# Causality and relativistic fluid dynamics

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Causality and relativistic fluid dynamicsStefan Florchinger
(Heidelberg U.)

based on:

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

Relativistic Navier-Stokes

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

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

with constraints

πµν = −2ησµν ,

set of equations not hyperbolic

1 / 23

crucial insight: time derivatives of πµν and πbulk needed

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

τbulk u ρ∂ρπbulk + πbulk + ζ∇µuµ + . . . = 0.

Causality and linear stability of linear perturbations around equilibrium 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 dispersion relation ω(k)

causality and stability condition in terms of asymptotic group velocity

vas = lim k→∞

Aij(Φ) ∂

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

know also internal derivative Dj(x)

−(∂1)∂0Φj + (∂0)∂1Φj = (∂0)Dj ,

can write

with velocity λ = −∂0/∂1.

4 / 23

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

= 0

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

∂0J (m) + λ(m) ∂1J

characteristic curves define the causality structure

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

6 / 23

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)u ν∇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

for shear stress

] + πµν [1 + δππ∇ρuρ − 6 πbulk] = 0

for bulk viscous pressure

+ δΠΠπbulk∇µuµ − 1π 2 bulk − λΠππ

µν∇µuν − 3π µ νπ

ν µ = 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 dropped terms of order O(Kn2) because they are at odds with quasi-linear structure

8 / 23

Radial expansion

assume Bjorken boost invariance and azimuthal rotation symmetry

fluid equations reduce effectively to 1 + 1 dimensional partial differential equations

5 independent fluid fields

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

η(τ, r) and πφ φ(τ, r)

bulk viscous pressure πbulk(τ, r)

combine fields Φ = (T, v, πηη , π φ φ , πbulk)

equations of quasi-linear form

λ(1) = v + c

1 + cv , λ(2) =

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

c = √ c2s + d

c2s = ∂p

η

Numerical examples use

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

neglect bulk viscosity

τshear = η 2(2− ln (2))

ε+ p , δππ =

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

compare different possibilities for initial shear stress and τ0

11 / 23

πµν = 0

0.7

0.8

0.9

1.0

1.1

1.2

c

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

12 / 23

πµν = −η σµν

0 5 10 15 0.6

0.7

0.8

0.9

1.0

1.1

1.2

c

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

13 / 23

πµν = −η σµν

0 5 10 15 0.6

0.7

0.8

0.9

1.0

1.1

1.2

c

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

14 / 23

0 2 4 6 8 10 12 14

2

4

6

8

10

12

14

15 / 23

preserves light-cones

16 / 23

preserves light-cones

17 / 23

Conformal diagrams

0.2

0.4

0.6

0.8

1.0

ρ

σ +

i0

i+

Σ0

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

characteristic curves must remain within light cone

18 / 23

Conformal diagrams

0.2

0.4

0.6

0.8

1.0

ρ

σ +

i0

i+

Σ0

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

characteristic curves must remain within light cone

19 / 23

Conformal diagrams

0.2

0.4

0.6

0.8

1.0

ρ

σ +

i0

i+

Σ0

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

characteristic curves must remain within light cone

20 / 23

2 4 6 8 10 12 14

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

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

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

also negative longitudinal pressure within this regime

21 / 23

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

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

Θ = ∇µuµ, Fµ = ∂µp(T ).

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

τΘu µ∇µΘ + Θ−∇µuµ = 0,

τFu ν∇νFµ + Fµ − ∂µp = 0.

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

22 / 23

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 of characteristic polynomial Q

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

λ(m) ≤ c = 1

causality poses a bound on range of applicability of relativistic fluid dynamics

23 / 23

based on:

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

Relativistic Navier-Stokes

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

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

with constraints

πµν = −2ησµν ,

set of equations not hyperbolic

1 / 23

crucial insight: time derivatives of πµν and πbulk needed

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

τbulk u ρ∂ρπbulk + πbulk + ζ∇µuµ + . . . = 0.

Causality and linear stability of linear perturbations around equilibrium 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 dispersion relation ω(k)

causality and stability condition in terms of asymptotic group velocity

vas = lim k→∞

Aij(Φ) ∂

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

know also internal derivative Dj(x)

−(∂1)∂0Φj + (∂0)∂1Φj = (∂0)Dj ,

can write

with velocity λ = −∂0/∂1.

4 / 23

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

= 0

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

∂0J (m) + λ(m) ∂1J

characteristic curves define the causality structure

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

6 / 23

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)u ν∇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

for shear stress

] + πµν [1 + δππ∇ρuρ − 6 πbulk] = 0

for bulk viscous pressure

+ δΠΠπbulk∇µuµ − 1π 2 bulk − λΠππ

µν∇µuν − 3π µ νπ

ν µ = 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 dropped terms of order O(Kn2) because they are at odds with quasi-linear structure

8 / 23

Radial expansion

assume Bjorken boost invariance and azimuthal rotation symmetry

fluid equations reduce effectively to 1 + 1 dimensional partial differential equations

5 independent fluid fields

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

η(τ, r) and πφ φ(τ, r)

bulk viscous pressure πbulk(τ, r)

combine fields Φ = (T, v, πηη , π φ φ , πbulk)

equations of quasi-linear form

λ(1) = v + c

1 + cv , λ(2) =

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

c = √ c2s + d

c2s = ∂p

η

Numerical examples use

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

neglect bulk viscosity

τshear = η 2(2− ln (2))

ε+ p , δππ =

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

compare different possibilities for initial shear stress and τ0

11 / 23

πµν = 0

0.7

0.8

0.9

1.0

1.1

1.2

c

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

12 / 23

πµν = −η σµν

0 5 10 15 0.6

0.7

0.8

0.9

1.0

1.1

1.2

c

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

13 / 23

πµν = −η σµν

0 5 10 15 0.6

0.7

0.8

0.9

1.0

1.1

1.2

c

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

14 / 23

0 2 4 6 8 10 12 14

2

4

6

8

10

12

14

15 / 23

preserves light-cones

16 / 23

preserves light-cones

17 / 23

Conformal diagrams

0.2

0.4

0.6

0.8

1.0

ρ

σ +

i0

i+

Σ0

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

characteristic curves must remain within light cone

18 / 23

Conformal diagrams

0.2

0.4

0.6

0.8

1.0

ρ

σ +

i0

i+

Σ0

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

characteristic curves must remain within light cone

19 / 23

Conformal diagrams

0.2

0.4

0.6

0.8

1.0

ρ

σ +

i0

i+

Σ0

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

characteristic curves must remain within light cone

20 / 23

2 4 6 8 10 12 14

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

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

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

also negative longitudinal pressure within this regime

21 / 23

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

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

Θ = ∇µuµ, Fµ = ∂µp(T ).

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

τΘu µ∇µΘ + Θ−∇µuµ = 0,

τFu ν∇νFµ + Fµ − ∂µp = 0.

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

22 / 23

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 of characteristic polynomial Q

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

λ(m) ≤ c = 1

causality poses a bound on range of applicability of relativistic fluid dynamics

23 / 23