Causal first-order hydrodynamics from kinetic theory and holography

We show how causal relativistic Navier-Stokes equations arise from the relativistic Boltzmann equation: the causality is preserved via a judicious choice of the zero modes of the collision operator. A completely analogous procedure may be used to extract causal hydrodynamics from the fluid-gravity correspondence: again, causality of the hydrodynamic equations is preserved by a suitable choice of zero modes of the corresponding differential operators in the bulk. We give examples of zero modes which give rise to causal hydrodynamic equations for non-conformal fluids with a conserved U(1) global symmetry current.


Introduction
Relativistic viscous hydrodynamics is by now over eighty years old, starting with the classic works by Eckart [1] in 1940, and by Landau and Lifshitz in the second edition of their book [2] in 1953. Following the treatment of non-relativistic fluids, the classic theories of relativistic hydrodynamics introduced dissipative effects via terms which have derivatives of the standard hydrodynamic variables: fluid velocity u α , temperature T , and the chemical potential µ. Schematically, the energy-momentum tensor T αβ and the particle number current J α of the classic theories take the form of the following constitutive relations: T αβ , J α = O(T, u, µ) + O(∂T, ∂u, ∂µ) , (1.1) where the first term in the right-hand side corresponds to perfect fluids, and the second term encodes dissipative corrections due to viscosity and heat conductivity. The hydrodynamic equations are the conservation laws which give partial differential equations for T , u, and µ. The classic theories are often called "first-order" theories, because the above T αβ and J α contain only up to one derivative of the hydrodynamic variables. It was understood soon after that the classic hydrodynamic theories suffer from violations of causality. From a mathematical point of view, the hydrodynamic equations of the classic theories are not hyperbolic. Related to that, the classic theories predict that the uniform equilibrium state of a non-gravitating fluid in flat space is unstable. See Refs. [3] and [4] for an extensive discussion of instability and acausality in the classic theories.
The derivation of first-order causal hydrodynamics from kinetic theory has been addressed previously in refs. [9,11]. Our treatment here differs in several respects. We do not ignore the conserved particle number current, and we do not impose any "matching conditions" on the higher moments of the distribution function. We show how first-order causal hydrodynamics arises from the simplest derivative expansion (the Hilbert expansion) of the solutions to the Boltzmann equation, and that "good frames" arise simply as a "good" choice of the free functions which parametrize the zero modes of the linearized collision operator in real space.
The method by which hydrodynamics is extracted from classical gravity in asymptotically anti-de Sitter spacetimes via the fluid/gravity correspondence [18] is analogous in many ways to the Hilbert expansion in kinetic theory. The methods used to extract causal hydrodynamics from kinetic theory apply equally well to holography, allowing the construction of causal hydrodynamic theories with holographic duals.
The structure of the paper is as follows. In section 2, we show how causal first-order hydrodynamics arises from the Hilbert expansion in relativistic kinetic theory. In section 3, we show how an analogous procedure leads to causal hydrodynamics via the fluid/gravity correspondence. In appendices A and B we discuss examples of causal hydrodynamic frames, illustrating a causal choice of the zero modes. Finally, in appendix C, we comment on the "matching conditions" on the higher moments of the distribution function.

Boltzmann equation
Relativistic kinetic theory is an established subject [19,20]. The fundamental object of the theory is the one-particle distribution function f (x, p), where both the particle's location x and momentum p are four-vectors. The particles are on-shell, with p 0 ≡ (p 2 + m 2 ) 1/2 . The distribution function counts the particles, and is normalized so that the number density is The particle number current is the covariant version of the above: where p · · · ≡ d 3 p/[(2π) 3 p 0 ] . . . denotes the Lorentz-invariant integration measure. Similarly, the energy-momentum tensor is

Conservation laws
The Boltzmann equation is the evolution equation for the distribution function, where the right-hand side is the collision term which is an integral operator (in momentum) that acts on f p ≡ f (x, p), and is at least quadratic in f . The details of C depend on the interactions and on the statistics of the particles, and we will assume that the inter-particle interactions conserve energy, momentum, and particle number. A simple form to keep in mind is 2-to-2 elastic collisions 1 The transition rates obey W (p, p 1 |p 2 , p 3 ) = W (p 2 , p 3 |p, p 1 ) = W (p 1 , p|p 2 , p 3 ) = W (p, p 1 |p 3 , p 2 ), and are proportional to δ(p + p 1 − p 2 − p 3 ). It then follows that Alternatively, the conservation laws (2.7) can be viewed as the zeroth and first "moments" of the Boltzmann equation, i.e. they follow by applying p . . . and p p ν . . . to eq. (2.4), and are true for any f (x, p). The property (2.6) of the collision operator is a manifestation of the conservation of energy, momentum, and particle number. Approximate forms of the collision operator which violate eq. (2.6) (such as the relaxation-time approximation) are in general inconsistent with the conservation laws (2.7) which form the basis of hydrodynamics. 2

