A $k$-contact Geometrical Approach to Pseudo-Gauge Transformation

TL;DR

This paper introduces a novel geometric framework using $k$-contact geometry to describe pseudo-gauge transformations in relativistic hydrodynamics, linking thermodynamic state redefinitions to gauge freedom.

Contribution

It develops the first $k$-contact geometric formulation of relativistic hydrodynamics, connecting thermodynamic redefinitions to pseudo-gauge ambiguity.

Findings

Pseudo-gauge transformations correspond to non-uniqueness in $k$-contact solutions.

The framework demonstrates invariance of physical dissipation under pseudo-gauge redefinitions.

Application to a Bjorken-like expansion model confirms the approach's validity.

Abstract

We propose a starting point to the geometric description for the pseudo-gauge ambiguity in relativistic hydrodynamics, showing that it corresponds to the freedom to redefine the thermodynamic equilibrium state of the system. To do this, we develop for the first time a description of a relativistic hydrodynamic-like theory using $k$-contact geometry. In this approach, thermodynamic laws are encoded in a $k$-contact form, thermodynamical states are described via $k$-contact Legendrian submanifolds, and conservation laws emerge as a consequence of Hamilton-de Donder-Weyl (HdDW) equations. The inherent non-uniqueness of these solutions is identified as the source of the pseudo-gauge freedom. We explicitly demonstrate how this redefinition of equilibrium works in a model of a Bjorken-like expansion, where a pseudo-gauge transformation is shown to leave the physical dissipation invariant.