# Field redefinitions and evolutions in relativistic Navier-Stokes

Yago Bea  Departament de Física Quàntica i Astrofísica and Institut de
Ciències del Cosmos (ICC), Universitat de Barcelona, Martí i Franquès 1,
ES-08028, Barcelona, Spain  Pau Figueras  School of Mathematical Sciences,
Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

###### Abstract

In recent years the equations of relativistic first-order viscous
hydrodynamics, that is, the relativistic version of Navier-Stokes, have been
shown to be well posed and causal under appropriate field redefinitions, also
known as hydrodynamic frames. We perform real-time evolutions of these
equations for a conformal fluid and explore, quantitatively, the consequences
of using different causal frames for different sets of initial data. By
defining specific criteria, we make precise and provide evidence for the
statement that the arbitrarily chosen frame does not affect the physics up to
first order, as long as the system is in the effective field theory regime.
Motivated by the physics of the quark-gluon plasma created in heavy-ion
collisions we also explore systems which are marginally in the effective field
theory regime, finding that even under these circumstances the first order
physics is robust under field redefinitions.

###### Contents

  1. 1 Introduction
  2. 2 First-order hydrodynamics and effective field theory
  3. 3 Real-time evolutions
    1. 3.1 Small perturbation of a homogeneous thermal state
      1. 3.1.1 The criteria
      2. 3.1.2 A solution marginally in the effective field theory regime
      3. 3.1.3 The choice of frame
      4. 3.1.4 Changing frame in the initial data
    2. 3.2 Large amplitude Gaussian perturbation
      1. 3.2.1 Changing frame in the initial data
    3. 3.3 Shockwaves
  4. 4 Discussion
  5. A Convergence tests

##  1 Introduction

Relativistic hydrodynamics is an effective field theory that provides the
real-time description of microscopic theories when the system is locally in
thermal equilibrium and gradient corrections are small. If the characteristic
macroscopic scale of the system is much larger than the microscopic scale of
the underlying theory, then ideal hydrodynamics should provide a good
description. However, if these two scales are not well separated, then viscous
effects are expected to become relevant.

Relativistic viscous hydrodynamics is well known to play a fundamental role in
the experimental description of the quark gluon-plasma (QGP) created in heavy-
ion collision experiments at RHIC and LHC [1, 2]. In this case the size of the
QGP droplet is comparable to the microscopic scale of quantum chromodynamics
(QCD). The data from the Beam Energy Scan experiment at RHIC [3] is currently
being analysed, and more experiments are planned for the near future at FAIR
and NICA. Relativistic viscous hydrodynamics also plays a relevant role in
astrophysical systems. For instance, in neutron star mergers, weak processes
during the post merger dynamics can give rise to an effective bulk viscosity
relevant for the scales of the system [4]. Moreover, magnetorotational
instabilities may act as the onset of turbulence, whose dynamics can be
effectively captured by introducing an effective viscosity [5]. Evaluating the
effect of viscosity in numerical evolutions of neutron star mergers is an
active area of research [6, 7, 8, 9, 10, 11, 12].

Both the QGP and neutron star mergers provide two experimental windows into
the hot and dense properties of the strongly interacting matter. They may
contribute to provide access to uncharted regions of the QCD phase diagram,
and relativistic viscous hydrodynamics is a key element to obtain the relevant
real-time dynamical evolutions. Moreover, the transport properties might be
useful to distinguish between different phases of matter.

The equations of relativistic first-order viscous hydrodynamics have their
origins in the works of Eckart [13] and Landau and Lifshitz [5]. From a more
modern perspective in the language of effective field theory, the two
different sets of equations proposed by Eckart and Landau and Lifshitz can be
obtained by using different field redefinitions of the fundamental variables
(temperature, velocity…) in a gradient expansion of the constitutive relations
of the stress tensor up to first order in derivatives.111In the Landau frame
the local fluid velocity is chosen such that there is no energy flow in the
local rest frame of the fluid, and in the Eckart frame the local fluid
velocity is chosen in such a way that there is no charge flow in the local
rest frame of the fluid. The fundamental variables are not uniquely defined
away from thermodynamic equilibrium and in this case the field redefinitions
of the local fluid velocity correspond to different extensions of these
quantities away from equilibrium with different physical meanings. Different
choices of field redefinitions are referred in the literature as ‘hydrodynamic
frames’. Since then, working in a specific hydrodynamic frame such as Eckart’s
frame or Landau’s frame became the usual approach. Written in this form, the
equations of first-order viscous hydrodynamics were observed to present
causality issues and instabilities [14]. That is, the relativistic
generalization of the Navier-Stokes equations, a theory that is meant to
provide the effective description of any relativistic viscous fluid, would
seem to be unphysical. This puzzling theoretical question remained unsolved
for many years.

Meanwhile the experimental analysis of the QGP created in heavy-ion collisions
required of some effective viscous relativistic hydrodynamical description. An
approach that provides such a description was introduced by Müller, Israel and
Stewart (MIS) [15, 16, 17], who proposed a set of equations for which the
causality and stability problems are absent. The strategy consists in
including second order terms in the gradient expansion and introduce extra
variables (and their corresponding evolution equations), in such a way that
the global set of equations has good causality and stability properties. There
are several formulations based on the same idea that we denote by MIS-type
theories, for example: BRSSS [18], DNMR [19], divergence-type theories [20,
21, 22], etc.. This approach has been very successful describing the
experimental data in heavy-ion collision experiments. In fact, Geroch famously
showed that all these different theories of relativistic dissipative
hydrodynamics provide the same physical predictions and should have the same
physical content as the Navier-Stokes equations as long as they are in the
regime of validity of effective field theory [23]. In spite of this success,
the question of why the relativistic generalization of Navier-Stokes appeared
to be unphysical remained unsolved. A solution was proposed in recent years.

Bemfica, Disconzi, Noronha [24, 25, 26] and Kovtun [27] proposed a formulation
of the relativistic Navier-Stokes equations which is well-posed, causal and
stable. Such a formulation has become known as the ‘BDNK equations’. The key
insight was to realize that by performing field redefinitions up to first
order in the hydrodynamic gradient expansion, one can change the highest
derivative terms in the equations of motion (i.e., the principal part) and
achieve local well-posedness of the initial value problem (Cauchy problem),
relativistic causality and dynamical stability of the homogeneous thermal
states. Thus, the solution to obtain a mathematically and physically well
behaved relativistic version of the Navier-Stokes equations is to perform an
appropriate field redefinition, or choice of hydrodynamic frame. Traditionally
used frames such as the Landau or Eckart’s frame do not belong to this set of
allowed frames.

Having at our disposal this novel theory, we may wonder: why do we need
another formulation of relativistic viscous hydrodynamics if the MIS-type
theories were successful describing the experimental data? We provide
arguments why first-order viscous hydrodynamics might be a good alternative to
MIS-type theories.

An important advantage of first-order viscous hydrodynamics theories over MIS-
type theories are the well posedness properties of the initial value problem.
Local well posedness of the initial value problem has been well established
and studied for many physically relevant sets of equations such as the Maxwell
equations, ideal hydrodynamics or the Einstein equations of general
relativity. However, there are limited results in the context of relativistic
viscous hydrodynamics. It was not until very recently that a study was
performed for a set of MIS-like equations [28]. The result is that these
theories present some limitations, for example, local well posedness may
depend on the specific state under consideration. Thus, the conditions must be
checked pointwise in spacetime for each solution. These conditions were shown
to fail in realistic simulations of the QGP created in heavy-ion collisions
experiments [29]. In contrast, for the relativistic Navier-Stokes equations,
once a causal frame has been chosen, local well posedness is satisfied
independently of the specific state. Thus, this is a clear advantage of
relativistic Navier-Stokes over MIS.

