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 Workshop’07, Budapest

Nonrelativistic Relativistic

Local equilibrium (1st) Fourier, Navier-Stokes Eckart

Beyond 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 – acausalsecond order is good - causal

Stability – first order is bad – instablesecond order is good - stable

Introduction:

Causality hyperbolic or parabolic? (Fichera 1992, Kostädt and Liu 2000)

Well-posedness Speed of signal propagation

0),,,,(2 TTTtxFTCTBTA txttxtxx

Second order linear partial differential equation:

02(*) 22 ttxx CBA

Corresponding equation of characteristics:

i) Hyperbolic equation: two distinct families of real characteristicsParabolic equation: one family of real characteristicsElliptic 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: ))()0,( xfxT

,0 TT xxt (1)

iii) The outer real characteristics that pass through a given point give its domain of influence .

),( 00 tx0

02)(

,0

~~4

2

~~2~~~~

Tc

v

c

vTv

TT

tttxxxxt

xxt

(1)

ii) (*) is transformation invariant ),(~~),,(~~ txtttxxx

t

x

t

x

?

0

cv

vtx

E.g.

)()0,(

,0

xxT

TT xxt

t

x

et

AtxT

4

2

2),(

-4 -2 2 4

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Infinite speed of signal propagation?physics - mathematics

Hydrodynamic range of validity:ξ – mean free pathτ – collision time

TT

TT tx

,

More complicated equations, more spacetime dimensions, ….

Water at room temperature:

Fermi gas of light quarks at :

t

x

et

AtxT

4

2

2),(

Vcv max

s

m

cv

V

14max

cTs

mmT

c

vmc

cv

V

V

V

31max 10

-4 -2 2 4

0.25

0.5

0.75

1

1.25

1.5

1.75

2

,0 TT xxt

homogeneous equilibrium (thermodynamics = theory of stability of …)

linear and nonlinearlinear – necessary condition

Eckart theory:instable – due to heat conduction

Stability of what and in what sense?

scpc

T 3422

10)(

water

Israel-Stewart theory: strange condition

relaxation to the first order theory (Geroch 1995, Lindblom 1995)

1)(1 p ...),(),,( 1 aa

ea qqnsqns

...),(),,,( 21 ababa

ae

aba qqnsqns

0 aa

aa

aa JussS

...1 T

q

T

qJ b

abaa

.....

0)( aaa

aa

aab

ba uqqupTu

Irreversible thermodynamics (standard method, e.g. B. Lukács):

Structure of dissipative hydrodynamic theories:

0)()(1

2 aa

a

ababab

aa

s uTTT

qupP

TTj

Eckart term

Complete Eckart system

.2

,

,

,

,0)(

,0

,0

ab

a

bv

cc

a

av

caca

ccaca

acbb

cac

ab

bbb

aacbb

ac

abab

aaa

aa

aab

ba

aa

aa

aa

uP

uP

Tj

uTTq

PquququT

uPuqquTu

junnN

.0,,,.,,, a

bvaaa Pqjconstun Equilibrium:

< > - symmetric traceless spacelike part

x

x

x

nn

en

en

j

q

u

n

ik

Tik

Tik

TTTikTik

ikppikpik

ikpik

ikikn

~1

0000

01

00

00

0)(

00)(0

000

2

Q

xy

y

y

q

u

ik

T

ikp

10

01

)(

R

0)()det( 22 kpTT R

Stability condition for transverse modes:

)exp(0 ikxtAA exponential plane-waves (Hiscock and Lindblom, 1985)

/4/3~

root with a positive real part instabilitycoupling of shear viscosity and heat conduction

Landau frame? aaq 0

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 fluidsboth 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)

.0)ˆ(ˆ

,0ˆ

,0

acbb

cac

ab

bbb

aacbb

ac

abab

aaa

aa

aab

ba

aabb

PquququT

uPuqquTu

T

Separation condition:

0ˆ,ˆˆˆ aa

aaa ququeE

ababab

bab uquueuE 0)ˆˆ()ˆ(

Separation of dissipation (PV and TSB arXiv:0704.2039)

flow energy

ijj

iab

aba

aa

abbababaab

Pq

qeeT

PuquPuqquuuT

ˆ

ˆ

0,0,ˆ

Something more…

.0ˆˆˆˆ)(

,0ˆˆˆˆ)(aab

baca

cbc

bac

aaa

aba

ba

ueuqquEb

uqueeuEua

0ˆˆ bb

aa uEE.

