Causality and relativistic fluid dynamics

Stefan Florchinger (Heidelberg U.)

MIAPP Programme Heavy Ions, Sep 4, 2018

based on:

Stefan Floerchinger & Eduardo Grossi, Causality of fluid dynamicsfor high-energy nuclear collisions [JHEP 08 (2018) 186].

Relativistic Navier-Stokes

evolution equations for energy density and fluid velocity

uµ∂µε+ (ε+ p+ πbulk)∇µuµ + πµν∇µuν =0,

(ε+ p+ πbulk)uν∇νuµ + ∆µν∂ν(p+ πbulk) + ∆µν∇ρπρν =0,

with constraints

πµν = −2ησµν ,

πbulk = −ζ∇µuµ.

is not causal in relativistic sense

set of equations not hyperbolic

1 / 23

Muller-Israel-Stewart

[Muller (1967), Israel & Stewart (1979)]

crucial insight: time derivatives of πµν and πbulk needed

Pµναβ τshear uρ∇ρπαβ + πµν + 2ησµν + . . . = 0,

τbulk uρ∂ρπbulk + πbulk + ζ∇µuµ + . . . = 0.

Causality and linear stability of linear perturbations aroundequilibrium states [Hiscock & Lindblom (1983)]

see also more recent discussions [Denicol, Kodama, Koide & Mota

(2008); Pu, Koide & Rischke (2010)]

analysis conveniently done in Fourier space, leads to dispersionrelation ω(k)

causality and stability condition in terms of asymptotic groupvelocity

vas = limk→∞

∣∣∣∣∂ Reω(k)

∂k

∣∣∣∣ ≤ c2 / 23

Hyperbolic differential equations

[R. Courant & D. Hilbert (1962)]

First order partial differential equations

Aij(Φ)∂

∂x0Φj +Bij(Φ)

∂

∂x1Φj + Ci(Φ) = 0

set of fields Φj with j = 1, . . . , n.

time coordinate x0, space coordinate x1 (radius)

linear for Aij and Bij independent of Φ and Ci(Φ) linear

semi-linear for Aij and Bij independent of Φ

quasi-linear for general Aij(Φ) and Bij(φ)

3 / 23

Cauchy initial value problem

x1

x0

assume Φj(x) given on a Cauchy surface / curve

Cauchy curve specified by ϕ(x) = 0 with ∂0ϕ(x) 6= 0

know also internal derivative Dj(x)

−(∂1ϕ)∂0Φj + (∂0ϕ)∂1Φj = (∂0ϕ)Dj ,

can write

[λAij(Φ)−Bij(Φ)] ∂0Φj + λCi(Φ) + λBij(Φ)Dj = 0

with velocity λ = −∂0ϕ/∂1ϕ.

4 / 23

Well-posedness of initial value problem

can solve for ∂0Φj if characteristic polynomial non-vanishing

Q(λ) = det [λAij(Φ)−Bij(Φ)] 6= 0

Cauchy surface needs to be free: Q(λ) 6= 0

initial conditions can be extended into strip

solution to PDE can be constructed this way

system of equations is hyperbolic: all zero crossings λ(m) are real

Relativistic causality: Cauchy curve can be any curve with |λ| > c→ zero crossings of Q(λ) should all be at |λ| ≤ c

5 / 23

Characteristic curvesreal solution

λ(m) of Q(λ) = 0

is a characteristic velocity

solution of differential equation

dx1

dx0= λ(m)

is called characteristic curve

left eigenvectors

w(m)i

[λ(m)δij −A−1

ik (Φ)Bkj(Φ)]

= 0

can be used to define new variables dJ (m) = w(m)j dΦj such that

∂0J(m) + λ(m) ∂1J

(m) + w(m)j (A−1C)j = 0

characteristic curves define the causality structure

relativistic causality: |λ(m)| ≤ c = 1

6 / 23

Fluid dynamic evolution equations

energy-momentum conservation

∇µTµν = 0

decomposition of energy-momentum tensor

Tµν = εuµuν + (p+ πbulk)∆µν + πµν

with p = p(T ) and ε = ε(T )

evolution equations for energy density and fluid velocity

uµ∂µε+ (ε+ p+ πbulk)∇µuµ + πµν∇µuν =0,

(ε+ p+ πbulk)uν∇νuµ + ∆µν∂ν(p+ πbulk) + ∆µν∇ρπρν =0.

only terms linear in first derivatives of T , uν , πµν and πbulk appear

7 / 23

Constitutive relations

[Denicol, Niemi, Molnar, Rischke, PRD 85, 114047 (2012)]

for shear stress

Pµ ρν σ

[τshear

(uλ∇λπσρ − 2πσλωρλ

)+ 2η∇ρuσ − ϕ7 π

λρπ

σλ

+τππ πσλσ

λρ − λπΠ πbulk∇ρuσ

]+ πµν [1 + δππ∇ρuρ − ϕ6 πbulk] = 0

for bulk viscous pressure

τbulk uµ∂µ πbulk + πbulk + ζ∇µuµ

+ δΠΠπbulk∇µuµ − ϕ1π2bulk − λΠππ

µν∇µuν − ϕ3πµνπ

νµ = 0

only terms linear in first derivatives of T , uν , πµν and πbulk appear

we included terms of order O(Re−2) and O(Re−1Kn) but droppedterms of order O(Kn2) because they are at odds with quasi-linearstructure

8 / 23

Radial expansion

choose coordinates τ , r, φ and η

assume Bjorken boost invariance and azimuthal rotation symmetry

fluid equations reduce effectively to 1 + 1 dimensional partialdifferential equations

5 independent fluid fields

temperature T (τ, r)radial fluid velocity v(τ, r)shear stress components πη

η(τ, r) and πφφ(τ, r)

bulk viscous pressure πbulk(τ, r)

combine fields Φ = (T, v, πηη , πφφ , πbulk)

equations of quasi-linear form

Aij(Φ)∂

∂x0Φj +Bij(Φ)

∂

∂x1Φj + Ci(Φ) = 0

9 / 23

Characteristic velocitiesdirect calculation leads to

λ(1) =v + c

1 + cv, λ(2) =

v − c1− cv

, λ(3) = λ(4) = λ(5) = v.

