Equivalence of first-order stable hydrodynamics and Israel-Stewart theory for boost-invariant systems with constant relaxation times

We show that the recently formulated first-order stable hydrodynamics and the Israel-Stewart theory are strictly equivalent for boost-invariant systems with a conformal equation of state, however, with a non-conformal regulating sector determined by constant relaxation times.


Introduction
Relativistic hydrodynamics has become nowadays the basic theoretical tool for modeling relativistic heavy-ion collisions [1,2]. It forms the main ingredient of the so-called standard model of such processes, which essentially includes three segments: modeling of the early stage, hydrodynamic description of the space-time evolution of matter, and freeze-out of hadrons [3,4,5,6]. Detailed comparisons of theoretical predictions based on the hydrodynamic approach with the data, allow for the determination of various properties of strongly interacting matter such as its equation of state [7] and kinetic coefficients [8,9,10]. The latter include the shear and bulk viscosities. The presence of the shear viscosity affects the response of the hydrodynamic flow to the initial space-time anisotropies of colliding matter [11,12]. Development of hydrodynamic models applied for description of heavyion collisions, triggered broad studies of formal aspects of hydrodynamics treated as an effective theory describing systems approaching local thermodynamic equilibrium, for a recent review see [13]. Already in 1970s, it was realized that the relativistic dissipative hydrodynamics in the Landau and Eckart formulations was not causal [14,15,16] and they were replaced by the so-called second order hydrodynamics of Israel and Stewart (IS) [17]. The IS theory has been extensively used to describe heavy-ion collisions studied at the Relativistic Heavy Ion Collider (RHIC) at BNL and the Large Hadron Collider (LHC) at CERN. At the same time, more advanced hydrodynamic approaches has been developed, which removed some of the disadvantages of the IS formulation (for example, see [18,19,20,21,22]). Formal studies of hydrodynamics has led to very interesting observations such as an asymptotic character of the hydrodynamic gradient expansion [23,24,25] or the existence of attractors [26,27,28,29,30].
The IS theory treats the shear stress tensor π µν and the bulk pressure Π as independent hydrodynamic variables, in the way similar to the treatment of the local temperature T (x) and the hydrodynamic flow vector u µ (x). Only during the space-time evolution of the system, π µν and Π approach the Navier-Stokes values π µν = 2ησ µν and Π = −ζ∂ µ u µ (where η and ζ are the shear and bulk viscosity coefficients, respectively, and σ µν is the shear flow tensor constructed from the derivatives of u µ ).
Only very recently, a new stable hydrodynamic approach based on the linear expansion in gradients has been proposed in the works of F. S. Bemfica, M. M. Disconzi, J. Noronha, and P. Kovtun [31,32,33]. It is based on a more general choice of the hydrodynamic frame and introduction of a new set of the kinetic coefficients that play role of the regulators of the theory and allow to make it causal and stable.
A natural question can be asked how the new formulation (dubbed below shortly as FOS, for first-order and stable) compares to the traditional IS framework. In Ref. [31] it was shown, that the two approaches lead to very similar equations, if applied to boost-invariant and conformal systems. In this work we extend this study. We assume that the system's equation of state is conformal but allow for non-conformality in the regulating sector of the theory. We show that if the kinetic coefficients are expressed by constant relaxation times then it is possible to achieve exact matching between the FOS and IS formulations.
Throughout the paper we use natural units.

Israel-Stewart and first-order stable hydrodynamics
We consider the boost-invariant version of the Israel-Stewart theory which, in this case, is reduced to the two equations: where ε and p are the energy density and pressure, π is the rapidity-rapidity component of the shear stress tensor (should not be mistaken with the bulk pressure Π that is zero in our case), η is the shear viscosity coefficient, τ R is the relaxation time, and the parameter λ [34] is related to the τ ππ coefficient in the DNMR approach [35]. The evolution parameter τ = √ t 2 − z 2 is the longitudinal proper time. We note that the form of hydrodynamic flow for boost-invariant systems is dictated by the symmetry, u µ = (t/τ, 0, 0, z/τ ), hence it is independent of the choice of the hydrodynamic frame. We also note that all scalar function depend only on τ .
For the FOS approach [31,33,32], the evolution equations are reduced to the formula where the following constitutive relations are assumed, Here we have used the properties ∂ µ u µ = 1/τ and u µ ∂ µ f (τ ) = df /dτ , where f is an arbitrary function of the proper time τ . Assuming the conformal equation of state where a is a constant (and usually proportional to the number of internal degrees of freedom of particles forming a fluid), we rewrite Eq. (1) as Introducing the variable and taking the derivative of Eq. (6) with respect to τ we obtain This allows us to rewrite Eq. (2) as Equations (7) and (9) are two coupled differential equations for the functions T and y, which are completely equivalent to the two original IS equations. We note that Eq. (9) has the form of Ricatti's equation (ay ′ +by 2 +cy+d = 0, with b/a = 0 and c/a = 0) analyzed recently in more detail in [34].