Equilibrium
Consider distribution functions of the form with arbitrary β µ (x), α(x). These are local Bose/Fermi distributions for a fluid at temperature T = 1/ −β 2 , with velocity u α = β α / −β 2 , and chemical potential µ = α/ −β 2 . The collision integral is such that it satisfies C[f ] = 0. If we want the function (2.8) to actually solve the Boltzmann equation, we need p µ p ν ∂ µ β ν + p µ ∂ µ α = 0, hence the functions β µ (x) and α(x) must satisfy Generalized to curved space, the first equation would say that β µ is a Killing vector.

Derivative expansion in hydrodynamics
Let us forget about kinetic theory for a moment, and consider hydrodynamics per se, with the constitutive relations written in the derivative expansion:

10)
1 Upper sign is for bosons, lower is for fermions. 2 A way to fix the relativistic relaxation-time approximation was discussed recently in ref. [21].
where γ is a formal derivative-counting parameter, and β µ = u µ /T are hydrodynamic variables: T is the temperature, and u µ is the fluid velocity. Truncating the above expansion at O(γ n ) gives n-th order hydrodynamics: T µν 0 = O(β) is the perfect fluid, T µν 1 = O(∂β) contains the viscosity, T µν 2 = O(∂ 2 β, (∂β) 2 ) etc. Let us look for the solutions using the same derivative expansion: We expect β n+1 = O(∂β n ). Expanding the constitutive relations, we have The hydrodynamic variables β are determined by solving ∂ µ T µν [β] = 0, order by order in γ.
At the leading order, the variables β 0 are determined by the perfect-fluid hydrodynamics: The first correction β 1 is then determined by At the next order, determines the correction β 2 , and the chain continues. The expansion (2.13), (2.14), (2.15) etc. naturally arises from the derivative expansion in both kinetic theory, and in the fluidgravity duality. Note, however, that this is not how the hydrodynamic equations are normally solved for hydrodynamic variables. In practice, the hydrodynamic constitutive relations are given to the desired order in γ, and then the conservation equations are solved "all at once" for β µ , as opposed to finding the order-by-order contributions β 0 , β 1 , etc. Such a procedure may lead to solutions which violate the small-derivative assumption of the expansion. The breakdown of the derivative expansion is a separate subject which we will not explore here.

Derivative expansion for the Boltzmann equation
The distribution function (2.8) with arbitrary non-constant β µ (x) and α(x) does not satisfy the Boltzmann equation. Approximate solutions to the Boltzmann equation may be constructed in the derivative expansion. To do so, we write the Boltzmann equation as with an auxiliary parameter ε (to be set to one at the end), and aim to find the solution as a power series in ε: This is sometimes called the Hilbert expansion [20]. The energy-momentum tensor (2.3) and the current (2.2) then take the form These expansions for T µν and J µ are however not necessarily the derivative expansions of the hydrodynamic constitutive relations. In order to talk about the constitutive relations, we need the hydrodynamic variables T , u λ , and µ, or equivalently β λ = u λ /T and α = µ/T . In kinetic theory, the hydrodynamic variables arise as arbitrary functions of x (or "integration constants" in momentum space) in the solutions of the Boltzmann equation. The x-dependence of these functions is then fixed by the consistency conditions for the Boltzmann equation at each order in the expansion. These consistency conditions are exactly the hydrodynamic conservation laws. Each order in the ε-expansion generates its own arbitrary functions, namely where the leading-order hydrodynamic variables β µ (0) = β µ and α (0) = α are the free functions that appear in the equilibrium distribution (2.8), and the corrections β µ (n) and α (n) appear as undetermined functions in the solution for f (n) (x, p). Connecting to the earlier discussion of the derivative expansion in hydrodynamics, we expect to find in the Hilbert expansion etc., with analogous expressions for the current J µ . We expect the conservation equations to hold at each order in the expansion, This is indeed what happens.