Related to the previous point there is another argument regarding the
principles of relativity. For the MIS equations the ability to propagate
information (more precisely, the characteristic velocities of the PDEs) might
be larger than the speed of light, and again this depends on the specific
state and must be checked pointwise in every evolution. In contrast, for the
relativistic Navier-Stokes equations the characteristic velocities are fixed
once the frame has been chosen, and they do not depend on the specific state.
Choosing a frame for which the characteristic velocities are not larger than
the speed of light ensures that every evolution performed with those equations
respects the principles of relativity. In MIS it cannot be ensured that the
solution respects the principles of relativity, and this has to be checked for
every solution.

Another important advantage of relativistic Navier-Stokes is that it is able
to capture arbitrarily strong shockwave solutions, while MIS-theories may
present limitations evolving large shocks; this may be an advantage in
astrophysical applications. The ability to evolve large shocks is related to
the characteristic velocities, and in MIS-type theories, these depend on the
state and might be less than the speed of light. A shock with velocity larger
than the characteristic velocities cannot be evolved with the MIS equations
and hence their predictive power is lost in these situations. In relativistic
Navier-Stokes, once the frame has been fixed, also the characteristic
velocities are fixed. Moreover, frames can be chosen such that these
characteristic velocities are equal to the speed of light. Mathematical
results in this direction were obtained in [30] and numerical studies were
performed in [31], providing evidence that arbitrarily large shocks can be
evolved with the relativistic Navier-Stokes equations.

An advantage of MIS-type formulations is that the numerical infrastructure is
already implemented in the context of the QGP. For relativistic Navier-Stokes
this infrastructure is still under construction and this paper aims to
contribute towards this goal.

The above arguments suggest that first-order viscous hydrodynamics might be a
very good alternative to MIS-type theories in aspects which are relevant for
an appropriate description of physical systems of interest like neutron star
mergers or the QGP created in heavy-ion collisions. However, for a practical
implementation of the first-order viscous hydrodynamics equations it is
relevant to address the following question. Now that we are not using
traditionally preferred frames like the Landau frame, what frame should we use
in actual numerical evolutions? In principle, one can use any frame within the
set of causal frames. Actually, if the system is within the effective field
theory regime, then it should not matter which specific frame is being used,
and using different frames should lead to an equivalent physical description.
Equivalently, if the system is within the effective field theory regime, the
physics to first order should be independent of the arbitrarily chosen frame.
If not, the equations would not have predictive power. Numerical studies of
the first-order viscous hydrodynamics equations were performed in [32, 31, 33,
34]. These papers constitute a proof of principle that first-order viscous
hydrodynamics produces physically sensible real-time evolutions. Moreover,
numerical techniques have been studied, with extensions to non conformal and
charged systems.

In this paper we go beyond the state of the art by performing a detailed,
quantitative study of the effect of field redefinitions in numerical
evolutions of the relativistic Navier-Stokes equations for a conformal theory.
For this purpose we consider different sets of initial data: a small amplitude
perturbation of a homogeneous thermal state, a large amplitude Gaussian
perturbation of a homogeneous state and shockwave solutions. We define a
specific set of criteria to assess different conditions on the numerical
solutions. By using these criteria, we make precise and provide evidence for
the statement that the arbitrarily chosen frame does not affect the physics to
first order, as long as the system is in the effective field theory regime.
Motivated by the physics of the QGP, we also study the effect of field
redefinitions in situations where the system is marginally in the effective
field theory regime. Furthermore, we also study the effect of changing the
frame in the initial data.

In this paper we use equivalently the terminology ‘relativistic first-order
viscous hydrodynamics equations’, ‘relativistic Navier-Stokes equations’ and
‘BDNK equations’. From the perspective of effective field theory it is natural
to use ‘relativistic Navier-Stokes equations’ [27, 35]. We define it as a
collection of sets of partial differential equations (PDEs), where each set of
PDEs is obtained by choosing a hydrodynamic frame and truncating the
hydrodynamics gradient expansion of the constitutive relations up to first
order in derivatives and plugging it into the conservation equation for the
stress tensor. By using different frames the truncations are different,
obtaining a different set of PDEs for each frame. Each set of PDEs can be
considered as a different version of relativistic Navier-Stokes, like Eckart’s
or Landau’s versions; it is expected that they all describe equivalent first-
order physics as long as the solutions are in the effective field theory
regime. The main results of this paper provide a quantitative assessment of
this point of view.222See [36] for a detailed and rigorous study of the
validity of effective field theory in the context of the Abelian Higgs model,
where similar issues arise.  The reader interested in a summary and
conclusions may jump to Section 4 Discussion.

##  2 First-order hydrodynamics and effective field theory

We consider a conformal fluid in 3+1 dimensions in Minkowski spacetime.
Conformal symmetry fixes the equation of state

$p=\frac{\epsilon}{3}\,,$ (2.1)

where $\epsilon$ is energy density and $p$ is pressure. It also imposes that
the energy is proportional to $T^{4}$, $\epsilon=\mathcal{C}\,T^{4}$, where
$T$ is the temperature and $\mathcal{C}$ is a constant that we choose
$\mathcal{C}=\frac{3}{4}\pi^{4}$.333We choose this constant motivated by
holographic fluids. The conservation of the stress-energy tensor

$\nabla_{\mu}T^{\mu\nu}=0\,,$ (2.2)

provides the evolution equations for the relativistic fluid. Considering the
4-velocity of the fluid $u^{\mu}$, normalized such that $u^{2}=-1$, we define
$\Delta^{\mu\nu}\equiv g^{\mu\nu}+u^{\mu}\,u^{\nu}$, $\dot{\epsilon}\equiv
u^{\mu}\nabla_{\mu}\epsilon$,
$\nabla_{\perp}^{\mu}\equiv\Delta^{\mu\nu}\nabla_{\nu}$, $\nabla\cdot
u\equiv\nabla_{\rho}u^{\rho}$ and $\sigma^{\mu\nu}\equiv
2\nabla^{\langle\mu}u^{\nu\rangle}$, where

$\displaystyle
A^{\langle\mu\nu\rangle}\equiv\frac{1}{2}\Delta^{\mu\alpha}\Delta^{\nu\beta}(A_{\alpha\beta}+A_{\beta\alpha})-\frac{1}{3}\Delta^{\mu\nu}\Delta^{\alpha\beta}A_{\alpha\beta}\,,$
(2.3)

is symmetric, traceless and transverse to $u^{\mu}$.

We start by presenting the equations of ideal hydrodynamics. The constitutive
relations of ideal hydrodynamics are

$T^{\mu\nu}=\epsilon\left(u^{\mu}\,u^{\nu}+\frac{1}{3}\,\Delta^{\mu\nu}\right)\,.$
(2.4)

The conservation of the stress-energy tensor (2.2) with (2.4) provides
evolution equations for the dynamical variables, namely $\epsilon$ and the
independent components of the velocity vector $u^{\mu}$,

$\displaystyle\frac{3}{4}\frac{\dot{\epsilon}}{\epsilon}+\nabla\cdot u$
$\displaystyle=0\,\,,$ (2.5a)
$\displaystyle\dot{u}^{\mu}+\frac{1}{4}\frac{\nabla_{\perp}^{\mu}\epsilon}{\epsilon}$
$\displaystyle=0\,\,.$ (2.5b)

The constitutive relations of relativistic hydrodynamics up to first order in
the gradient expansion in the Landau frame444The Landau frame is defined by a
local fluid velocity that satisfies $u_{\mu}T^{\mu\nu}=\epsilon\,u^{\nu}$,
where $\epsilon$ is the energy density in the local rest frame of the fluid.
can be written as

$T^{\mu\nu}=\epsilon\left(u^{\mu}\,u^{\nu}+\frac{1}{3}\,\Delta^{\mu\nu}\right)-\eta\,\sigma^{\mu\nu}+O(\partial^{2})\,,$
(2.6)

