Relativistic hydrodynamics stability and causality

1
Relativistic hydrodynamics stability and
causality

• P. Ván1,2 and T. S. Bíró1
• RMKI, Budapest1 and University of Bergen2

• Introduction
• Causality parabolic equations
• Stability Eckart problem
• Separation of dissipative and nondissipative
  parts
• Conclusions

Zimányi 75 Memorial Workshop07, Budapest
2
Introduction
Nonrelativistic Relativistic Local
equilibrium (1st) Fourier, Navier-Stokes Eckart B
eyond local equilibrium Cattaneo-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
3
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.
(1)
4
ii) () is transformation invariant
E.g.
5
Infinite speed of signal propagation? physics -
mathematics
Hydrodynamic range of validity ? mean free
path t collision time
Water at room temperature Fermi gas of light
quarks at
More complicated equations, more spacetime
dimensions, .
6
Stability of what and in what sense?
homogeneous equilibrium (thermodynamics
theory of stability of ) linear and
nonlinear linear necessary condition Eckart
theory instable due to heat conduction
water
Israel-Stewart theory ? strange
condition ? relaxation to the first order
theory (Geroch 1995, Lindblom 1995)
7
Structure of dissipative hydrodynamic theories
Irreversible thermodynamics (standard method,
e.g. B. Lukács)
8
Complete Eckart system
lt gt - symmetric traceless spacelike part
Equilibrium
9
exponential plane-waves (Hiscock and Lindblom,
1985)
Stability condition for transverse modes

• root with a positive real part ? instability
• coupling of shear viscosity and heat conduction

Landau frame?
10
First or second (or higher) order theory?
Causality speed of the VALIDITY lt speed
of light both for first and second order
Stability Landau choice (q0) 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)
11
Separation of dissipation (PV and TSB
arXiv0704.2039)
flow energy
Separation condition
12
Something more
(a) energies total internal flow (mass?) (b)
velocity momentum (heat) flow energy heat
flux
13
Thermodynamics
normal with internal energy e, or
Statics
q dependence
14
(No Transcript)
15
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
16
Thank you for your attention!
17
(No Transcript)
18
(No Transcript)
19
Balance of entropy
Stable!
Net balances
20
Linearization
21
Routh-Hurwitz
thermodynamic stability
hydrodynamic stability
22
Nonrelativistic experience a four vector
formalism
Energy units of mass
mass velocity (momentum ?) internal
energy velocity-momentum (relativistic?)
23
Nonrelativistic spacetime there is time
(absolute)
spacelike, timelike, vectors and
covectors, substantial time derivative
energy-momentum tensor
?
24
mass-momentum vector
total energy-momentum tensor
separation of dissipative and nondissiaptive
parts