Onset of superradiant instabilities in the hydrodynamic vortex model

TL;DR

This paper analytically investigates the conditions under which superradiant instabilities occur in a hydrodynamic vortex model, revealing the critical state that separates stability from instability and confirming previous numerical findings.

Contribution

It provides an analytical formula for the marginally-stable state of the hydrodynamic vortex, advancing understanding of superradiant instabilities in acoustic geometries.

Findings

Analytical formula matches numerical data for critical vortex state.

Identifies the boundary between stable and unstable configurations.

Enhances understanding of superradiant instabilities in ergoregions.

Abstract

The hydrodynamic vortex, an effective spacetime geometry for propagating sound waves, is studied analytically. In contrast with the familiar Kerr black-hole spacetime, the hydrodynamic vortex model is described by an effective acoustic geometry which has no horizons. However, this acoustic spacetime possesses an ergoregion, a property which it shares with the rotating Kerr spacetime. It has recently been shown numerically that this physical system is linearly unstable due to the superradiant scattering of sound waves in the ergoregion of the effective spacetime. In the present study we use analytical tools in order to explore the onset of these superradiant instabilities which characterize the effective spacetime geometry. In particular, we derive a simple analytical formula which describes the physical properties of the hydrodynamic vortex system in its critical (marginally-stable) state, the state which marks the boundary between stable and unstable fluid configurations. The analytically derived formula is shown to agree with the recently published numerical data for the hydrodynamic vortex system.