First order: The equation
At first order in the expansion we have We expand the collision operator to linear order in ε.
where L is the linearized collision operator. Its explicit form depends on the details of the full collision operator C, and in general one has with arbitrary a(x), b µ (x). The existence of these zero modes is a consequence of C[f ] = 0, reflecting the microscopic conservation laws of energy, momentum, and particle number. For 2-to-2 elastic collisions (2.5), the explicit form is First order: The constraint Given two functions g p ≡ g(x, p), h p ≡ h(x, p), the linearized collision operator satisfies In other words, at first order in the ε-expansion, the functions β µ (x) and α(x) that appear in the local-equilibrium distribution function (2.8) must obey These are the perfect-fluid conservation equations. The above T µν and J µ can be written as where ∆ µν ≡ g µν + u µ u ν , and the coefficients are corresponding to the ideal-gas particle number density, energy density, and pressure. In the notation of eq. (2.22a), T µν = T µν 0 , J µ = J µ 0 . The conservation equations (2.30) are where all quantities are of order O(ε 0 ). The dot stands for u λ ∂ λ , and ∂ ⊥ µ ≡ ∆ µν ∂ ν . The vector conservation equation can be rewritten asu µ + ∂ ⊥ µ T /T + nT ǫ+p ∂ ⊥ µ α = 0. Another way to arrive at eqs. (2.30) is to note that in the Boltzmann equation (2.25) the linearized collision operator L has zero modes, and therefore is not invertible. In general, the linear equation H = L[φ] can only be solved for φ if the left-hand side H is orthogonal to the zero-modes of the operator L in the right-hand side. For the linearized Boltzmann equation, the zero-modes are 1 and p λ , and the consistency conditions amount to This again gives eq. (2.30). In other words, the equations of 0 th -order (perfect-fluid) hydrodynamics arise as constraint equations at 1 st -order in the expansion.

First order: Homogeneous solution
At the first order in the expansion we have to solve eq. (2.25) which we write as where a(x) and b µ (x) are arbitrary, and the inhomogeneous solution Φ satisfies Φ| H→0 = 0.
Alternatively, when we evaluate the energy-momentum tensor T µν and the current J µ using the distribution function (2.37), the only effect of the "integration constants" b µ (x) and a(x) is a linearized redefinition of β µ (x) and α(x) in the perfect-fluid T µν and J µ . We thus identify the correction to the hydrodynamic variables in (2.20) and (2.21) as β µ (1) = b µ , α (1) = a, keeping in mind that b µ and a are arbitrary, hence the fluid velocity, temperature, and the chemical potential at O(ε) are intrinsically ambiguous quantities. Explicitly, the function (2.38) leads to the shift of T = T (0) , α = α (0) , and u µ = u µ (0) in the perfect-fluid expressions (2.31) by The resulting energy-momentum tensor and the current evaluated with the first-order distribution function (2.37) are: where the corrections T µν 1 , J µ 1 are due to the inhomogeneous solution Φ(x, p) in eq. (2.37).

First order: Inhomogeneous solution
The hard part is to find the inhomogeneous solution Φ which satisfies In general, for any timelike β µ (x) and α(x) we have the identity where the functions F σ , F u , F α depend on T , α, (p·u), and are fixed by the ideal-gas equation of state. In particular, F σ = 1 2T , F α = 1 + n ǫ+p (p·u). For massless particles, F u would vanish (at order ε), as a consequence of scale-invariant thermodynamics, p(T , α) = T d+1 g(α). Note that ∂ ⊥ µ T does not appear in the left-hand side of eq. (2.46), onceu µ has been eliminated. Now from eq. (2.46), the unknown Φ can be parametrized as where the coefficients K η , K ζ , K α in general depend on T , α, and p·u, and can in principle be found by solving the linearized Boltzmann equation (2.46). Let us write the first-order distribution function in terms of the first-order hydrodynamic variables β µ = β The first term has both O(1) and O(ε) contributions. In the second term, the O(ε) contributions in u, T , and α give O(ε 2 ) contributions to f p which can be neglected at first order.

