Correspondence between quasinormal modes and grey-body factors of Schwarzschild--Tangherlini black holes

TL;DR

This work extends the QNM–grey-body factor correspondence, long established in four dimensions, to Schwarzschild--Tangherlini black holes in higher dimensions by classifying gravitational perturbations into scalar, vector, and tensor types. Using continued fraction methods and integration-through-midpoints, it computes accurate QNMs across D=6–8 and employs a sixth-order WKB-based relation to derive grey-body factors from QNM data, comparing with numerical grey-body factors. The results show that the correspondence holds for vector and tensor perturbations in all studied cases, while it breaks down for the l=2 scalar mode in D≥7 due to multiple potential barriers. The findings highlight the crucial role of the potential’s barrier structure in the validity of the correspondence and suggest practical pathways to infer grey-body factors from QNM spectra, especially in the eikonal regime. Overall, the tensor-type perturbations exhibit the highest accuracy, and the work extends prior D=5 results to a broader dimensional landscape of higher-dimensional gravity theories.

Abstract

We investigate the correspondence between the quasinormal modes and grey-body factors of Schwarzschild--Tangherlini black holes. The gravitational perturbations in higher-dimensional black holes can be classified into scalar, vector, and tensor types. Considering the dimension-dependent forms of their effective potentials, the correspondence was examined for each dimension and perturbation mode. The accurate quasinormal modes were computed by suitably adopting the continued fraction and integration-through-midpoints methods, depending on the structure of the singularity. The grey-body factor can be obtained through its correspondence with the quasinormal mode, and its accuracy was analyzed by calculating its difference from the numerically computed grey-body factor. The correspondence failed for $l=2$ scalar gravitational perturbations in $D\ge7$ because the form of the potential is markedly different from that in four dimensions. The vector and tensor perturbation types exhibited good correspondence accuracies in all cases. The breakdown of the correspondence was rigorously demonstrated to stem from multiple potential barriers, and its applicability to each mode in higher dimensions was assessed.

Figures (21)

• Figure 1: Left: Grey-body factors obtained by the correspondence with the quasinormal modes for scalar gravitational perturbations for $l=2$ (blue), $l=3$ (yellow), and $l=4$ (green) in $D=6$. Right: Differences between the grey-body factors calculated by using the correspondence and the numerical method.
• Figure 2: Left: Grey-body factors obtained by using the correspondence with the quasinormal modes for scalar gravitational perturbations for $l=2$ (blue), $l=3$ (yellow), and $l=4$ (green) in $D=7$. Right: Differences between the grey-body factors calculated by using the correspondence and the numerical method.
• Figure 3: Left: Grey-body factors obtained by using the correspondence with the quasinormal modes for scalar gravitational perturbations for $l=2$ (blue), $l=3$ (yellow), and $l=4$ (green) in $D=8$. Right: Differences between the grey-body factors calculated by using the correspondence and the numerical method.
• Figure 4: Effective potentials for scalar-type of gravitational perturbations of Schwarzschild--Tangherlini black holes ($r_H=1$) for $l=2$ (left), $l=3$ (center), and $l=4$ (right).
• Figure 5: Left: Grey-body factors calculated by using the correspondence with the quasinormal modes for $l=2$ scalar gravitational perturbations in $D=5$ (blue), $D=6$ (yellow), $D=7$ (green), and $D=8$ (red). Right: Differences between the values obtained using the correspondence and the accurate values.