Simulating gravity in rotational flow

TL;DR

This paper explores how classical rotational fluid flows can be used to simulate gravitational effects by deriving an analogue space-time through conservation equations of perturbations, highlighting limitations and specific conditions for such analogies.

Contribution

It demonstrates that an analogue space-time can be derived from conservation equations in rotational flows, even when a direct wave equation similar to a massless scalar field is not always obtainable.

Findings

Conservation equations can yield an analogue space-time in rotational flows.

High frequency limit simplifies perturbation equations to a massless scalar field.

Specific systems allow for an exact analogue space-time through current conservation.

Abstract

We consider classical fluids in non-relativistic framework. The flow is considered to be barotropic, inviscid and rotational. We study the linear perturbations over a steady state background flow. We find the acoustic metric from the conservation equation of a current constructed from linear perturbation of first order derivatives (in position and time coordinate) of Bernoulli's constant (scalar field) and vorticity (a vector field). We have rather shown that the conservation equation of current reduces to a massless scalar field equation in the high frequency limit. In contrast to the contemporary works, our work shows that even if we can not find a wave equation (in rotational flow) which is structurally similar to a massless scalar field equation in curved space-time, but still an analogue space-time exists through a conservation equation. Considering velocity potential and Clebesch coefficients, we find that only for some specific systems current conservation equation can be found yielding the same analogue space-time. We conclude that for rotational flows, it is wise to study linear perturbation of Bernoulli's constant over the velocity potential and Clebsch coefficients.