[Paper Review] Acoustic superradiance in a slightly viscous fluid

Oindrila Ganguly|arXiv (Cornell University)|May 14, 2017

Quantum Electrodynamics and Casimir Effect8 references3 citations

TL;DR

This paper investigates acoustic superradiance in a rotating sonic black hole formed in a slightly viscous fluid, showing that viscosity breaks Lorentz invariance and imposes an upper frequency bound on superradiant modes. The bound exactly matches the standard superradiance condition ω < mΩ_H, confirming that all superradiant waves are amplified while viscosity suppresses higher-frequency modes.

ABSTRACT

The acoustic analogue of the geometry of curved spacetime realised in a classical fluid becomes Lorentz violating in the presence of viscosity. We study how this effective Lorentz violation affects acoustic superradiance from the ergosphere of a rotating sonic black hole formed in such a fluid. It turns out that Lorentz violation imposes an upper bound on frequencies of acoustic perturbations that can get scattered from the ergosphere. Incidentally, this upper bound is same as that on the spectrum of superradiant frequencies. This study also reveals how superradiance is in general modified when the wave propagates through a dispersive and dissipative medium. Our result is valid only upto linear order in the coefficient of viscosity and is thus a first approximation to the full solution, focussing on the key qualitative features.

Motivation & Objective

• To understand how viscosity-induced Lorentz violation affects acoustic superradiance in a rotating sonic black hole.
• To determine whether viscosity restricts the range of superradiant frequencies in a classical fluid.
• To assess the impact of dispersion and dissipation on superradiance in a weakly viscous medium.
• To provide a first-order approximation valid up to linear order in viscosity, focusing on qualitative features.
• To bridge theoretical models of acoustic black holes with real experimental systems where viscosity is unavoidable.

Proposed method

• Modeling the fluid flow using a draining bathtub metric with radial and tangential velocity components, v_b = -A/r r̂ + B/r ϕ̂.
• Deriving the acoustic wave equation in the presence of viscosity, introducing a viscous correction term proportional to ν (kinematic viscosity).
• Applying the ansatz ψ_a = R(r)e^(-iωt + imϕ) to separate variables and reduce the wave equation to a radial ODE.
• Approximating the radial equation by neglecting third-order radial derivatives (d_r^3R) and retaining only dominant second-order terms.
• Solving the resulting equation perturbatively in ν, up to linear order, to analyze the scattering of acoustic waves from the ergosphere.
• Using the Killing vector symmetries (∂_t and ∂_ϕ) to define conserved quantities and analyze energy extraction via superradiance.

Experimental results

Research questions

• RQ1 How does the presence of viscosity in a classical fluid break Lorentz invariance in the acoustic metric?
• RQ2 What is the effect of viscosity on the range of frequencies that can undergo superradiance in a rotating sonic black hole?
• RQ3 Does the upper frequency bound imposed by viscosity coincide with the standard superradiance condition ω < mΩ_H?
• RQ4 How do dispersion and dissipation modify the superradiant amplification process in a weakly viscous fluid?
• RQ5 Can the results from a viscous model be extrapolated to predict behavior in ideal, inviscid fluids used in laboratory analogues?

Key findings

• Viscosity breaks Lorentz invariance in the acoustic metric, leading to a modified wave equation that no longer respects relativistic symmetry.
• The upper bound on superradiant frequencies due to viscosity is given by ω < (1 + ν/A)mΩ_H - (m/A)√(2ν/(3A)), which reduces to ω < mΩ_H in the inviscid limit.
• This frequency bound exactly matches the standard superradiance condition ω < mΩ_H, implying all superradiant modes remain amplified despite viscosity.
• The suppression of high-frequency modes is consistent with the assumption that ν << 1 and B/A << 1, ensuring the perturbative approximation holds.
• The analysis confirms that superradiance persists in viscous fluids as long as ω < mΩ_H, with no additional constraints from viscosity beyond frequency filtering.
• The results suggest that viscous effects can be systematically subtracted in real experiments to infer behavior in the ideal, inviscid limit, aiding experimental validation of superradiance.