First order: Constitutive relations
Beyond leading (perfect-fluid) order, the energy-momentum tensor and the current will no longer have the simple form (2.31). For any normalized timelike vector u µ , the energymomentum tensor and the current may be decomposed as [22] T µν = Eu µ u ν + P∆ µν + Q µ u ν + Q ν u µ + T µν , (2.49) where Q·u = J ·u = T ·u = 0, and T µν is symmetric and traceless. These decompositions define E, P, Q, T , N and J , for a given u µ . At first order in the ε-expansion, u µ = β µ / −β 2 , where β µ = β µ (0) + εβ µ (1) , as in Eq. (2.41). Similarly, at first order T = T (0) + εT (1) , and α = α (0) + εα (1) . The first-order corrections to β µ and α are arbitrary, and one can always Here ǫ, p, and n are functions of (ε-corrected) T and α. The angular brackets stand for · · · = p f p · · · . These are the constitutive relations for a viscous relativistic fluid at first order in the derivative expansion. The energy-momentum tensor and the current given by these constitutive relations (in terms of O(ε)-corrected hydrodynamic variables) must obey the standard conservation equations (2.7), which are true for any distribution function.
First order: Hydrodynamic "frames" One might be tempted to ignore the "integration constants" b ′ µ (x) and a ′ (x) altogether. However, they have a simple physical meaning: the hydrodynamic variables T , u λ , and α that appear in the O(ε) (i.e. Navier-Stokes) hydrodynamic equations can differ from the hydrodynamic variables that appear in the distribution function (2.8) by derivative corrections, reflecting the ambiguity in what one chooses to mean by "fluid velocity", "fluid temperature" and "fluid chemical potential" beyond the perfect-fluid approximation. The most general parametrization of such arbitrary one-derivative corrections is a ′ = a 1Ṫ /T + a 2 ∂·u + a 3α , (2.52b) with arbitrary coefficients b n (T, α) and a n (T, α). In relativistic hydrodynamics, one's choice of a particular form of these derivative corrections is often called a choice of "frame". The parametrization (2.52) contains the most general one-derivative corrections with arbitrary coefficients b n and a n . One could further demand that the redefinitions of T , α and u λ (provided by b ′ µ and a ′ ) are such that they vanish in equilibrium, even when the fluid is subject to a static external gravitational field. In equilibrium, one can choose the fluid velocity as the normalized timelike Killing vector. In zero-derivative hydrodynamics (perfect fluids) this is manifested by eq. (2.9), however such a choice of the fluid velocity in equilibruim of course extends beyond zero-derivative hydrodynamics, and has non-trivial consequences [23]. The Killing equation (2.9) for β µ impliesu µ + ∂ ⊥ µ T /T = 0, even thoughu µ and ∂ ⊥ µ T may separately be non-zero in external gravitational field. Thus demanding that (2.52a), (2.52b) vanish in equilibrium, we have b 4 = b 5 . Such a choice was called a "thermodynamic frame" in ref. [23]. The choice amounts to demanding that the hydrostatic limit of the constitutive relations (2.51) follows by varying the equilibrium grand canonical free energy with respect to the external metric (for T µν ), or with respect to the external gauge field (for J µ ).
The popular frame adopted by Landau and Lifshitz [2] is obtained in the following way.
The arbitrary coefficients b 2 and a 2 are fixed by demanding that E = ǫ+O(ε 2 ), N = n+O(ε 2 ). Following the constitutive relations (2.51), this determines b 2 and a 2 in terms of (p·u) 2 K ζ and (p·u)K ζ . After that, the non-equilibrium pressure takes the form P = p − εζ(∂·u) + O(ε 2 ), where ζ is the bulk viscosity, and we have used the on-shell relation m 2 = (p·u) 2 − p 2 ⊥ . For massless particles, we have m 2 = 0, ǫ = d p, and the above expression gives ζ = 0. Finally, the coefficient b 6 is fixed by demanding Q µ = O(ε 2 ). Following the constitutive relations (2.51), this determines b 6 in terms of p 2 ⊥ (p·u)K α . The particle number flux takes the form J µ = −εσT ∂ µ ⊥ α + O(ε 2 ), where σ is the particle number conductivity (which would become electrical conductivity if the particles were to carry electric charge), The frame of Eckart [1] is obtained in a similar manner. One chooses b 4 = b 5 (consistent with the thermodynamic frame), and sets b 1 = b 3 = b 6 = a 1 = a 3 = 0, so that The arbitrary coefficients b 2 and a 2 are fixed by demanding that E = ǫ+O(ε 2 ), N = n+O(ε 2 ), while b 4 is fixed by demanding J µ = O(ε 2 ). The bulk viscosity ζ again arises as the nonequilibrium correction to pressure, while the conductivity σ arises as the non-equilibrium contribution to the energy flux Q µ . The transport coefficients ζ and σ are physical observables, and do not depend on how one chooses to fix the arbitrary coefficients in eq. (2.52). For example, one could choose a frame where the bulk viscosity arises as a non-equilibrium correction to the energy density, while the pressure stays uncorrected to first order, P = p + O(ε 2 ). The actual values of ζ and σ are of course unchanged by where they appear in the constitutive relations [10].
In the above examples of Landau-Lifshitz and Eckart frames, the arbitrary coefficients b n (T, α) and a n (T, α) in eq. (2.52) were fixed by a choice of aesthetics. For example, in the Landau-Lifshitz frame the fluid velocity u µ appears as an eigenvector of the energymomentum tensor, while in the Eckart frame the equations resemble the historical formulation of the non-relativistic equations of compressible dissipative hydrodynamics. The idea behind BDNK hydrodynamics is: rather than being guided by aesthetics, the arbitrary coefficients b n and a n need to be chosen in a way that makes the resulting hydrodynamical equations mathematically well-posed. It is a non-trivial statement that it is in fact possible to choose the coefficients a n , b n such that the hydrodynamic equations are hyperbolic and causal. We illustrate this in appendices A and B.

