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Critical masses and numerical computation of massive scalar quasinormal modes in Schwarzschild black holes

Matheus F. S. Alves, Bruno P. Pônquio, L. G. Medeiros

TL;DR

This work analyzes quasinormal modes of a massive scalar field in Schwarzschild spacetime using two complementary numerical methods: the Hill-determinant approach and Leaver’s continued-fraction method. It systematically compares their convergence, stability, and efficiency across a wide range of masses $m$ and angular momenta $l$, and discovers three critical mass scales $m_{ m lim}$, $m_{ m max}$, and $m_{zd}$. As $m$ increases, the spectrum experiences qualitative changes, with long-lived, zero-damping modes emerging at $m_{zd}$, even beyond the traditional disappearance point at $m_{ m max}$. The results establish the robustness and complementarity of the two methods and lay groundwork for extending the analysis to rotating or charged black holes and to quasi-bound states, with potential implications for gravitational-wave phenomenology of massive fields around black holes.

Abstract

We present a comprehensive analysis of the quasinormal modes (QNMs) of a massive scalar field in Schwarzschild spacetime using two complementary numerical techniques: the Hill-determinant method and Leaver continued-fraction method. Our study systematically compares the performance, convergence, and consistency of the two approaches across a wide range of field masses and angular momenta. We identify three critical mass thresholds, $m_{\rm lim}$, $m_{\rm max}$, and $m_{zd}$, which govern qualitative changes in the QNM spectrum. In particular, long-lived modes emerge at $m_{zd}$, where the imaginary part of the frequency vanishes and the mode becomes essentially non-decaying. This phenomenon is robust across multipoles and may have important implications for the phenomenology of massive fields around black holes. Our results provide a detailed numerical characterization of massive scalar QNMs and highlight the complementary strengths of the Hill-determinant and continued-fraction methods, paving the way for future studies of rotating or charged black holes and quasi-bound states.

Critical masses and numerical computation of massive scalar quasinormal modes in Schwarzschild black holes

TL;DR

This work analyzes quasinormal modes of a massive scalar field in Schwarzschild spacetime using two complementary numerical methods: the Hill-determinant approach and Leaver’s continued-fraction method. It systematically compares their convergence, stability, and efficiency across a wide range of masses and angular momenta , and discovers three critical mass scales , , and . As increases, the spectrum experiences qualitative changes, with long-lived, zero-damping modes emerging at , even beyond the traditional disappearance point at . The results establish the robustness and complementarity of the two methods and lay groundwork for extending the analysis to rotating or charged black holes and to quasi-bound states, with potential implications for gravitational-wave phenomenology of massive fields around black holes.

Abstract

We present a comprehensive analysis of the quasinormal modes (QNMs) of a massive scalar field in Schwarzschild spacetime using two complementary numerical techniques: the Hill-determinant method and Leaver continued-fraction method. Our study systematically compares the performance, convergence, and consistency of the two approaches across a wide range of field masses and angular momenta. We identify three critical mass thresholds, , , and , which govern qualitative changes in the QNM spectrum. In particular, long-lived modes emerge at , where the imaginary part of the frequency vanishes and the mode becomes essentially non-decaying. This phenomenon is robust across multipoles and may have important implications for the phenomenology of massive fields around black holes. Our results provide a detailed numerical characterization of massive scalar QNMs and highlight the complementary strengths of the Hill-determinant and continued-fraction methods, paving the way for future studies of rotating or charged black holes and quasi-bound states.

Paper Structure

This paper contains 11 sections, 44 equations, 5 figures, 3 tables.

Table of Contents

  1. INTRODUCTION
  2. MASSIVE SCALAR FIELD IN SCHWARZSCHILD SPACETIME
  3. THE HILL DETERMINANT METHOD
  4. The Leaver continued fraction method
  5. Effective Potential and Critical Mass
  6. Roots and behavior near horzion
  7. Critical masses
  8. RESULTS
  9. Massless limit
  10. Quasinormal modes
  11. Conclusions

Figures (5)

  • Figure 1: The effective potential $V$ as a function of $r$ for $l = 0$ (left panel) and $l=1$ (right panel), for different values of mass $m$.
  • Figure 2: Quasifrequency spectra for the fundamental mode with $l=1$. Left panel: massless case ($m=0$), showing symmetric QNMs with respect to $\text{Re}(\omega)$. Right panel: massive case ($m = 10^{-5}$), where the symmetry is broken.
  • Figure 3: Fundamental mode for $l=0$ QNMs of the scalar field as a function of the field mass, computed using the Leaver method (black curve) and the Hill determinant method (blue curve). Left panel: real part of the frequency. Right panel: imaginary part of the frequency.
  • Figure 4: Fundamental mode for $l=1$ QNMs of the scalar field as a function of the field mass, computed using the Leaver method (black curve) and the Hill determinant method (blue curve). Left panel: real part of the frequency. Right panel: imaginary part of the frequency.
  • Figure 5: Fundamental mode for $l=2$ QNMs of the scalar field as a function of the field mass, computed using the Leaver method (black curve) and the Hill determinant method (blue curve). Left panel: real part of the frequency. Right panel: imaginary part of the frequency.