Waves and null congruences in a draining bathtub

TL;DR

This paper investigates wave propagation in a fluid-mechanical black hole analogue called the draining bathtub, using analytical and numerical methods to explore how disturbances trace the effective spacetime geometry and reveal features like horizons and ergospheres.

Contribution

It combines the eikonal approximation with numerical simulations to map out the light-cone structure and wave dynamics in the draining bathtub analogue, highlighting effects related to null orbits and ergospheres.

Findings

Wavefronts follow null geodesics, tracing the light-cone structure.

Numerical simulations show wavefront intersections and interference effects.

Features like frame-dragging are observed in wave propagation.

Abstract

We study wave propagation in a draining bathtub: a fluid-mechanical black hole analogue in which perturbations are governed by a Klein-Gordon equation on an effective Lorentzian geometry. Like the Kerr spacetime, the draining bathtub geometry possesses an (effective) horizon, an ergosphere and null circular orbits. We propose that a `pulse' disturbance may be used to map out the light-cone of the effective geometry. First, we apply the eikonal approximation to elucidate the link between wavefronts, null geodesic congruences and the Raychaudhuri equation. Next, we solve the wave equation numerically in the time domain using the method of lines. Starting with Gaussian initial data, we demonstrate that a pulse will propagate along a null congruence and thus trace out the light-cone of the effective geometry. Our numerical results reveal features, such as wavefront intersections, frame-dragging, winding and interference effects, that are closely associated with the presence of null circular orbits and the ergosphere.