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Exceptional line and pseudospectrum in black hole spectroscopy

Li-Ming Cao, Ming-Fei Ji, Liang-Bi Wu, Yu-Sen Zhou

TL;DR

This work reveals a continuous exceptional line (EL) in the Gaussian-bump-modified Regge-Wheeler potential for black-hole perturbations, characterized by loops encircling the EL that exhibit vorticity $\nu=\pm\tfrac{1}{2}$ and Berry phase $\gamma=\pi$. Through matrix perturbation theory, it proves that the $\epsilon$-pseudospectrum contour near an EP scales as $\epsilon^{1/q}$ where $q$ is the largest Jordan-block order, yielding $\epsilon^{1/2}$ scaling at a second-order EP and linear scaling away from EPs. Numerical results corroborate these predictions, demonstrating enhanced spectral instability near EPs in non-Hermitian BH perturbations. The findings provide a universal framework for pseudospectrum analysis in gravitational settings and have potential implications for BH spectroscopy and gravitational-wave modeling.

Abstract

We investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters, $(\varepsilon,d,σ_0)$, a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. We find that the vorticity $ν=\pm1/2$ and the Berry phase $γ=π$ for loops encircling the EL, while $ν=0$ and $γ=0$ for those do not encircle the EL. Through matrix perturbation theory, we prove that the $ε$-pseudospectrum contour size scales as $ε^{1/q}$ at an EP, where $q$ is the order of the largest Jordan block of the Hamiltonian-like operator, contrasting with the linear $ε$ scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.

Exceptional line and pseudospectrum in black hole spectroscopy

TL;DR

This work reveals a continuous exceptional line (EL) in the Gaussian-bump-modified Regge-Wheeler potential for black-hole perturbations, characterized by loops encircling the EL that exhibit vorticity and Berry phase . Through matrix perturbation theory, it proves that the -pseudospectrum contour near an EP scales as where is the largest Jordan-block order, yielding scaling at a second-order EP and linear scaling away from EPs. Numerical results corroborate these predictions, demonstrating enhanced spectral instability near EPs in non-Hermitian BH perturbations. The findings provide a universal framework for pseudospectrum analysis in gravitational settings and have potential implications for BH spectroscopy and gravitational-wave modeling.

Abstract

We investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters, , a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. We find that the vorticity and the Berry phase for loops encircling the EL, while and for those do not encircle the EL. Through matrix perturbation theory, we prove that the -pseudospectrum contour size scales as at an EP, where is the order of the largest Jordan block of the Hamiltonian-like operator, contrasting with the linear scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.

Paper Structure

This paper contains 8 sections, 1 theorem, 45 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Topological structures of QNMs associated with exceptional line
  3. Pseudospectrum at exceptional points
  4. Conclusions and discussion
  5. The hyperboloidal framework
  6. Proof of the Theorem \ref{['theorem1']}
  7. A toy model
  8. Horizontal and vertical sample points of pseudospectra for EP and non-EP configurations

Key Result

Theorem 1

For a matrix perturbation problem $A(\varkappa)=A^{(0)}+\varkappa A^{(1)}$ with $\lVert A^{(1)}\rVert<\infty$, a degenerate eigenvalue $\lambda_i^{(0)}$ of $A^{(0)}$ bifurcates into several groups of eigenvalue $\{\lambda_1,\lambda_2,\cdots,\lambda_{p_1}\},\{\lambda_{p_1+1},\lambda_{p_1+2},\cdots,\l

Figures (8)

  • Figure 1: The blue line is the EL in the $3$D parameter space $(\varepsilon,d,2\sigma_0^2)$. The symbol $+$ marks the EP when $2\sigma_0^2=1$. The three closed curves $C_1(s)$ (yellow), $C_2(s)$ (green) and $C_3(s)$ (red). The $\bigtriangleup$ is the point where $C_1(0)$ and $C_2(0)$ coincide, while the $\Box$ represents the point $C_3(0)$.
  • Figure 2: Trajectories of $\omega_{+}(s)$ and $\omega_{-}(s)$ along the three parameter-space curves $C_1(s)$, $C_2(s)$, and $C_3(s)$. The positions of $\omega_{\pm}$ are explicitly marked by $\circ$ and $\vartriangle$, respectively, at discrete parameter values $s = 0, 0.125, 0.25, \cdots, 1$.
  • Figure 3: $\Delta\Phi(s)$ for $s$ varying from $0$ to $1$ for the three parameter-space curves $C_1(s), C_2(s)$, and $C_3(s)$.
  • Figure 4: The real and imaginary part of the integral \ref{['integral']} for the two near modes and the three curves, where the blue, orange, green and red lines denote $\mathrm{Re}(\phi_+),\,\mathrm{Im}(\phi_+),\,\mathrm{Re}(\phi_-)$ and $\mathrm{Im}(\phi_-)$ respectively.
  • Figure 5: Pseudospectrum of Regge-Wheeler potential with a bump modification near the fundamental mode at non-EP $\varepsilon=0.005, \,d=15, 2\sigma^2_0=1$ (left panel) and at the EP $\varepsilon=0.005083,\,d=15.6976, 2\sigma^2_0=1$ (right panel) by plotting $-\ln(\epsilon)$. We fit the smallest circle inscribed in the innermost contour, the red dashed lines and the blue lines are a group of circles centered in the QNM spectrum, and their radius are proportional to $\epsilon^{1/2}$ and $\epsilon$ respectively.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1