Wave equation of the scalar field and superfluids

TL;DR

This paper explores the analogy between superfluid systems and cosmology, deriving hydrodynamical equations from scalar wave equations and connecting them to General Relativity in a four-dimensional framework.

Contribution

It introduces a formal analogy that incorporates vacuum back-reaction, justifies a GR-based derivation of superfluid hydrodynamics, and links these equations to scalar wave equations in four-dimensional space.

Findings

Hydrodynamical equations can be derived from scalar wave equations.

A formal analogy between superfluids and cosmology is established.

Conditions for a GR derivation of superfluid equations are identified.

Abstract

The new formal analogy between superfluid systems and cosmology, which emerges by taking into account the back-reaction of the vacuum to the quanta of sound waves \cite{noi}, enables us to put forward some common features between these two different areas of physics. We find the condition that allows us to justify a General Relativity (GR) derivation of the hydrodynamical equation for the superfluid in a four-dimensional space whose metric is the Unruh one \cite{Unruh}. Furthermore we show how, in the particular case taken into account, our hydrodynamical equation can be deduced within a four-dimensional space from the wave equation of a massless scalar field.