Second order
Going to order O(ε 2 ) the Boltzmann equation becomes (2.57) Recall that the linearized collision operator is defined as where φ (n) ≡ f (n) /f (1 ± f ), and C (2) is quadratic in φ (1) , but does not contain φ (2) . Without specifying the explicit form of C (2) , it follows that for arbitrary a(x), b µ (x) we have where f (1) in the left-hand side is known from the O(ε) calculation in eqs. (2.37), (2.47), As before, the linear equation (2.60) can only be solved for φ (2) if the left-hand side is orthogonal to the zero-modes of the operator L in the right-hand side. The quadratic part C (2) drops out from the orthogonality condition thanks to eq. (2.59), and the constraint becomes p (a + b ν p ν ) p µ ∂ µ f (1) = 0, or equivalently Here J µ (1) and T µν (1) are given by eqs. (2.2), (2.3), evaluated with f (1) p in eq. (2.61). Connecting these expressions to T µν (1) in eq. (2.22b), the first term in (2.61) gives T µν 0,1 , the second term in (2.61) gives T µν 1 , and similarly for the current J µ (1) . In other words, the equations of 1-st order (Navier-Stokes) hydrodynamics arise as constraint equations at 2-nd order in the expansion. The same happens to all orders: the equations (2.23) of n th -order hydrodynamics arise as constraint equations at (n+1) th -order in the expansion.

Fluid/Gravity correspondence
In the preceding section, we have outlined a procedure to derive causal hydrodynamics from kinetic theory. There is an analogous procedure to derive hydrodynamic equations from classical gravity in asymptotically anti-de Sitter spacetimes. This is done via the fluid/gravity correspondence [18,24,25], see [26] for a review.

Einstein-Maxwell equations and Hilbert expansion
Following the original fluid-gravity discussion, we focus on the simplest holographic model of a 3+1 dimensional quantum field theory with a conserved global U(1) symmetry: the Einstein-Maxwell theory in AdS 5 , where latin indices M, N are bulk indices; greek indices, raised and lowered by the Minkowski metric η µν , will be used for the boundary directions. The AdS radius of curvature has been set to one, hence the cosmological constant is Λ = −6. The Einstein-Maxwell equations are The solution of (3.2) that corresponds to the equilibrium state in the dual field theory at nonzero temperature and non-zero U(1) charge density is the electrically charged black brane, (3.3b) The solution contains three constant parameters: a timelike covector u µ (normalized such that u µ u µ = −1), a charge Q, and a mass parameter b. As before, ∆ µν = η µν + u µ u ν is the spatial projector on the boundary. This metric is written in infalling Eddington-Finkelstein coordinates. The vector u µ defines the rest frame of the fluid on the boundary. The parameters b and Q are (somewhat unilluminating) functions of the temperature T and the U(1) chemical potential µ of the boundary fluid. The explicit expressions for b(T, µ) and Q(T, µ) can be obtained from refs. [24,25], in particular b(T, µ→0) = 1/πT and Q(T, µ→0) = 0. Drawing an analogy with kinetic theory, the equilibrium metric g and the equilibrium gauge field A of eq. (3.3) are the holographic analogues of the equilibrium distribution function f . If the parameters b, Q, and u µ are promoted to be functions of the boundary coordinates, i.e. b(x), u µ (x), Q(x), then (3.3) is no longer a solution to (3.2). However, in analogy with kinetic theory, we may construct approximate solutions through a Hilbert expansion of the form Similarly, the parameters themselves get corrected order-by-order as well: where the operator H, like the linearized collision operator L, depends only of the equilibrium metric g and equilibrium gauge field A, is the same at all orders in ε, and (crucially) has zero modes. In the same way that L involves integrals of p, the operator H involves derivatives with respect to r (compare with the interpretation of the r-direction as the energy scale in the dual field theory). The source term depends only on the lower-order corrections to the metric and the gauge field. The explicit expressions for H and s 0 , s 1 , s 2 may be found in [18,24,25]. The constraint equations in the bulk give rise to where T µν (n−1) and J µ (n−1) are the (n−1) th -order correction to the boundary stress-energy tensor and the U(1) charge current, respectively. Again, this is the exact same constraint that one finds in kinetic theory: the perfect-fluid equations come about as a constraint at first order, the Navier-Stokes equations arise as a constraint at second order, etc.
By direct comparison, one can see that b (1) = b(x), Q (1) = q(x), and u (1) µ = u µ (x). As the hydrodynamic variables β µ = u µ /T and α = µ/T are functions of b, Q, and u µ , the "integration constants" b, q, and u λ will set the hydrodynamic "frame". The corrections to the "conventional" hydrodynamic variables α = µ/T and β µ = u µ /T are given by The partial derivatives can be evaluated by inverting the known equilibrium functions b(T, µ) and Q(T, µ) to find T (b, Q) and µ(b, Q). Thus fixing b, q, and u µ is equivalent to fixing the definitions of the hydrodynamic variables β µ and α at O(ε). The original fluid-gravity references [18,24,25] adopted the Landau-Lifshitz convention, however tuning b, q, and u µ may be used to generate other conventions. In particular, hydrodynamic field redefinitions can be used to arrive at stable and causal first-order hydrodynamics as described in [10].

