Stable and causal relativistic Navier-Stokes equations
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Springer
Received: April 29, 2020 Accepted: May 14, 2020 Published: June 9, 2020
Raphael E. Hoult and Pavel Kovtun Department of Physics & Astronomy, University of Victoria, P.O. Box 1700 STN CSC, Victoria, BC, V8W 2Y2, Canada
E-mail: [email protected], [email protected] Abstract: Relativistic Navier-Stokes equations express the conservation of the energymomentum tensor and the particle number current in terms of the local hydrodynamic variables: temperature, fluid velocity, and the chemical potential. We show that the viscous-fluid equations are stable and causal if one adopts suitable non-equilibrium definitions of the hydrodynamic variables. Keywords: Holography and quark-gluon plasmas, Thermal Field Theory ArXiv ePrint: 2004.04102
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)067
JHEP06(2020)067
Stable and causal relativistic Navier-Stokes equations
Contents 1
2 Constitutive relations
3
3 Small fluctuations in equilibrium
5
4 Real-space causality
8
5 Conclusions
11
A Connection to the Landau-Lifshitz frame
12
1
Introduction
Hydrodynamics is a classical effective description of macroscopic states of matter with small deviations from local thermal equilibrium. Hydrodynamics is conventionally formulated by starting from thermodynamics, and then promoting the constant parameters of global thermal equilibrium (temperature T , fluid velocity v, etc.) to slowly varying functions in space and time: T (t, x), v(t, x), etc. The evolution of these hydrodynamic variables is then determined by the local conservation laws of energy, momentum, and possibly other conserved quantities such as mass or particle number [1]. Intrinsic to hydrodynamics is thus a fundamental ambiguity: one must (somewhat arbitrarily) make a choice as to how to define “local temperature”, “local fluid velocity”, etc., out of equilibrium. For example, the non-equilibrium “fluid velocity” may be chosen to correspond to the flow of particles, or to the flow of energy, or to the flow of entropy, etc., with each choice resulting in different hydrodynamic equations. In non-relativistic Navier-Stokes equations, the standard convention is to define the “fluid velocity” through the flow of mass [1]. Relativistic hydrodynamics was presented originally in two different formulations, one pioneered by Eckart [2] and one by Landau and Lifshitz [1]; both are still widely discussed. The non-equilibrium conventions differ between the two formulations. Consequently, the hydrodynamic equations of Eckart and of Landau and Lifshitz are different, mathematically inequivalent, equations. More generally, the arbitrariness in adopting different non-equilibrium definitions implies that there is simply no such thing as “the” equations of hydrodynamics. Still, one expects that conventions should not be physically relevant, and that different hydrodynamic equations must give rise to the same physical predictions within the domain of applicability of hydrodynamics. The standard formulation of relativistic hydrodynamics
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