Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mykhailo Hontarenko, Javier de Lucas, Adam Maskalaniec

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.

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

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 -contact geometry. In this approach, thermodynamic laws are encoded in a -contact form, thermodynamical states are described via -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.

Paper Structure

This paper contains 8 sections, 11 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Basics on k-contact geometry
  3. k-contact Hamiltonian systems
  4. Legendrian submanifolds of k-contact manifolds
  5. k-contact approach to "extensive relativistic hydrodynamics"
  6. Analysis of pseudo-gauge degrees of freedom
  7. Physical Interpretation of the Pseudo-Gauge
  8. Conclusions and outlook

Key Result

Theorem 2.1

Let $(M,\bm\eta)$ be a $k$-contact manifold. Then, there exist vector fields $R_1,\ldots,R_k\in\mathfrak{X}(M)$, defined uniquely by called the Reeb vector fields of $(M,{\bm \eta})$. The Reeb vector fields span the Reeb distribution and commute with each other, namely

Theorems & Definitions (36)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.2
  • Theorem 2.2
  • Example 2.3
  • ...and 26 more