On the origin of quasinormal modes in semi-open systems

Abstract

Astrophysical black holes are open systems which, when perturbed, radiate quasi-normal modes (QNMs) to infinity. By contrast, laboratory analogues are necessarily finite-sized, presenting a potential obstacle to exciting QNMs in experiments. We explore how the QNM spectrum of a toy-model black hole changes when enclosed by a partially reflecting wall with adjustable reflectivity. Our results reveal a continuous connection between the QNM spectra of open and finite-sized systems. Additionally, we demonstrate that QNMs in this setup are easily excited by incoherent background noise. This work opens new avenues for studying QNMs of black holes and compact objects in laboratory settings, where finite-size effects and noise are unavoidable.

Figures (3)

• Figure 1: a. Trajectories in the complex plane of the first 11 resonances of the PT potential with a partially-reflective barrier relative to the fundamental QNM $\omega_{n=0}^{\varepsilon=0}\equiv \tilde{\omega}_r +i \tilde{\omega}_i$. Here we take $\alpha = 0.4$ and the position of the wall is fixed at $x_b = 10$. The large yellow dots represent QNMs of the open system, whereas the blue lines are obtained varying $\varepsilon$. Coloured markers label the resonances at particular values of reflectivity. b. Trajectories of the resonances as reflectivity increases, for $\alpha = 0.3$ (Green, dashed), $=0.4$ (Blue, solid), $=0.5$ (Orange, dotted) and $x_b = 10$. The yellow dots are $\omega_{n=0,1}^{\varepsilon=0}$ and the lines end at $\varepsilon = 1$. c. The migration of the resonance is compared against the vector field $\mathrm{H}(\omega)$ for $\alpha=0.4$. The red cross is placed on the repelling point, i.e. the pole of $\mathrm{H}$.
• Figure 2: a. Ringdown signal in time and frequency domain for the same $x_b$ and $\alpha$ in Fig. \ref{['fig:QNMdrift']}a. The logarithm of the absolute value of the signal close to the reflective wall is shown on the left panels. For the open-system $\varepsilon=0$, we observe the characteristic QNM response whilst for $\varepsilon\neq 0$, the signal is characterised by a fast (slowly) decaying signal at early (late) times. b. We show Fourier transform the of the signal at $\bar{x}_1$ near the wall (dotted line) and on the open side of the system at $\bar{x}_2$ (solid). The first overtone (the peak to the right of $\mathrm{Re}[\omega]>1$) becomes more pronounced for larger $\varepsilon$ since the damping is reduced relative to $\varepsilon=0$, and corresponds to the short-lived part of the time domain signal. c. We plot the Fourier transform of the signal at each point to illustrate the spatial features for the resonant modes against the PT potential (thick white line). The signal is normalised such that the colour bar goes from 0 (black) to 1 (white). The dotted and the solid lines label $\bar{x}_1$ and $\bar{x}_2$ respectively. The spectrum averaged over the $x$ axis is showed on the right side.
• Figure 3: The systems response in the frequency domain when stimulated with mechanical noise. We take the average over 500 runs (black lines) and display the standard deviation as the shaded regions. The semi-open system is characterised by the presence of well defined resonance peaks. In particular, we manage to resolve the first overtone above the noise level. The data is for a potential with $\alpha=1$ and a wall at $x_b=10$.