Conclusions
Physically, hydrodynamics is a theory of local densities of conserved quantities (energy, momentum, etc) which can not disappear through microscopic interactions, but rather spread out through the corresponding fluxes. On the other hand, when derived from a more fundamental microscopic description such as the kinetic theory or holography, classical hydrodynamics may be viewed as a theory of zero modes. In kinetic theory, the zero modes are those of the linearized collision operator L. In the fluid-gravity correspondence, the zero modes are those of the operator H. While the bulk fields in the fluid-gravity correspondence are the analogues of the distribution function, the operator H is the analogue of the linearized collision operator. Indeed, as was emphasized in ref. [26], the equations of bulk dynamics may be considered as a strong-coupling analogue of the Boltzmann equation. The freedom of choosing the zero modes at each order of the derivative expansion translates to the freedom of field redefinitions of the hydrodynamic variables. While in kinetic theory the zero modes are naturally associated with the shifts of β µ = u µ /T and α = µ/T which parametrize the equilibrium distribution function, the zero modes in the fluid-gravity correspondence are naturally associated with the shifts of b(T, α), u µ , and Q(T, α) which parametrize the equilibrium bulk metric and the gauge field. Still, hydrodynamic field redefinitions work in exactly the same way in both setups: neither the Hilbert expansion in kinetic theory nor the analogous expansion in fluid-gravity come with a preferred "frame". In both kinetic theory and in fluid-gravity one may obtain causal hydrodynamic equations through a judicious choice of zero modes at one-derivative order. We plan to return to further exploring the connections between the Botlzmann equation and the fluid-gravity duality in the future.
to the zero modes of the linearized collision operator, with the exception of the physical hydrodynamic field redefinitions can not generate causal frames because the latter require non-zero transport parameters in the scalar sector. Put differently, conditions (C.6) imply that once the frame-invariants f i vanish, the transport parameters ε i , π i , ν i must vanish as well, which is inconsistent with causal frames in first-order conformal hydrodynamics.
Similarly, for massive particles with K ζ = 0, eqs. (C.7) and (C.6) imply a = (0)Ṫ + (non-zero) ∂·u + (0)α , (C.8a) b·u = (0)Ṫ + (non-zero) ∂·u + (0)α . (C.8b) Again, we see that by fixing the zero modes via matching conditions (C.4), one is unable to generate a suitable mix of time-and space-derivatives required for hyperbolicity and causality. The only way to generate a causal frame via the matching conditions (C.4) would be if Φ p contained independent functions multiplyingṪ /T , ∂·u,α in the scalar sector. However, as the source term of the linearized Boltzmann equation must obey the perfect-fluid constraints, only one of these three functions is allowed in the inhomogeneous solution. The issue may be alleviated by taking moments of the full Boltzmann equation, as was done in [28] to study non-hydrodynamic contributions.