Regulating sector in FOS
Let us discuss in more detail the regulating sector of the FOS approach. In natural units, the coefficient functions ε 1 , ε 2 , π 1 , and π 2 have dimension of energy cubed, so for conformal systems they should scale as T 3 . Similarly, in this case the IS relaxation time τ R should be inversely proportional to T , while η should scale with T 3 , yielding a dimensionless ratio of the shear viscosity η to the entropy density s. For strictly conformal systems one requires also that E = 3P, which implies that the π i coefficients are one third of the ε i coefficients. This choice has been made in [31], with an additional constraint that ε 1 = 3ε 2 .
In this work, we want to discuss yet another case, where the functions ε 1 , ε 2 , π 1 , and π 2 are expressed by constant relaxation times. This leads to the parametrizations of the type x i = x 0 i T 4 , where x i stands for any of the FOS coefficients mentioned above and x 0 i has dimension of time (fm). We think that this assumption is interesting from the point of view where the terms containing ε 1 , ε 2 , π 1 , and π 2 are interpreted as regulators. We know that the regularization or renormalization of the scale-invariant theory may introduce a certain scale, as it happens in the case of scale-invariant field theories such as QCD.
To include and discuss different cases together we rewrite Eq. (4) as where ε 0 1 , ε 0 2 , π 0 1 , and π 0 2 are dimensionless (n = 3) or dimensionful (n = 3) constants. The power n can take different values depending on the case we want to discuss. Substituting Eqs. (10) into Eq. (3) and using Eq. (7) we find Equations (7) and (11) are two first-order differential equations that can be treated as the basis of the FOS formulation for our geometry.

Comparison of the two frameworks
Using the parametrizations defined above we can formulate the IS and FOS frameworks in terms of the two differential equations for the functions T and y. A natural question is if these two formulations are identical. Since Eq. (7) is common for the two approaches, one has to check if Eqs. (9) and (11) are equivalent. After equating the terms standing at the same derivatives of the function y in Eq. (9) and (11) we find: One can easily notice that in the strictly conformal case, n = 3, it is impossible to match exactly the FOS and IS equations, however, they have the same mathematical structure of Ricatti's equation. The parametrization of Ricatti's equation in [31] uses a function χ that is related to our parametrization through the formula Moreover, in [31] one uses the relation π 0 1 = (1/3)ε 0 1 = ε 0 2 . A very interesting situation takes place for n = 4. In this case Eqs. (12) and (13) are fully consistent and the kinetic coefficient ε 0 1 has dimension of fm, so can be treated as a fixed relaxation time related to τ R that is also constant. Equations (14) and (15) determine the values of π 0 1 and π 0 2 in terms of the IS relaxation time, ε 0 1 and ε 0 2 . Although for n = 4 the system of equations (12)-(15) is underdetermined (only the expression π 0 1 + 4ε 0 2 is constrained), it can be adjusted to exactly match the IS system of equations.

Conclusions
In this work we have compared the recent first-order stable formulation of relativistic hydrodynamics with conventional Israel-Stewart theory. To make such a comparison feasible, we have restricted our study to boostinvariant, baryon-free systems with a conformal equation of state. In the strictly conformal case, where the regulator sectors of the theories are also conformal, the two approaches cannot be exactly matched, although they are based on the same system of differential equations. If the regulator sectors of the theories are determined by constant relaxation times, there exist a mapping between the FOS and IS approaches that makes them exactly the same.
Our results help to clarify mutual relations between FOS and more traditional formulations of relativistic dissipative hydrodynamics. Further investigations of more general systems are of course mandatory in this respect. Although for more complex system simple relations connecting FOS with second order hydrodynamic frameworks may not exist (since FOS yields four second-order equations which are in general equivalent to eight first-order equations, while IS is based on ten equations describing the time evolution of ten independent components of the symmetric energy-momentum tensor), it is in our opinion very interesting to identify the cases where such constructions are possible. This may help to understand why the first-order stable formulation may become an attractive alternative for more traditional hydrodynamic frameworks.
Acknowledgements. WF and RR would like to thank the participants of the Banff Workshop on Theoretical Foundations of Relativistic Hydrodynamics (19w5048) for inspiring discussions during the meeting, which triggered these investigations. This work was supported in part by the Pol-