In the spirit of effective field theory we consider the most general field
redefinition of the fundamental variables $\\{\epsilon,u^{\mu}\\}$ by first
order terms compatible with Poincaré and conformal symmetries. There are two
Lorentz scalars $\dot{\epsilon}$, $\nabla\cdot u$ and two Lorentz vectors
$\dot{u^{\mu}}$, $\nabla_{\perp}^{\mu}\epsilon$ that we can construct with the
first derivatives of the dynamical fields. However, conformal symmetry
uniquely determines the most general field redefinition [27]

$\displaystyle\epsilon$
$\displaystyle\rightarrow\epsilon+a_{1}\,\eta\left(\frac{3}{4}\frac{\dot{\epsilon}}{\epsilon}+\nabla\cdot
u\right)+O(\partial^{2})\,\,,$ (2.7a) $\displaystyle u^{\mu}$
$\displaystyle\rightarrow
u^{\mu}+\frac{3\,a_{2}\,\eta}{4\epsilon}\left(\dot{u}^{\mu}+\frac{1}{4}\frac{\nabla_{\perp}^{\mu}\epsilon}{\epsilon}\right)+O(\partial^{2})\,\,.$
(2.7b)

In these equations, the overall factor multiplying the terms in brackets has
dimensions fixed by conformal symmetry, and we choose it to be proportional to
the shear viscosity $\eta$ times a constant, which we denote by $a_{1}$ or
$a_{2}$ respectively. By performing a general field redefinition (2.7) of the
constitutive relations in the Landau frame (2.6) and neglecting higher order
terms, we obtain the stress tensor up to first order in the gradient expansion
in a general hydrodynamic frame

$\displaystyle T^{\mu\nu}=$
$\displaystyle~{}\epsilon\left(u^{\mu}\,u^{\nu}+\frac{1}{3}\,\Delta^{\mu\nu}\right)-\eta\,\sigma^{\mu\nu}$
(2.8)
$\displaystyle+a_{1}\eta\left(\frac{3}{4}\frac{\dot{\epsilon}}{\epsilon}+\nabla\cdot
u\right)\left(u^{\mu}\,u^{\nu}+\frac{1}{3}\,\Delta^{\mu\nu}\right)$
$\displaystyle+a_{2}\eta\left[\left(\dot{u}^{\mu}+\frac{1}{4}\frac{\nabla_{\perp}^{\mu}\epsilon}{\epsilon}\right)u^{\nu}+(\mu\leftrightarrow\nu)\right]+O(\partial^{2})\,.$

The two real numbers $\\{a_{1},a_{2}\\}$ in (2.8) specify the hydrodynamic
frame. The Landau frame constitutive relations (2.6) correspond to
$\\{a_{1},a_{2}\\}=\\{0,0\\}$. The stress tensor (2.8) can be equivalently
obtained by considering the most general stress tensor up to first order in
the gradient expansion compatible with Poincaré and conformal symmetries [26,
27]. The first order terms in (2.8) are the shear term
$-\eta\,\sigma^{\mu\nu}$, which is field redefinition independent, and the
terms in second and third lines of (2.8), which we refer to as the ‘$a_{1}$
term’ and ‘$a_{2}$ term’ respectively and depend on the choice of frame. We
use the notation $T_{\mu\nu}^{(0)}$ to refer to the ideal part and
$T_{\mu\nu}^{(1)}$ to the first order part in (2.8). The equations of first-
order viscous hydrodynamics are obtained by plugging (2.8) into the
conservation equation for the stress tensor (2.2), and are of second order in
derivatives.

Inspired by the works of Van and Biro [37] and Freistuhler and Temple [38, 39,
40], Bemfica, Disconzi and Noronha [24, 25, 26] and Kovtun [27], showed that
in some frames the equations of first-order hydrodynamics present good
causality and stability properties. The key observation is that different
hydrodynamic frames give rise to different evolution equations at the level of
two derivatives. More precisely, the principal part of the equations, that is,
the highest derivative terms, depends on the constants $a_{1}$, $a_{2}$ and
hence on the choice of frame. Local well posedness and the causality
properties of the PDEs are determined by the principal part of the equations
and these references found that in some frames the equations have these good
properties. More specifically, for the conformal fluid it was proven [24, 41,
42] that if

$a_{1}\geq 4\,,~{}~{}~{}~{}a_{2}\geq\frac{3a_{1}}{a_{1}-1}\,,$ (2.9)

then the PDEs (2.2) with (2.8) are strongly hyperbolic and the initial value
problem is locally well posed.555By ‘well posed’ we mean that the solution
locally exists, is unique and it depends continuously on the initial data,
which should be in a suitable Sobolev space. Also, the characteristic
velocities can be chosen to be no larger than the speed of light, thus
respecting the principles of relativity.

These results provide an answer to the longstanding question of why first-
order viscous hydrodynamics had apparent ill causality properties: it is just
a matter of choice of hydrodynamic frame and it can be solved by an
appropriate field redefinition. The Landau frame $\\{a_{1},a_{2}\\}=\\{0,0\\}$
lies outside the region (2.9), and there is a gap between the causal frames
and the Landau frame.

Another important aspect of the first-order viscous hydrodynamics equations is
that in the Landau frame the equilibrium states are unstable [14]. Thus, apart
from the local well posedness and causality properties, it is also important
to discuss the stability of equilibrium states under the conditions (2.9), and
this was shown in [24, 27]. A relevant result in this direction was recently
proven in [26], who showed that under the conditions (2.9), if a thermal
equilibrium state is stable in one Lorentz frame, then it is stable in all
Lorentz frames.

A natural question that arises if one is familiar with working in the Landau
frame is the physical meaning of the ‘extra terms’ $a_{1}$ and $a_{2}$ in
(2.8). We now provide intuition about the significance of these terms.

We start by emphasizing that the field redefinition dependent terms in (2.8),
that is, the $a_{1}$ and $a_{2}$ terms, are proportional to the lower order
equations of motion, namely the ideal hydrodynamics equations (2.5). Thus,
upon using the equations of motion, these terms are equivalent to second order
terms [26]. Actually, this is a general principle in effective field theory:
when a term at a given order in the derivative expansion is proportional to
the lower order equations of motion, it can be pushed to a higher order by a
field redefinition. Therefore, on shell, the $a_{1}$, $a_{2}$ terms are
effectively of second order and hence one should expect that their
contribution to the first order physics is negligible, as long as the system
is in the effective field theory regime. In other words, in the effective
field theory regime they are smaller than the shear viscosity term, which is
of first order and field redefinition independent. Thus, changing frame should
leave the physics to first order invariant, as it is a second order
effect.666In charged fluids the physical interpretation of the local fluid
velocity is different in the Landau and Eckart frames, and this is because the
change of frame is first order but not proportional to the ideal equations of
motion. In the conformal and uncharged fluids studied in this paper, all field
redefinitions are second order on shell and so the physical interpretation of
the fundamental variables does not change under field redefinitions. This is a
formal statement in effective field theory, where gradients are assumed to be
infinitesimally small. However, in realistic situations the gradients are
finite, and hence it would be desirable to have some specific quantitative
criteria to assess whether the higher order gradients are indeed negligible
compared to the first order ones. In the following we suggest some criteria
that capture different aspects relevant for this discussion; more precise
definitions of these criteria will be given in Section 3.1.

  * •

Criterion A The ‘size’ of each term evaluated on a solution of the viscous
equations at given order in the hydrodynamic gradient expansion is smaller
than any of the lower order ones.

  * •

Criterion B The ‘size’ of the $a_{1}$ and $a_{2}$ terms in (2.8) evaluated on
a solution of the viscous equations are much smaller than the shear term.

  * •

Criterion C The difference between two solutions of the viscous equations
obtained in two different frames is much smaller than the difference between
the solution of the ideal equations and any of the solutions of the viscous
equations.

