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
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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)
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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.
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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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= 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
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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
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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
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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
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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
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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
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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
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0 2 4 6 8 10 12 14
2
4
6
8
10
12
14
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preserves light-cones
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preserves light-cones
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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
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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
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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
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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
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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
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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
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