[Paper Review] Numerical analysis of backreaction in acoustic black holes

Roberto Balbinot, Alessandro Fabbri+2 more|arXiv (Cornell University)|Jan 20, 2006

Aerodynamics and Acoustics in Jet Flows5 citations

TL;DR

This paper presents a numerical investigation of quantum backreaction effects in acoustic black holes using quantum field theory in curved spacetime. It computes first-order $ackslash$hbar corrections to fluid flow due to phonon emission, employing upwind and implicit finite-difference schemes to solve coupled hydrodynamic equations, revealing significant modifications to density and velocity profiles near the sonic horizon, especially in non-equilibrium and cavity-confined configurations.

ABSTRACT

Using methods of Quantum Field Theory in curved spacetime, the first order in hbar quantum corrections to the motion of a fluid in an acoustic black hole configuration are numerically computed. These corrections arise from the non linear backreaction of the emitted phonons. Time dependent (isolated system) and equilibrium configurations (hole in a sonic cavity) are both analyzed.

Motivation & Objective

• To compute first-order quantum corrections ($\hbar$-order) to classical fluid dynamics in acoustic black hole configurations.
• To extend previous analytical studies near the sonic horizon to a full numerical description across the entire fluid domain.
• To analyze backreaction effects in two physical setups: isolated transonic flow and a fluid in a sonic cavity at equilibrium.
• To validate numerical methods with convergence tests and ensure stability using explicit (FTCS) and implicit (BTCS) schemes.
• To model the Unruh state for phonons via outgoing boundary conditions, ensuring physical consistency in numerical simulations.

Proposed method

• Formulates the acoustic black hole using a one-dimensional, stationary, inviscid, irrotational fluid flow in a de Laval nozzle with a sonic horizon at $z_H$.
• Derives effective curved spacetime metric for phonons from fluid variables ($\rho$, $v$, $c$), enabling quantization of the phonon field.
• Applies quantum field theory in curved spacetime to compute first-order $\hbar$ corrections to the mean flow via coupled equations for $\psi_1$ (velocity perturbation) and $\rho_1$ (density perturbation).
• Uses a standard upwind method for the $\psi_1$ equation and both explicit FTCS and implicit BTCS schemes for the $\rho_1$ equation to ensure stability and accuracy.
• Imposes outgoing boundary conditions at both ends of the domain to model the Unruh state, with $u_{i-1}^n = u_i^n$ at $i=1$ and $u_{i+1}^n = u_i^n$ at $i=i_{\rm max}$.
• Performs convergence tests using resolutions from 500 to 4000 grid points, finding a convergence rate $\sigma \approx 1.3$ for both $\psi_1$ and $\rho_1$.

Experimental results

Research questions

• RQ1 How do quantum backreaction effects modify the classical fluid flow in an acoustic black hole beyond the immediate vicinity of the sonic horizon?
• RQ2 What are the numerical characteristics and stability properties of finite-difference schemes (FTCS and BTCS) when applied to the quantum-corrected hydrodynamic equations?
• RQ3 How does the inclusion of a confining sonic cavity alter the quantum backreaction effects compared to an isolated system?
• RQ4 To what extent do the numerical results converge with increasing spatial resolution, and what is the effective order of convergence?
• RQ5 How do the boundary conditions consistent with the Unruh state affect the numerical solution of the backreaction equations?

Key findings

• The numerical scheme successfully computes first-order $\hbar$ corrections to the fluid density and velocity across the entire acoustic black hole system, not just near the horizon.
• The convergence rate of the numerical method is approximately $\sigma \approx 1.3$ for both $\psi_1$ and $\rho_1$, indicating first-order accuracy with a suboptimal but stable convergence behavior.
• The use of implicit BTCS and explicit FTCS schemes ensures numerical stability, with a time step chosen as $\Delta t = \alpha \Delta z^2 / \max(\rho)$ and $\alpha = 10^{-4}$.
• For a resolution of $\Delta z = 2.5 \times 10^{-5}$ cm (2000 grid points), the full simulation completes in about 11 seconds on a single-processor machine.
• The boundary treatment with outgoing conditions at both ends ensures physical consistency with the Unruh state, crucial for modeling phonon emission in the quantum regime.
• The results demonstrate that quantum backreaction induces measurable, non-trivial modifications to the classical fluid profile, especially near the sonic horizon, validating the need for numerical treatment beyond local analytical approximations.