[Paper Review] Fermion Quasi-normal modes of the Kerr Black-Hole

Warren Carlson, Alan S. Cornell+1 more|arXiv (Cornell University)|Jan 16, 2012

Particle Accelerators and Free-Electron Lasers3 citations

TL;DR

This paper computes fermion quasi-normal modes (QNMs) of the Kerr black hole using third- and sixth-order WKB(J) approximations and the Asymptotic Iteration Method (AIM), comparing results with numerical solutions. It finds that both semi-analytic methods show good agreement with numerical data for low overtone and angular quantum numbers, with the sixth-order WKB(J) method being significantly faster and more accurate than AIM at higher orders.

ABSTRACT

In this paper we study the fermion quasi-normal modes of a 4-dimensional rotating black-hole using the WKB(J) (to third and sixth order) and the AIM semi-analytic methods in the massless Dirac fermion sector. These semi-analytic approximations are computed in a pedagogical manner with comparisons made to the numerical values of the quasi-normal mode frequencies presented in the literature. It was found that The WKB(J) method and AIM show good agreement with direct numerical solutions for low values of the overtone number $n$ and angular quantum number l.

Motivation & Objective

• To compute quasi-normal mode (QNM) frequencies for massless Dirac fermions in a Kerr black hole spacetime.
• To evaluate the accuracy and efficiency of semi-analytic methods—WKB(J) to third and sixth order, and the Asymptotic Iteration Method (AIM)—against direct numerical solutions.
• To extend prior work on scalar and gravitational perturbations to the fermionic sector in Kerr spacetime.
• To provide a pedagogical derivation of the Dirac equation in the Newman-Penrose formalism for Kerr black holes.
• To compare the performance of WKB(J) and AIM in terms of convergence, accuracy, and computational speed for low-lying QNM modes.

Proposed method

• Uses the Newman-Penrose (NP) formalism to derive the master equation for massless spin-1/2 fields in Kerr spacetime.
• Applies the WKB(J) method up to sixth order to approximate the QNM frequencies, with iterative corrections to the potential barrier.
• Employs the Asymptotic Iteration Method (AIM) to solve the second-order differential equation governing the perturbations.
• Compares results from WKB(J) and AIM with numerically computed QNM frequencies from existing literature (Jing et al., 2005).
• Sets c = ℏ = G = 1 and M = 1 for all computations, standardizing units for consistency.
• Evaluates accuracy via relative percentage differences between methods and numerical benchmarks.

Experimental results

Research questions

• RQ1 How accurately do the third- and sixth-order WKB(J) approximations reproduce the QNM frequencies of massless Dirac fermions in Kerr black holes?
• RQ2 How does the Asymptotic Iteration Method (AIM) compare in accuracy and computational cost to the WKB(J) method for fermionic QNMs?
• RQ3 What is the dependence of QNM frequency accuracy on the black hole spin parameter a, overtone number n, and angular quantum number l?
• RQ4 How do the semi-analytic methods perform in the absence of numerical benchmarks, particularly for l = 2 and l = 3 modes?
• RQ5 Which method—WKB(J) or AIM—offers the best trade-off between accuracy and computational efficiency for low-lying QNM modes?

Key findings

• The sixth-order WKB(J) method achieves sub-0.01% relative error compared to numerical solutions for l = 0, n = 0, a = 0.80, with (0.02%, <0.01%) accuracy at l = 2, n = 0.
• For a = 0, the WKB(J) method shows improved accuracy with increasing l and n, reaching (0.02%, <0.01%) at l = 2, n = 0.
• The AIM method exhibits degraded accuracy with increasing l and n at a = 0, reaching (1.46%, 4.32%) at l = 2, n = 2, but performs best at low n and high l.
• At a = 0.80, the AIM method shows relative errors of (-0.62%, -2.13%) for l = 0, n = 0, improving to (-0.39%, 1.44%) for l = 2, n = 0.
• The sixth-order WKB(J) method is significantly faster than AIM, with the latter requiring over 15 iterations and showing slower convergence.
• For l = 2 and l = 3, AIM results are most accurate when compared to sixth-order WKB(J), particularly at low n and high l, such as (a,l,n) = (0,3,0), (0.60,3,0), and (0.80,3,0).