In practice, to check Criterion A we will evaluate the ideal and first order
terms in (2.8) on our numerical solutions. Criteria B and C are two different
and complementary criteria to make precise when the physics to first order in
the gradient expansion is independent of the choice of hydrodynamic frame. For
Criterion C to be meaningful, we need to require that the different causal
frames are sufficiently distinct from each other, otherwise, by continuity and
Cauchy stability, two different frames corresponding to values of $a_{1}$ and
$a_{2}$ that are sufficiently close will give solutions that are trivially
close to each other even if they are not in the effective field theory regime.
For this reason, in this paper we will compare situations in which the frames
$\\{a_{1},a_{2}\\}$ differ by a factor of 2 at least. Typically used values
will be $\\{a_{1},a_{2}\\}=\\{5,5\\}$ and $\\{a_{1},a_{2}\\}=\\{10,10\\}$.

We will consider different sets of initial data to obtain solutions to the
viscous equations that we will use to analyse in detail the different criteria
above. In particular, we study situations in which the system is marginally in
the effective field theory regime. This is motivated by the physics of the QGP
created in heavy-ion collision experiments, where at the initial stages of the
hydrodynamic evolution gradients might not be small and hence it is not clear
that the system is within the effective field theory regime. We will explore
the behaviour under change of frames by employing criteria B and C in this
limiting situation in which Criterion A might be violated. Also motivated by
the physics of the QGP, we will address the following question: given some
initial data, how do we change from the usual Landau frame to the chosen
causal frame? To do so, we will study the effect of the change of frame in the
initial data.

The equations of first-order relativistic hydrodynamics (2.2) with (2.8) in a
frame satisfying (2.9) can be understood as a UV completion of hydrodynamics
in the following sense. The equations are obtained after truncating the
gradient expansion of the stress-energy tensor to first order in derivatives
(2.8), and then plugging the latter into (2.2). In general, solutions to these
truncated equations will not be in the effective field theory regime: they
could be arbitrarily far from equilibrium, for example setting initial data
with large gradients.777In particular, the theory (2.8) contains non-
hydrodynamic modes that depend on the choice of frame. Similarly, MIS-like
theories also contain non-hydrodynamic modes and can be thought of as
different UV completions.  The fact that there are solutions to the viscous
hydrodynamics equations that are not in the effective field theory regime is a
generic feature in truncated effective field theories. If we want that a
solution of the truncated theory to correspond to the long distance
description of a solution of the microscopic theory, the former must be in the
effective field theory regime at all times. Let us emphasize that this is the
case not only for this particular set of equations; the same happens with the
equations of ideal hydrodynamics and other theories of viscous hydrodynamics
such as MIS-type equations. The fact that we construct solutions to the
equations of (viscous) hydrodynamics does not necessarily mean that such
solutions are consistent; for instance, there may be physical situations in
which solutions explore the UV of theory, e.g., turbulence in 3+1 dimensions,
and preventing a flow of energy to UV in an ad-hoc way in the truncated theory
may be unphysical. We also emphasize that when we state that the system is in
the ‘regime of hydrodynamics’ it may sound trivial because we are solving the
hydrodynamics equations, but it is not: we mean that the system is within the
effective field theory regime, and this is in general not true for certain
solutions of the equations of hydrodynamics.

To assess if a solution is in the regime of hydrodynamics one would typically
compare ideal and first order terms evaluated on the solution, that is,
Criterion A. However, we can try to do better. One motivation is the
following: there might be situations in which the system is away from the
regime of hydrodynamics and still first order terms are smaller than ideal
terms, so Criterion A would fail. One example is adding a perturbation to a
homogeneous thermal state with a very small amplitude and large momentum: the
small amplitude can make the first order terms small even if the momentum is
large compared to the temperature. In that case the ratio of second order
terms to first order terms would be large, indicating that the system is not
in the regime of hydrodynamics, but the ratio of the first order terms to
ideal terms is small. Another example is the Gaussian initial data that we
explore in section 3.2: at $t=0$ gradients might be arbitrarily large (for
sufficiently small width of the Gaussian) and yet the shear term vanishes.
Then, can we improve our practical assessment of whether the system is in the
regime of hdyrodynamics? We now provide one possible idea.

Nothing prevents us from measuring the size of the higher order gradients of
our specific solutions; in particular, we can evaluate the second derivative
terms in the gradient expansion of the stress-energy tensor on the solutions
of the first order theory. Note that doing so is perfectly consistent in our
scheme since we are solving the classical equations of motion, which are of
second order in derivatives; therefore, the second order terms in the gradient
expansion are well-defined in our solutions.888The presence of viscosity
smoothes out shock solutions, which therefore are continuous. For this
purpose, we collect here the constitutive relations of relativistic viscous
hydrodynamics up to second order in the derivative expansion in the Landau
frame for a conformal fluid [18]

$\displaystyle T^{\mu\nu}=$
$\displaystyle~{}\epsilon\,u^{\mu}\,u^{\nu}+p\,\Delta^{\mu\nu}-\eta\,\sigma^{\mu\nu}$
$\displaystyle+\eta\,\tau_{\pi}\left(\dot{\sigma}^{\langle\mu\nu\rangle}+\frac{1}{3}\,\sigma^{\mu\nu}\,\nabla\cdot
u\right)+{\lambda_{1}}\,{\sigma^{\langle\mu}}_{\rho}\sigma^{\nu\rangle\rho}+{\lambda_{2}}\,{\sigma^{\langle\mu}}_{\rho}\Omega^{\nu\rangle\rho}+\lambda_{3}\,{\Omega^{\langle\mu}}_{\rho}\Omega^{\nu\rangle\rho}\,,$
(2.10)

where $\eta$ is the shear viscosity, $\tau_{\pi}$, $\lambda_{1}$,
$\lambda_{2}$, $\lambda_{3}$ are second order transport coefficients and
$\Omega_{\mu\nu}=\Delta_{\mu}^{\phantom{\mu}\alpha}\Delta_{\nu}^{\phantom{\nu}\beta}\partial_{[\alpha}u_{\beta]}$
is the vorticity tensor. We will use the second line of (2.10) to measure the
size of the second order gradients in our solutions. Note that (2.10) is in
the Landau frame, and our solutions will be in a different (causal) frame. We
could change the hydrodynamic frame of our solutions to Landau frame and
evaluate (2.10) in the Landau frame. However, the contribution of the terms
corresponding the the change of frame to the second line in (2.10) is of
higher order, and hence negligible if the solution is in the hydrodynamic
regime. Having a measurement of the size of the second order terms allows for
a more detailed assessment of when the solution is in the regime of
hydrodynamics since we can compare these second order terms with the ideal and
first order terms. Hence, we will state that the system is in the effective
field theory regime when there is a clear hierarchy between the ‘sizes’ of the
ideal, first order and second order terms respectively. In practice, when
considering Criterion A, we will include the second order terms even though we
do not use them to obtain the solutions.

##  3 Real-time evolutions

In this section we perform a detailed study of real-time numerical evolutions
of the relativistic first-order viscous hydrodynamics equations of a conformal
fluid in 3+1 dimensions. These are obtained by plugging (2.8) into the
conservation equation (2.2). We perform evolutions in Minkowski spacetime and
we restrict the dynamics in 1+1 dimensions, assuming homogeneity along the
other two spatial dimensions. We use Cartesian coordinates $(t,x,y,z)$ and we
choose the dynamics to take place along the $x$-direction. Our evolution
variables are $\\{\epsilon,u_{x}\\}$, where $u_{x}$ is the $x$ component of
the 4-velocity $u^{\mu}$. The numerical code that we use was presented in
[32], see Appendix A of that paper for details.999Even if our code evolves the
equations in a 2+1 dynamical set up, in this paper we present evolutions with
dynamics in 1+1 dimensions for presentation purposes: it is enough for the
physics that we want to explore and easier to present. We do not expect that
the qualitative conclusions obtained in this paper depend on the
dimensionality of the dynamics of the problem. In Appendix A we include some
convergence tests for the runs presented in this paper.

