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Quasinormal behavior of the D-dimensional Schwarzshild black hole and higher order WKB approach

R. A. Konoplya

TL;DR

The paper develops a sixth-order WKB framework for computing quasinormal modes of D-dimensional Schwarzschild black holes, extending prior 3rd-order results. It shows that the real parts of QNM frequencies scale approximately as $\omega_{Re} \sim D/r_{0}$ and provides large-$l$ asymptotics, offering an efficient and accurate method for higher-dimensional perturbations. Validation against 4D numerical results (Leaver) demonstrates improved accuracy of the 6th-order scheme, particularly for low overtones, and the work raises questions about the underlying origin of the dimensional scaling and its applicability to other fields or backgrounds.

Abstract

We study characteristic (quasinormal) modes of a $D$-dimensional Schwarzshild black hole. It proves out that the real parts of the complex quasinormal modes, representing the real oscillation frequencies, are proportional to the product of the number of dimensions and inverse horizon radius $\sim D r_{0}^{-1}$. The asymptotic formula for large multipole number $l$ and arbitrary $D$ is derived. In addition the WKB formula for computing QN modes, developed to the 3rd order beyond the eikonal approximation, is extended to the 6th order here. This gives us an accurate and economic way to compute quasinormal frequencies.

Quasinormal behavior of the D-dimensional Schwarzshild black hole and higher order WKB approach

TL;DR

The paper develops a sixth-order WKB framework for computing quasinormal modes of D-dimensional Schwarzschild black holes, extending prior 3rd-order results. It shows that the real parts of QNM frequencies scale approximately as and provides large- asymptotics, offering an efficient and accurate method for higher-dimensional perturbations. Validation against 4D numerical results (Leaver) demonstrates improved accuracy of the 6th-order scheme, particularly for low overtones, and the work raises questions about the underlying origin of the dimensional scaling and its applicability to other fields or backgrounds.

Abstract

We study characteristic (quasinormal) modes of a -dimensional Schwarzshild black hole. It proves out that the real parts of the complex quasinormal modes, representing the real oscillation frequencies, are proportional to the product of the number of dimensions and inverse horizon radius . The asymptotic formula for large multipole number and arbitrary is derived. In addition the WKB formula for computing QN modes, developed to the 3rd order beyond the eikonal approximation, is extended to the 6th order here. This gives us an accurate and economic way to compute quasinormal frequencies.

Paper Structure

This paper contains 7 sections, 98 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Sixth order WKB analysis
  3. Quasinormal modes of the D-dimensional Schwarzshild black hole
  4. Conclusion
  5. Conclusion
  6. Appendix 1: Correction terms for WKB formula
  7. Appendix 2: QNMs of a 4-dimensional Schwarzshild black hole

Figures (6)

  • Figure 1: $Re \omega$ for different dimensions $D$; $l=1$ (bottom), $2$, $3$, $4$ (top); $n=0$.
  • Figure 2: $Im \omega$ for different dimensions $D$; $l=1$ (bottom), $2$, $3$, $4$ (top); $n=0$.
  • Figure 3: $\omega_{Re}$ (bottom) and $\omega_{Im}$ (top) as a function of WKB order of the formula with which it was obtained for $l=1$, $n=2$, $D=4$ modes, and the corresponding numerical value. We see how the WKB values converge to an accurate numerical value as the WKB order increases.
  • Figure 4: $\omega_{Re}$ (top) and $\omega_{Im}$ (bottom) as a function of WKB order of the formula with which it was obtained for $l=0$, $n=0$, $D=12$ modes.
  • Figure 5: $\omega_{Re}$ (top) and $\omega_{Im}$ (bottom) as a function of WKB order of the formula with which it was obtained for $l=0$, $n=0$, $D=6$ modes.
  • ...and 1 more figures