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A strongly hyperbolic viscous relativistic hydrodynamics theory with first-order charge current

Federico Schianchi, Fernando Abalos

Abstract

We extend the first order dissipative relativistic hydrodynamics model of Bemfica-Disconzi-Noronha- Kovtun (BDNK) in order to include the charge number current in full first order expansion with out-of-equilibrium contribution proportional to the evolution equation of the ideal fluid. We obtain a fully second order system of partial differential equation (PDE) that can be casted in a fully conservative way. We analyze the hyperbolicity of this model coupled to Einstein field equations using a newly developed technique that allows for hyperbolicity studies without explicit first order reduction. Furthermore, we identify a frame choice where our formulation is causal, stable and with positive entropy generation for a wide range of equations of state (EoS). Our analysis shows that the inclusion of an out-of-equilibrium correction to the charge current, plays an important role in guaranteeing the strong hyperbolicity and, therefore, the well-posedness of the system. If such correction is not applied, an extra frame restriction must be added to the present in the literature in order to obtain a strongly hyperbolic system.

A strongly hyperbolic viscous relativistic hydrodynamics theory with first-order charge current

Abstract

We extend the first order dissipative relativistic hydrodynamics model of Bemfica-Disconzi-Noronha- Kovtun (BDNK) in order to include the charge number current in full first order expansion with out-of-equilibrium contribution proportional to the evolution equation of the ideal fluid. We obtain a fully second order system of partial differential equation (PDE) that can be casted in a fully conservative way. We analyze the hyperbolicity of this model coupled to Einstein field equations using a newly developed technique that allows for hyperbolicity studies without explicit first order reduction. Furthermore, we identify a frame choice where our formulation is causal, stable and with positive entropy generation for a wide range of equations of state (EoS). Our analysis shows that the inclusion of an out-of-equilibrium correction to the charge current, plays an important role in guaranteeing the strong hyperbolicity and, therefore, the well-posedness of the system. If such correction is not applied, an extra frame restriction must be added to the present in the literature in order to obtain a strongly hyperbolic system.
Paper Structure (14 sections, 8 theorems, 142 equations, 1 figure)

This paper contains 14 sections, 8 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. New Hydrodynamic formulation, hyperbolicity, causality, entropy and stability
  3. Setting and evolution equations
  4. Hyperbolicity and Causality
  5. Entropy production
  6. Linear Stability
  7. Choice of frame
  8. Coupling to Einstein field equations
  9. Summary
  10. Acknowledgments
  11. Hyperbolicity without imposing 4-velocity normalization
  12. The case $\beta_n=0$
  13. positivity of $s_2$
  14. Piecewise polytropic EoS with thermal contribution

Key Result

Lemma 2

If the following inequalities are satisfied, then Eqs. eq:evolution1-eq:evolution3, with dissipative terms expressed as Eqs. eq:A-eq:J, are strongly hyperbolic. Moreover, if the additional conditions are also satisfied, then the system is causal.

Figures (1)

  • Figure 1: Hyperbolicity, Causality, and stability conditions as function of $\hat{\tau}$ for $\alpha=5/3$, $c_s^2=0.3$, $\omega=0.1$, $\hat{\sigma}=-0.05$, $\tilde{\sigma}=0.15$.

Theorems & Definitions (9)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Proposition 7
  • Lemma 8
  • Lemma 9