho}\rightarrow\nobreak \infty$), we recall a property of the hypergeometric function

$$ \lim_{x\rightarrow1}{}_2F_1[a,b,c;x]= \frac{\varGamma(c)\varGamma (c-a-b)}{\varGamma(c-a)\varGamma(c-b)}. $$

(A.7)

The Dirichlet boundary condition can be achieved when choosing

$$ c-a=-n\quad {\rm or}\quad c-b=-n $$

(A.8)

with n=0, 1, 2,… . Therefore, using (A.8) together with (A.5), the QNFs are given by

$$ \omega_n=-i\tilde{\kappa}(2n+l+1),\qquad \omega_n=-i\tilde{\kappa}(2n+l+2), $$

(A.9)

which are combined to give a single expression of purely imaginary QNFs

$$ \tilde{\omega}_n=-i\tilde{\kappa}(n+l+1). $$

(A.10)

This is the exactly same form as found in the QNFs (66).