Relativistic hydrodynamics – stability and causality P. Ván 1,2 and T. S. Bíró 1 RMKI, Budapest 1 and University of Bergen 2 – Introduction – Causality – parabolic equations – Stability – Eckart problem – Separation of dissipative and nondissipative parts – Conclusions Zimányi 75 Memorial Workshop’07, Budapest

NonrelativisticRelativistic Local equilibrium (1st)Fourier, Navier-StokesEckart Beyond local equilibriumCattaneo-Vernotte, Israel-Stewart, (2nd) gen. Navier-Stokes Müller-Ruggieri Öttinger, Carter, etc.. Conceptual issues plaguing relativistic hydrodynamics: Causality – first order is bad – acausal second order is good - causal Stability – first order is bad – instable second order is good - stable Introduction:

Causality hyperbolic or parabolic? (Fichera 1992, Kostädt and Liu 2000)  Well-posedness  Speed of signal propagation Second order linear partial differential equation: Corresponding equation of characteristics: i)Hyperbolic equation: two distinct families of real characteristics Parabolic equation: one family of real characteristics Elliptic equation: no real characteristics Well-posedness: existence, unicity, continuous dependence on initial data. A characteristic Cauchy problem of (1) is well posed. (initial data on the characteristic surface: ) (1)

iii) The outer real characteristics that pass through a given point give its domain of influence. (1) ii)(*) is transformation invariant t x t x E.g.

Infinite speed of signal propagation? physics - mathematics Hydrodynamic range of validity: ξ – mean free path τ – collision time More complicated equations, more spacetime dimensions, …. Water at room temperature: Fermi gas of light quarks at :

homogeneous equilibrium (thermodynamics = theory of stability of …) linear and nonlinear linear – necessary condition Eckart theory: instable – due to heat conduction Stability of what and in what sense? water Israel-Stewart theory:  strange condition  relaxation to the first order theory (Geroch 1995, Lindblom 1995)

Irreversible thermodynamics (standard method, e.g. B. Lukács): Structure of dissipative hydrodynamic theories: Eckart term

Complete Eckart system Equilibrium: - symmetric traceless spacelike part

Stability condition for transverse modes: exponential plane-waves (Hiscock and Lindblom, 1985)  root with a positive real part  instability coupling of shear viscosity and heat conduction Landau frame?

First or second (or higher) order theory? Causality: speed of the VALIDITY < speed of light both for first and second order Stability: Landau choice (q=0) is a temporary escape - entropy production, multicomponent fluids both for first and second order Origin of stability problem: wrong separation of dissipative and non dissipative terms and effects e.g. the choice of velocity field is not free (e.g. entropy production)

Separation condition: Separation of dissipation (PV and TSB arXiv: ) flow energy

Something more… (a) energies: total= internal+ flow (mass?) (b) velocity – momentum (heat) flow energy – heat flux

Thermodynamics: Statics: q dependence: normal with internal energy e, or:

Summary – momentum density = but ≡ heat flow – energy = internal energy + flow energy ADDS: – entropy flux and can ben justified (thermodynamic theory construction – Liu procedure) – linear stability of homogeneous equilibrium Thermodynamics  stability of matter

Thank you for your attention!

Net balances: Balance of entropy: Stable!

Linearization

Routh-Hurwitz: thermodynamic stability hydrodynamic stability

Nonrelativistic experience – a four vector formalism Energy units of mass: mass velocity (momentum ?) internal energy velocity-momentum (relativistic?)

spacelike, timelike, vectors and covectors, substantial time derivative Nonrelativistic spacetime: there is time (absolute) ? energy-momentum tensor

mass-momentum vector total energy-momentum tensor separation of dissipative and nondissiaptive parts 