In our evolutions we use hydrodynamic frames $\\{a_{1},a_{2}\\}$ that satisfy
the causality conditions (2.9). Typical values used in this paper will be
$\\{a_{1},a_{2}\\}=\\{5,5\\}$ or $\\{a_{1},a_{2}\\}=\\{10,10\\}$, but we will
also use other frames that will be specified in due course. We use the
conformal equation of state (2.1) and the shear viscosity

$\eta=\frac{s}{4\pi}~{},$ (3.1)

where $s$ is the entropy density. We use the value of shear viscosity (3.1) of
large-$N$ and large (infinite) coupling holographic theories, which is a
universal result within holographic theories with an Einstein gravity dual
[43, 44]. Another motivation for choosing (3.1) is that this value is very
close to the shear viscosity measured in the QGP created in heavy-ion
collision experiments [45].

Even if in this paper we only perform evolutions of the first-order viscous
hydrodynamics equations, in order to assess if our solutions are in the
effective field theory regime we will evaluate pointwise in spacetime the
second line of (2.10), as explained above. For this purpose, we use the
following second order transport coefficients [46, 18]

$\displaystyle\eta\,\tau_{\pi}$ $\displaystyle=\frac{s}{8\pi^{2}T}\left(2-\ln
2\right)~{},$ (3.2a) $\displaystyle\lambda_{1}$
$\displaystyle=\frac{s}{8\pi^{2}T}~{},$ (3.2b) $\displaystyle\lambda_{2}$
$\displaystyle=\frac{s}{8\pi^{2}T}\left(-2\ln 2\right)~{},$ (3.2c)
$\displaystyle\lambda_{3}$ $\displaystyle=0~{}.$ (3.2d)

These are the coefficients of a set of 3+1 dimensional holographic conformal
field theories, such as $\mathcal{N}=4$ Super Yang Mills, which admit a dual
gravitational description in terms the Einstein-Hilbert action plus a negative
cosmological constant. In planar 1+1 dynamics only $\tau_{\pi}$ and
$\lambda_{1}$ are relevant, as the tensors multiplying $\lambda_{2}$,
$\lambda_{3}$ vanish. We use these coefficients because in these holographic
theories these coefficients are known; in other frameworks, obtaining these
coefficients from a microscopic quantum field theory may be challenging.

We will start by performing evolutions of a small perturbation of a
homogeneous thermal state. We consider small amplitude sinusoidal
perturbations with a large wavelength, and thus by construction this system is
close to the linear hydrodynamic regime. We evolve the equations at the full
non-linear level, but being close to the linear regime gives us good control
of the system. This is particularly useful when studying the effect of field
redefinitions. More specifically, by continuously varying the momentum we can
have a system that is well within the regime of hydrodynamics (low momentum),
or far from equilibrium (large momentum), thereby allowing us to assess the
effect of field redefinitions in situations where the system may be on the
verge of exiting the regime of hydrodynamics.

We will continue our studies by considering initial data corresponding to a
thermal state deformed by a Gaussian with a large amplitude to explore the
non-linear regime of the theory. By tuning the width of the Gaussian profile,
we can smoothly interpolate between the regime of hydrodynamics (large widths)
and the far from equilibrium regime (very small widths).

Finally, we continue exploring the non-linear regime of the theory in more
extreme situations by studying shockwave solutions. By changing the amplitude
of the shockwave we can take the system to be well in the regime of
hydrodynamics (small shocks) or far from equilibrium (large shocks).

###  3.1 Small perturbation of a homogeneous thermal state

We start by considering a physical system in which we can study the properties
of the first-order viscous hydrodynamics equations under well controlled
conditions: a small amplitude sinusoidal perturbation of a homogeneous thermal
state. By choosing large wavelength perturbations, the system is by
construction in the regime of (linear) hydrodynamics. Choosing a small
amplitude is useful because we can obtain some intuition from linear
hydrodynamics; we use this intuition to analyse our solutions, that are
obtained by solving the full non-linear equations.

We consider a thermal homogeneous state with temperature $\overline{T}$ and
energy $\overline{\mathcal{E}}=\frac{3}{4}\pi^{4}\,\overline{T}^{4}$, and a
sinusoidal perturbation of amplitude $0.01\overline{\mathcal{E}}$ and momentum
$k/\overline{T}\simeq 0.184$. Specifically, the initial data is as follows:

$\displaystyle\epsilon|_{t=0}$
$\displaystyle=\overline{\mathcal{E}}\left(1+0.01\cos(kx)\right)\,\,,$ (3.3a)
$\displaystyle\partial_{t}\epsilon|_{t=0}$ $\displaystyle=0\,\,,$ (3.3b)
$\displaystyle u_{x}|_{t=0}$ $\displaystyle=0\,\,,$ (3.3c)
$\displaystyle\partial_{t}u_{x}|_{t=0}$
$\displaystyle=\frac{0.01k\sin(kx)}{4(1+0.01\cos(kx))}\,\,.$ (3.3d)

The time derivative of the velocity at $t=0$ is chosen so that the ideal
hydrodynamics equations (2.5) are initially satisfied. If a system is expected
to be in the regime of hydrodynamics, the ideal equations should be nearly
satisfied, up to higher order terms which are supposed to be small. Therefore,
setting them to be exactly zero initially should be a good approximation to
this situation. Other options such as setting to zero the time derivatives at
$t=0$ would imply that the ideal equations would not be initially satisfied;
later on we will analyse this case in detail. Also, very importantly, note
that if the initial data satisfies the ideal equations, then the first order
change of frame vanishes at $t=0$. This means that we can study the effect of
using different frames along the evolution without having to worry about the
effect of changing frame in the initial data: we leave this as a another
exercise to explore in Section 3.1.4 of the paper. We choose
$\overline{\mathcal{E}}=1$ and spatial domain of size $L\,\overline{T}\simeq
34.2$ with periodic boundary conditions.

We perform the real-time evolution of the non-linear first-order viscous
hydrodynamics equations (2.2) with (2.8), initial conditions (3.3) and
hydrodynamic frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$. The resulting solution is
shown in Fig. 1 (left) for the $T_{tt}$ component of the stress tensor. This
evolution captures the well known physics of the sound mode, with frequency
and decay rate well approximated by [27]:

$\displaystyle\omega(k)$
$\displaystyle=c_{s}k-i\frac{\Gamma}{2(\epsilon+p)}k^{2}+O(k^{3})\,\,,$ (3.4)

with $c_{s}=1/\sqrt{3}$, $p=\epsilon/3$ and $\Gamma=4\eta/3$.

Figure 1: (Left): Spacetime evolution of the $T_{tt}$ component of the stress
tensor of a sinusoidal perturbation of a homogeneous thermal state with
initial data (3.3), momentum $k/\overline{T}\simeq 0.184$, box of size
$L\,\overline{T}\simeq 34.2$ and energy $\overline{\mathcal{E}}=1$, evolved
using the equations of first-order viscous hydrodynamics with frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$ and $N_{x}=2048$ points. (Right): For the
evolution on the left, we show the log plot of the absolute value of the
different contributions to the $T_{xx}$ component of the stress tensor (2.8)
at $x=0$ as a function of time: ideal term in solid black, shear term in solid
blue and $a_{1}$ term in solid red ($a_{2}$ term is vanishing at $x=0$ by
symmetry). We also include the result of the approximate expression (3.5a) in
dotted grey, which allows to understand the amplitude of the $a_{1}$ term.
Additionally, we include the result of evaluating the second order terms
(2.10) in solid purple. From this plot we can confirm that the system is well
within the regime of hydrodynamics, as the shear term is much smaller than the
ideal term and the second order term is much smaller than the shear term; more
specifically, Criterion A is satisfied. The most important conclusion from
this plot is that $a_{1}$, $a_{2}$ terms are much smaller than the shear term,
as expected as they are of higher order on shell, and thus negligible for the
physics up to first order; so, Criterion B is also satisfied.