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

ababab

bb

aabab uquueuEEuE 0)ˆˆ(ˆˆ)ˆ(

0ˆ0,0 euq aa

Thermodynamics:

,ˆ,)ˆ( pTsEeTdsEed aa

Statics:

TE

s

e

sa

1ˆ

)(ˆ2

1ˆˆˆˆ 42222 qqq O

eeeqqeE a

aa

q dependence:

normal with internal energy e, or:

...ˆ2

1)()

ˆ2

1ˆ()ˆ( 22 qq

eTs

eeesEes a

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

)ˆ( aEes

Thank you for your attention!

0)(

11

11)(

1

))((1

2

aa

a

aa

aa

a

aa

aa

aa

a

a

aa

aa

aa

aa

a

a

aa

a

aa

aa

aa

uTTT

q

Tquq

T

T

qqu

Tq

TuTsp

T

T

qusququp

T

T

qus

s

JussS

))(( aaa

aa

a uqqup

Tdsd

T

qJ

aa

pTs 0aaqu

01

1)(

1

))((1

)(

2

.

TT

q

Tq

T

qq

TuTsEep

T

T

qusquEpe

T

T

qusE

E

se

e

s

JusEesS

a

a

aa

a

aa

aa

a

a

aa

aa

aa

a

a

aa

a

aa

aa

aa

TdsEed )(

pTsEe

.0)(

,0)(a

baba

bbacb

bac

aa

aa

abb

a

puqupeT

qupeeTu

Net balances:

....

....

,Tq caca

Balance of entropy:

Stable!

0)(1

2 T

T

qupP

TTj a

a

ababab

aa

0ˆ bab

abb uET

aaa

aaaean

aaean

aa

aaaaean

aa

aa

aa

aa

jquen

quTT

eTT

nTT

jeT

nT

u

queepnp

qupee

junn

.0

,0

,0

,0

,0)(

,0

222

LinearizationAAA 0

Routh-Hurwitz:

0)(

0,0

n

pnpe

e

pTne

T thermodynamic stability

hydrodynamic stability

TT

TTk

pTpTnT

pT

ppeTk

TT

TTpek

TTTkpnpepTk

TTTepTk

peT

QDet

enne

enneenne

enne

nene

ne

2

24

2

2222

2222

23

)(

)(

))((

))((

)(

)(

Nonrelativistic experience – a four vector formalism

Energy units of mass: )1(][][ ce

,0)3(

,0)2(

,0)1(

ijiji

ii

i

iijj

i

ii

vPqvee

Pv

v

mass

velocity (momentum ?)

internal energy

velocity-momentum (relativistic?).)4( ii vq

,0)2(

0:)4()2()1(iij

jj

jii

ijj

jj

ii

Pvqqa

Pvvv

,0)4( iijj

ii vvqqa

,0)3( ijiji

ii

i vPqvee ,0)2( iijj

jj

ii Pvqqa

AAvAAv

Au

uqqvu

xtx

ii

titibb

aaia

Tia

aia

),(1

1);,0(;),1(

);0,0,0,1();,(

spacelike, timelike, vectors and covectors,

substantial time derivative

,0~

aa

baab

abb

bab

baab

ab uqqPqveeT

Nonrelativistic spacetime: there is time (absolute)

j

jb

a

iP

qeT

0~

?energy-momentum tensor

0)1( iiv ,0)4( iij

jii vvqqa

0

)(

ab

baab

baa

baab

bab

uuqquuu

uquuE

aaa quE

mass-momentum vector

j

i

jb

ab

ab

a

iPq

qeuETT

~

)4()3()2()1(0)(0 aauEandT bab

bab

total energy-momentum tensor

separation of dissipative and nondissiaptive parts