WKB method and quasinormal modes of string-theoretical d-dimensional black holes

TL;DR

This work analyzes quasinormal modes of $d$-dimensional, string-corrected black holes using analytic WKB methods in two regimes: the eikonal limit and the asymptotic highly damped limit. It derives explicit expressions for the eikonal QNM frequencies with first-order string corrections and shows how the spectrum encodes the horizon temperature $T_H$, while the asymptotic analysis employs the monodromy method to obtain a universal real part $\ln(3)$ plus dimension-dependent imaginary parts and string corrections. The results extend the QNM analysis to Callan–Myers–Perry black holes and offer insight into how higher-derivative string effects modify black hole ringdown signals. These analytical benchmarks can aid tests of string-inspired gravity in higher dimensions and inform gravitational-wave phenomenology in theories beyond Einstein gravity.

Abstract

After a brief introduction to quasinormal modes in dissipative systems, we review the WKB formalism in the context of the analytical calculation of quasinormal frequencies. We apply these results to the calculation of quasinormal frequencies associated with gravitational perturbations of d-dimensional spherically symmetric black holes with string corrections. We do this for two distinct limits: the eikonal limit and the asymptotic limit.

Figures (5)

• Figure 1: Pictorial representation of the closed contour needed to compute the integral over $s$ in (\ref{['15']}) as the blue dashed line. Furthermore, possible quasinormal frequencies are depicted by conjugated red dots. These conjugation pairs arise from conjugation of (\ref{['156']}).
• Figure 2: Graphical depiction of $Q$ as the blue line. Possible roots of $Q$ are denoted by $x_1$ and $x_2$. Moreover, the x-axis is subdivided in three regions, denoted by I, II and III.
• Figure 3: Numerical plot of the Stokes lines topology for different dimensions. The horizontal axis stands for $\Re(r/R_h)$ and the vertical axis stands for $\Im(r/R_h)$. We denoted the positions of the physical horizon $R_h$ and of the fictitious horizons by red dots.
• Figure 4: Schematic depiction of the big contour, as the blue dashed line. The Stokes lines are depicted as red curves. Naturally, not all Stokes lines are depicted. Furthermore, we marked by $D$ and $U$ the regions where the boundary condition (\ref{['74']}) may be imposed.
• Figure 5: Schematic depiction of the small and big contours as the orange and blue dashed lines respectively. The orange contour is to be interpreted as arbitrarily close to $R_H$. The Stokes lines are depicted by red curves. Naturally, not all Stokes lines are depicted.