####  3.1.1 The criteria

We start by examining if the solution is within the regime of hydrodynamics,
and for this we explain in detail our definition of Criterion A. We use the
numerical solution for $\\{\epsilon,u_{x}\\}$ to evaluate pointwise in
spacetime the different terms of the constitutive relations (2.8); in Fig. 1
(right) we show the absolute value of the different terms in the $T_{xx}$
component of the stress tensor (2.8),101010The reason why we analyse the
relative sizes of the terms of the $T_{xx}$ component of the stress tensor
(2.8) and not $T_{tx}$ or $T_{tt}$ is that in the shear tensor
$\sigma_{\mu\nu}$, because of being transverse to the velocity, the component
$\sigma_{tx}$ is suppressed by the velocity and the component $\sigma_{tt}$ by
the square of the velocity, and so these components are much smaller than
$\sigma_{xx}$ for this solution, in which velocities are small.  in a log
scale at $x=0$ as functions of time. Comparing the ideal term (solid black)
with the shear term (solid blue), we conclude that first order terms are much
smaller than ideal terms, indicating that the system is within the regime of
hydrodynamics. To be precise, we should compare the ideal term $T_{xx}^{(0)}$
with the full first order $T_{xx}^{(1)}$ (shear term + $a_{1}$ term + $a_{2}$
term), and not only the shear term; however, as we will describe in detail
below, the $a_{1}$, $a_{2}$ terms are much smaller than the shear term and so
$T_{xx}^{(1)}\simeq-\eta\,\sigma_{xx}$ is a very good approximation.

As explained in the previous section, even if in the theory under
consideration the stress tensor does not include second order terms, we will
measure the size of such terms in our solutions since nothing prevent us from
computing the required higher order derivatives. We evaluate the second line
of the constitutive relations (2.10) using our numerical solution for
$\\{\epsilon,u_{x}\\}$, obtaining the solid purple line in Fig. 1 (right). We
use the transport coefficients (3.2). Note that these second order expressions
in the second line of (2.10) are in the Landau frame, and our
$\\{\epsilon,u_{x}\\}$ are not in the Landau frame; however, as we will verify
below in detail, the difference between frames is of higher order, and hence
negligible in our solution. So, it is justified to use expression (2.10) as a
very good approximation to the second order terms.

By comparing ideal, first order and second order terms, we find that the
amplitude of each term in the gradient expansion is much smaller than the
previous one. Ideally we would like to provide a quantitative criterion that
is useful in a generic solution and that does not rely on the amplitude of
each term. For example, one possible criterion could be that the system is in
the regime of hydrodynamics if, pointwise in spacetime and for all components
of the stress tensor, each term in the hydrodynamic expansion is not larger
than a certain threshold value, e.g., 10%, of any of the lower order terms.
However, a local, pointwise criterion has some disadvantages. For instance, at
some specific points it might fail even if one considers a solution that is
clearly in the regime of hydrodynamics: in our previous example, the
definition above always fails when one of the relevant quantities crosses
zero, and they do because they are oscillating. We would like to capture the
fact that the system is well described by hydrodynamics in spite of these
regions.

One possible idea to define a criterion that avoids this problem of having
regions where the relevant quantities cross zero is to depart from the
pointwise analysis and consider some kind of integral version, i.e., a
suitable norm that captures the global aspects of the solution. The idea would
be to integrate in patches, possibly extended both in space and time. It would
be reasonable to use patches whose size is similar to the characteristic scale
of the problem: if they are too small we go back to the same problems as in
the pointwise analysis, and if they are too large, we wash out the details of
the system. It would be interesting to perform an exploration of different
such definitions, and we leave this systematic exploration for future work. In
this paper we will use the following definition: we compute the $L_{1}$ norm
(i.e., integral of the absolute value) of the relevant quantities over the the
full spatial domain and also in time, with a time integration domain of the
order of the characteristic timescale of the system.111111The choice of the
$L_{1}$ norm is only for convenience; other norms could be equally useful.
Also, the choice of $10\%$ as a threshold value for acceptance is completely
arbitrary. This value may depend on the system under consideration and the
choice of initial conditions.

Criterion A

The system is within the effective field theory regime at a given time $t$ if
the $L_{1}$ norm over the spatial domain and over times
$\\{t-\tilde{t},t+\tilde{t}\\}$, where $\tilde{t}$ is a characteristic time
scale of the problem, of the shear term is smaller than $10\%$ of the $L_{1}$
norm of the ideal term, and also the $L_{1}$ of the second order terms is
smaller than $10\%$ the $L_{1}$ norm of the shear term, for all components of
the stress tensor.

In the case of a sinusoidal perturbation, we consider $\tilde{t}$ as the
period of oscillation. Relevant data for this criterion is ploted in Fig. 1
(right) at $x=0$ and in Fig. 4 (top, left) in the whole spacetime domain,
where we plot the $10\%$ of the shear term (in blue) and the second order term
(in purple) in such a way that we can visually inspect their relative sizes.
We do not include a plot of the ideal term because the shear term is orders of
magnitude smaller, so it clearly satisfies the criterion. By computing the
corresponding norms we verified that Criterion A is well satisfied in our
solution at all times, and thus we conclude that the system is in the
effective field theory regime. The ratio of the $L_{1}$ norm of second order
and first order terms is $2.2\%$, and of first order and ideal terms is
$3.4\%$. We can then define the hydrodynamization time as the time that it
takes for a system that is initially away from equilibrium, violating
Criterion A, to relax to a regime satisfying Criterion A, and hence in the
hydrodynamic regime.

We now proceed with a detailed definition of our second criterion, denoted by
Criterion B. We examine the size of the field redefinition dependent $a_{1}$,
$a_{2}$ terms in (2.8) and compare them to the shear viscosity term in (2.8),
which is field redefinition independent. According to the discussion in
Section 2, by using the equations of motion, the terms $a_{1}$, $a_{2}$ can be
re-expressed, on shell, as second order terms, and thus in our solution they
should be much smaller than the shear viscosity term; if they are much
smaller, then the choice of frame will not affect the physics to first order.

Criterion B

We will say that the physics to first order is independent of the chosen
hydrodynamic frame at a time $t$ if the $L_{1}$ norm over the spatial domain
and over times $\\{t-\tilde{t},t+\tilde{t}\\}$ of the spatial components of
both the $a_{1}$ and $a_{2}$ terms in the stress energy tensor are smaller
than $10\%$ of the $L_{1}$ norm of the spatial components of shear term
(computed over the same spacetime domain).

Note that we only consider the spatial components of the stress tensor in this
definition. This is because the shear tensor is transverse to the velocity and
the temporal components are suppressed when the velocity is small, as in our
solution, while in the $a_{1}$, $a_{2}$ terms this suppression is not present.
Therefore, for the temporal components of the stress tensor, the shear term
may be smaller than the $a_{1}$, $a_{2}$ terms.

By computing the norms, we find that Criterion B is satisfied in our solution.
In Fig. 1 (right) we can observe that at $x=0$ the $a_{1}$ term is smaller
than the shear term well below a $10\%$. In Fig. 4 (middle, left) we present
the $a_{1}$ term (in red) and $a_{1}$ term (in orange) in the spacetime
domain, together with $10\%$ the shear term (in blue). As additional
information, we note that the ratio of amplitudes of the $a_{1}$ term and
shear term is $0.19\%$ and of the $a_{2}$ term shear term is $0.06\%$. Thus,
we can conclude that the physics up to first order in the hydrodynamic
gradient expansion is independent of the arbitrarily chosen frame used to
perform the actual numerical evolution, see Fig. 3.

