Causality and relativistic fluid dynamics Stefan Fl¨ orchinger (Heidelberg U.) MIAPP Programme Heavy Ions, Sep 4, 2018
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
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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)
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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