“relativistic sums” of fluid velocity v and modified sound velocity

c =√c2s + d

ideal velocity of sound from thermodynamics

c2s =∂p

∂ε=

∂p∂T∂ε∂T

dissipative correction

d =

4η3τshear

+ ζτbulk

−(

τππ3τshear

− δππτshear

+ λΠπτbulk

)(πφφ + πη

η

)+

(δΠΠτbulk

+ λπΠ3τshear

)πbulk

ε+ p+ πbulk − πφφ − πη

η

relativistic causality needs c ≤ 110 / 23

Numerical examples use

EOS from lattice [Borsanyi et al. (2016)]

temperature dependent η/s for Yang-Mills theory[Christiansen et al. (2015)]

neglect bulk viscosity

only two second order coefficients [BRSSS (2008)]

τshear = η2(2− ln (2))

ε+ p, δππ =

4

3τshear.

no initial flow v(τ0, r) = 0

Glauber initial conditions with T (τ0, 0) = 0.4 GeV

compare different possibilities for initial shear stress and τ0

11 / 23

Modified velocity of sound

initialized at τ0 = 0.6 fm/c with vanishing shear stress

πµν = 0

respects relativistic causality

0 5 10 150.6

0.7

0.8

0.9

1.0

1.1

1.2

r [fm]

c

τ=0.6 fm/c τ=1.5 fm/c τ=4 fm/c τ=15 fm/c

12 / 23

Modified velocity of sound

initialized at τ0 = 0.6 fm/c with Navier-Stokes shear stress

πµν = −η σµν

respects relativistic causality in central region

0 5 10 150.6

0.7

0.8

0.9

1.0

1.1

1.2

r [fm]

c

τ=0.6 fm/c τ=1.5 fm/c τ=4 fm/c τ=15 fm/c

13 / 23

Modified velocity of sound

initialized at τ0 = 0.3 fm/c with Navier-Stokes shear stress

πµν = −η σµν

breaks relativistic causality at early times

0 5 10 150.6

0.7

0.8

0.9

1.0

1.1

1.2

r [fm]

c

τ=0.1 fm/c τ=0.6 fm/c τ=4 fm/c τ=15 fm/c

14 / 23

Domain of dependence, domain of influence

0 2 4 6 8 10 12 14

2

4

6

8

10

12

14

r [fm]

τ[fm/c]

λ(1) λ(3) λ(2)

T=0.145 GeV

Γd

Γi

x

generalization of light-cone structure

space-time dependent for relativistic fluid dynamics

15 / 23

Penrose or conformal diagrams

conformal representation of space-time structure

preserves light-cones

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Penrose or conformal diagrams

conformal representation of space-time structure

preserves light-cones

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Conformal diagrams

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

ρ

σ+

i0

i+

Σ0

T=0.15 GeV

T=0.20 GeV

T=0.25 GeV

λ(1)

conformal map r − τ = h(ρ− σ), r + τ = h(ρ+ σ)

characteristic curves must remain within light cone

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Conformal diagrams

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

ρ

σ+

i0

i+

Σ0

T=0.15 GeV

T=0.20 GeV

T=0.25 GeV

λ(2)

conformal map r − τ = h(ρ− σ), r + τ = h(ρ+ σ)

characteristic curves must remain within light cone

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Conformal diagrams

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

ρ

σ+

i0

i+

Σ0

T=0.15 GeV

T=0.20 GeV

T=0.25 GeV

λ(3)

conformal map r − τ = h(ρ− σ), r + τ = h(ρ+ σ)

characteristic curves must remain within light cone

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Pressure ratios allowed by causality

2 4 6 8 10 12 14

-0.5

0.0

0.5

1.0

τ [fm/c]

Pη/Pr

2 4 6 8 10 12 14

-0.5

0.0

0.5

1.0

τ [fm/c]

Pη/Pr

relativistic fluid dynamics only applicable for |λ(m)| ≤ 1

poses a sharp bound on allowed range of shear stress and bulkviscous pressure

also negative longitudinal pressure within this regime

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Terms of second order in Knudsen number

exemplary bulk pressure relation with terms of order O(Kn2)

τbulk uµ∂µ πbulk + πbulk + ζ∇µuµ = ζ3 Θ2 + ζ8∇µFµ,

Θ = ∇µuµ,Fµ = ∂µp(T ).

non-linear term in derivatives ∼ Θ2

second derivative term ∼ ∇µFµ

constraint equations for Θ and Fµ similar as in Navier-Stokes

can be remedied by relaxation time terms of higher order

τΘuµ∇µΘ + Θ−∇µuµ = 0,

τFuν∇νFµ + Fµ − ∂µp = 0.

leads to quasi-linear system of equations that can be hyperbolic

22 / 23

Conclusions

relativistic fluid dynamics is not always causal

second order DNMR approximation with terms of order O(Re−2)and O(Re−1Kn) can be formulated as quasi-linear set of equations

well-posedness and causality can be investigated in term ofcharacteristic polynomial Q

for 1 + 1 dimensional dynamics: characteristic velocities need bebounded by speed of light

λ(m) ≤ c = 1

causality poses a bound on range of applicability of relativistic fluiddynamics

23 / 23