After this conclusion one could think to proceed in practical terms as
follows. If the system under consideration is well within the regime of
hydrodynamics, one could think of the terms $a_{1}$, $a_{2}$ in the first-
order hydrodynamic equations as mere regulators that render the equations
well-posed since, on shell, they are of second order, and thus negligible to
first order. Then we can work as if we were in the Landau frame, as changing
frame is a second order effect. However, this way of thinking of the equations
might be appropriate only when the system is well within the regime of
hydrodynamics, and one must be careful if it is not clear that the system is
in that regime, such as in systems that are marginally in the regime of
hydrodynamics, as the ones we study below.

Now we would like to find an easy way to understand the value of the amplitude
of the $a_{1}$, $a_{2}$ terms in this solution. A natural approach would be to
linearize the $a_{1}$, $a_{2}$ terms, but under linearization the detailed
cancellations between the addends are spoiled. Recall that the $a_{1}$,
$a_{2}$ terms are proportional to the equations of motion, which are nearly
satisfied if the solution is in the effective field theory regime, explaining
why there are detailed cancellations. Another option would be to use the
equations of motion to replace the ideal hydrodynamic equations in $a_{1}$,
$a_{2}$ by higher order terms. However, by linearising these higher order
terms also cancellations among them are spoiled.

A way forward is as follows. In the Landau frame, i.e.,
$\\{a_{1},a_{2}\\}=\\{0,0\\}$, the only contribution to the second order
derivatives in the equations of motion comes from the shear term, and this
contribution can be easily linearized. With this in mind, we can perform a
field redefinition (2.7) from the Landau frame to a generic frame
$\\{a_{1},a_{2}\\}$ in which we replace the ideal hydrodynamic equations by
the linearization of the contribution of the shear term to the terms with
second order derivatives in the equations of motion. With these expressions in
a general frame, we can now perform an expansion in amplitude of the relevant
part of the $a_{1}$, $a_{2}$ terms (without spoiling any detailed
cancellation) obtaining

$\displaystyle\frac{3}{4}\frac{\dot{\epsilon}}{\epsilon}+\nabla\cdot u$
$\displaystyle\simeq\frac{\eta}{\epsilon}(\partial_{x}u_{x})^{2}+\frac{3}{4}\,a_{2}\,\frac{\eta^{2}}{\epsilon^{2}}\partial_{x}^{3}u_{x}+...\,\,,$
(3.5a)
$\displaystyle\dot{u}^{\mu}+\frac{1}{4}\frac{\nabla_{\perp}^{\mu}\epsilon}{\epsilon}$
$\displaystyle\simeq\frac{\eta}{\epsilon}\partial_{x}^{2}u_{x}+\frac{1}{4}\,a_{1}\,\frac{\eta^{2}}{\epsilon^{2}}\partial_{t}\partial_{x}^{2}u_{x}+...\,\,.$
(3.5b)

The first term on the right hand side comes from the leading term in amplitude
of the shear part and the second term from the leading term in amplitude of
the $a_{1}$, $a_{2}$ contribution. Notice that in (3.5a) the term linear in
the shear is quadratic in the velocity, whilst the term proportional in
$a_{2}$ part is linear in the velocity. Thus the latter dominates in our
solution. In (3.5b) both the shear and $a_{1}$ part are linear in the
velocity, but the shear part is of lower order in derivatives, so it
dominates.

Expressions (3.5) are very useful because they allow to estimate the size of
the $a_{1}$, $a_{2}$ terms in the constitutive relations (2.8) in the
particular case of the sinusoidal perturbation of a thermal state. We can have
analytical control over these terms, with explicit dependence on the values of
the constants $a_{1}$ and $a_{2}$, amplitude, momentum, etc. Thus, we can
perform an analysis of the size of these terms without the need to perform
numerical simulations. In particular, it is interesting to analyze the
limiting situation where $k/\overline{T}$ is not small and the system is
starting to exit the regime of hydrodynamics and check to what extent the
$a_{1}$, $a_{2}$ terms are small or not.

In Fig. 2 we use the approximate expressions (3.5) to obtain the amplitude of
the $a_{1}$, $a_{2}$ terms of the $T_{xx}$ component of the stress tensor
(2.8) for a sinusoidal perturbation of a thermal state (3.3), as a function of
momentum: $a_{1}$ term in red, $a_{2}$ term in orange. We also include the
amplitude of the shear term in (2.8) and the second order terms in (2.10) that
we obtain by linearizing these expressions.

Figure 2: Log-log plot of the amplitude of the different terms of the $T_{xx}$
component of the stress tensor (2.8) for initial data (3.3) and frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$ as a function of momentum. To produce this plot
we did not do any numerical evolutions; we simply computed the amplitude of
each term in the initial data. For the shear term (solid blue) we linearize
the expression in (2.8) and for the $a_{1}$ and $a_{2}$ terms (solid red and
solid orange respectively) we use the approximate expressions (3.5). We also
include the amplitude of second order terms obtained by linearizing the second
line in (2.10), solid purple. We observe that the $a_{1}$, $a_{2}$ terms are
smaller than the shear term up to values $k/\overline{T}\simeq 1$ or even
beyond. The vertical line shows the case studied in Fig. 1, with
$k/\overline{T}\simeq 0.184$ and the dots correspond to the amplitudes
obtained from that real time evolution.

From Fig. 2 we can obtain the following conclusions. We observe that the
$a_{1}$, $a_{2}$ terms are smaller than the shear term up to values
$k/\overline{T}\simeq 1$ or even beyond. Thus, different frames provide the
same physical description to first order. If we consider the criterion that
the $a_{1}$, $a_{2}$ terms are smaller than the shear term by at least $10\%$,
that is, Criterion A, in the case of Fig. 2 this happens at
$k/\overline{T}\simeq 1.37$; however, this number will depend on the values of
$a_{1}$, $a_{2}$ and on the amplitude of the perturbation (and of course on
the choice of threshold). For larger values of $a_{1}$, $a_{2}$ this number
will be smaller. This suggests that the larger $a_{1}$, $a_{2}$ are, the
smaller the value of $k/\overline{T}$ where the ratio equals $10\%$ is. Thus,
if we are close to the boundary of the regime of applicability of
hydrodynamics, it is preferable to work with smaller values of $a_{1}$,
$a_{2}$ as this imposes a lower limit for the frame independence of the
physics up to first order according to Criterion B.

We now consider a third criterion, which we denote by Criterion C. The
motivation is the following. Above we have proposed Criterion B to determine
when the physics up to first order is independent of the choice of frame. We
can complement it with another criterion: perform two numerical evolutions in
different causal frames with similar initial data and compare the two
solutions after a certain time $t$. It could happen that even if Criterion B
indicates that the $a_{1}$, $a_{2}$ terms are small at all times, their
effects accumulate over time, resulting large differences at late times.
Another possibility is that even if the $a_{1}$, $a_{2}$ terms are not small,
the solutions obtained in different frames remain close to each other.

To asses this criterion, in addition to the solution in Fig. 1 in frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$, we now perform another evolution with similar
initial data (3.3), $k/\overline{T}\simeq 0.184$, but in the frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$;121212One may wonder if the initial data
should be changed to the corresponding working frame, but for this specific
initial data the change of frame at $t=0$ is exactly vanishing; this is
because it satisfies the ideal equations (2.5) at $t=0$. Below we will address
the question of changing frame in the initial data in cases where the change
of frame is non trivial. we denote the two solutions by
$T^{\text{visc1}}_{\mu\nu}$ and $T^{\text{visc2}}_{\mu\nu}$ respectively,
shown in Fig. 3 (left) in solid and dashed lines respectively. According to
the discussion in Section 2, we choose two frames that are well separated, in
this case by a factor of 2. In Fig. 3 (left) we present the difference of the
stress tensor component $T_{xx}$ of the two solutions,
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$, at $x=0$ as a function of time
in solid green.

