Pseudospectrum and black hole total transmission mode (in)stability

TL;DR

The paper investigates the spectral stability of total transmission modes (TTMs) in $d$-dimensional Tangherlini black holes using pseudospectrum analysis within a generalized eigenvalue framework. By reformulating the TTM problem in Eddington–Finkelstein coordinates and discretizing with Chebyshev–Lobatto methods, the authors reveal that TTMs are generally spectrally unstable, with higher overtones exhibiting stronger sensitivity to perturbations, similarly to quasinormal modes. A notable exception is a purely imaginary TTM for gravitational perturbations ($s=2$) that shows near-concentric pseudospectral contours and a small, grid-robust condition number, signaling enhanced stability. The onset of genuinely complex TTM families occurs at $d\ge 8$, extending prior claims, and the results have implications for controlled black-hole scattering and virtual absorption experiments, while motivating further work on rotating backgrounds and time-domain dynamics.

Abstract

Total transmission modes (TTMs) are modes with complex frequencies that propagate across a black hole spacetime without reflection. Recently, it is found that suitably tailored time-dependent scattering can excite these complex modes and suppress the reflected signal for the entire duration of the process, a phenomenon referred to as virtual absorption. Motivated by this, we present the study of the spectrum stability of TTMs using pseudospectrum and condition numbers. We focus on perturbations of $d$-dimensional Tangherlini black holes and recast the TTM problem as a generalized eigenvalue problem by utilizing the Eddington-Finkelstein coordinates. The results show that TTMs are generically spectrally unstable, with sensitivity increasing for higher overtones, in close analogy with quasinormal modes. A notable exception is a purely imaginary TTM whose pseudospectrum's contours are nearly concentric and whose condition number is orders of magnitude smaller than that of the overtones, indicating enhanced spectral stability. Additionally, we confirm that purely imaginary TTMs occur only for spin $s=2$, whereas genuinely complex TTM families appear only in sufficiently high dimensions, $d \geqslant 8$, extending earlier claims that placed the onset at $d \geqslant 10$.

Figures (3)

• Figure 1: Pseudospectra of TTMs with $s=2$ (Fig. \ref{['fig:pse s=2']}) compared with those of TTMs with $s=0$ (Fig. \ref{['fig:pse s=0']}) and QNMs (Fig. \ref{['fig:pse qnm']}). All results are computed with grid resolution $N=200$. The top row displays the overall landscape, while the bottom row zooms in on the $n=0, 1,$ and $2$ modes for the $s=2$ case. Red $\star$ symbols indicate the exact TTMs and QNMs, and the green contours mark the transition to an open structure. Due to large variation in the gradient of $-\ln\epsilon$, a specific contour spacing of $1/8$ is used within the following $-\ln\epsilon$ ranges: $[-3.5, -2.5]$ (Fig. \ref{['fig:pse s=2']}), $[-4.065, -2.44]$ (Fig. \ref{['fig:pse s=0']}), and $[-1.5, 1]$ (Fig. \ref{['fig:pse qnm']}).
• Figure 2: The TTMs $\omega_n$ (top panels) and their condition numbers $\kappa$ (bottom panels) as functions of $N$ for $s=0,2$ and $d=14,20$ with $\ell=2$ obtained within the resolution $N=300$. Dashed lines show linear fits to the condition numbers which align with the numerical values very accurately.
• Figure 3: Pseudospectra of TTMs adopting the third definition (\ref{['def3']}) of a generalized eigenvalue problem. All the parameters are the same as the Fig. \ref{['fig:pse']}. The top row displays the overall landscape, while the bottom row zooms in on the $n=0, 1,$ and $2$ modes for the $s=2$ case. Red $\star$ symbols indicate the exact TTMs, and the green contours mark the transition to an open structure. Due to large variation in the gradient of $-\ln\epsilon$, a specific contour spacing of $1/8$ is used within the following $-\ln\epsilon$ ranges: $[-3, -2]$ (Fig. \ref{['fig:def3 pse s=2']}) and $[-4.5, -2]$ (Fig. \ref{['fig:def3 pse s=0']}).