Figure 3:  Left: We plot the same information as in Fig. 1 (right) which was
performed in frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$ in continuous lines, and now
we add the data of an evolution with similar initial data (3.3) and momentum
$k/\overline{T}\simeq 0.184$, but now using frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$, in dashed lines and same color coding. The
red solid and dashed lines correspond to the $a_{1}$ terms in each case, with
an amplitude differing by a factor of around 4 as the approximate expression
(3.5a) indicates. We include the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ at $x=0$ of the two solutions in
different frames $\\{a_{1},a_{2}\\}=\\{5,5\\}$ and
$\\{a_{1},a_{2}\\}=\\{10,10\\}$, in solid green. Moreover, we also perform a
numerical evolution of the ideal equations with initial data (3.3a)–(3.3b) and
obtain the difference $T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ of one
viscous evolution and the ideal evolution, in solid grey. We include the
analytical results for the differences
$\left(T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}\right)_{\text{analytical}}$
and
$\left(T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}\right)_{\text{analytical}}$
in dotted green and dotted grey respectively, finding good agreement. Right:
Similar as (left) but with momentum ten times larger, $k/\overline{T}\simeq
1.84$, to analyse a situation which is marginally in the regime of
hydrodynamics. See Fig. 4 for plots in the spacetime domain and not just at
$x=0$ of the relevant quantities to evaluate Criteria A, B and C in these
runs.

$\displaystyle Criterion$$\displaystyle A$ 10%Shear Second order$\displaystyle
Criterion$$\displaystyle B$ 10%Shear $a_{1}$term $a_{2}$term$\displaystyle
Criterion$$\displaystyle C$
10%$\left(T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}\right)$
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$

Figure 4:  For the evolutions presented in Fig. 3 we show here relevant
quantities that allow for an easy visual inspection of the different criteria
A, B and C. For this, in the top panels we plot a $10\%$ of the shear term
(blue) and second order terms (purple). In middle panels we plot a $10\%$ of
the ideal term (blue) and $a_{1}$, $a_{2}$ terms (red, orange). In bottom
panels we plot a $10\%$ of $T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in
solid grey and $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in solid green.
Left plots correspond to the case $k/\overline{T}\simeq 0.184$, and right
plots to $k/\overline{T}\simeq 1.84$. All quantities are in absolute values.
Top an middle panels correspond to evolutions in frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$.

Having computed the difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ ,
we would like to find a relevant quantity to compare it with so that we can
establish a criterion to decide when the system is invariant under field
redefinitions to first order in the hydrodynamic gradient expansion. One
possibility could be to compare this difference with $T_{xx}$ itself; however,
this might not be sensible in our solution because the perturbation is small
compared to the average value and it does not tell us about the first order
physics. Another possibility would be to compare this difference to the shear
term; however, the former will typically increases with time while the shear
term will decrease in such a way that the difference might eventually become
larger than the shear term.

We propose the following approach. We consider a third solution, namely one
obtained by solving the ideal hydrodynamics evolution equations using the
initial data (3.3a)–(3.3b),131313Recall that ideal hydrodynamics equations are
of first order in derivatives, thus we do not need to specify the time
derivatives of the evolved quantities at the initial time. The latter were
chosen so that the equations of ideal hydrodynamics are satisfied initially,
so using this initial data for the ideal case is consistent.
$k/\overline{T}\simeq 0.184$. We denote the stress energy tensor obtained from
this third solution by $T^{\text{ideal}}_{\mu\nu}$. We can now compute the
difference of the stress tensors of one viscous solution and the ideal
solution, that is, $T^{\text{visc1}}_{\mu\nu}-T^{\text{ideal}}_{\mu\nu}$, and
compare it with the difference of the stress tensors of the two viscous
solutions, $T^{\text{visc1}}_{\mu\nu}-T^{\text{visc2}}_{\mu\nu}$. In this way
we are comparing the effect of changing frames with the effect of including
viscous terms in the hydrodynamics description, which is precisely what we
want.

Criterion C

We will say that the physics to up first order in the derivative expansion is
invariant under a change of frame at time $t$ when the $L_{1}$ norm of the
difference $T^{\text{visc1}}_{\mu\nu}-T^{\text{visc2}}_{\mu\nu}$ computed over
the spatial domain and time interval $\\{t-\tilde{t},t+\tilde{t}\\}$ is less
than $10\%$ of the $L_{1}$ norm of the difference
$T^{\text{visc1}}_{\mu\nu}-T^{\text{ideal}}_{\mu\nu}$ (and
$T^{\text{visc2}}_{\mu\nu}-T^{\text{ideal}}_{\mu\nu}$), for all components of
the stress tensor.141414Let us emphasize that both quantities
$T^{\text{visc1}}_{\mu\nu}-T^{\text{visc2}}_{\mu\nu}$ and
$T^{\text{visc1}}_{\mu\nu}-T^{\text{ideal}}_{\mu\nu}$ are expected to grow in
linearly in time (e.g., [36]), but their ratio should have a weaker time
dependence and hence this criterion could still be useful at much later times.

In Fig. 3 (left) we show the result of the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in solid green and
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey at $x=0$. In Fig.
4 (left, bottom) we show a plot in all spacetime of the quantities
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ and $10\%$ of
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$; this allows for a direct visual
inspection of the relevant quantities for Criterion C. Computing the $L_{1}$
norms, we find that the first one is smaller than $10\%$ of the second one at
all times, and thus we conclude that Criterion C is satisfied. The actual
ratio of the $L_{1}$ norm of terms
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ and
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ is $0.14\%$.

Thus, criterion C also confirms that the frame dependent terms have a higher
order effect that is negligible for the physics up to first order. Also, we
can conclude that the accumulated effect of the terms $a_{1}$, $a_{2}$ is
small in this solution, and having locally small $a_{1}$, $a_{2}$ terms also
implies having a small difference in the evolutions in different frames.

Now we would like to address the following question: would it be possible to
have control on the difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$?
We will obtain an analytical estimate of the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$, which will allow to understand
the size of this quantity. To this end, we use that the solution to the linear
problem is known and given by the dispersion relation of the sound mode [27]:

$\displaystyle\omega(k)$
$\displaystyle=c_{s}k-i\frac{\Gamma}{2(\epsilon+p)}k^{2}+\frac{\Gamma^{2}}{8c_{s}(\epsilon+p)^{2}}k^{3}-ia_{2}\eta\frac{\Gamma^{2}}{2(\epsilon+p)^{3}}k^{4}+O(k^{5})\,\,,$
(3.6)

up to $O(k^{4})$, with $c_{s}=1/\sqrt{3}$, $p=\epsilon/3$ and
$\Gamma=4\eta/3$. Recall that this is the dispersion relation of the first-
order hydrodynamics equations, that is, the truncated theory; beyond
$O(k^{2})$, the terms would be different if we had included second and/or
higher order terms in the expansion of the stress energy tensor. Recalling
that our initial data is $x\leftrightarrow-x$ symmetric, the general solution
to the linear problem is a sum of functions

$\displaystyle\left(\tilde{A}e^{\omega_{Im}t}\cos(\omega_{Re}t)+\tilde{B}e^{\omega_{Im}t}\sin(\omega_{Re}t)\right)\cos(kx)\,\,,$
(3.7)

in harmonics of $k$, even if here we will restrict to the lowest harmonic as
the other terms are very small in our solution. Here
$\\{\omega_{Re},\omega_{Im}\\}$ are the values obtained from evaluating (3.6).
Now, for the two solutions in Fig. 3 (left) we use the analytical expression
(3.7) to fit $T_{xx}$ and obtain the values of the constants $\tilde{A}$ and
$\tilde{B}$ in each case. We now consider the difference of the two analytical
expressions
$\left(T^{\text{visc1}}_{\mu\nu}-T^{\text{visc2}}_{\mu\nu}\right)_{\text{analytical}}$
and include it in Fig. 3 (left), in dotted green. We can conclude that we
obtain a very good estimate of the difference
$T^{\text{visc1}}_{\mu\nu}-T^{\text{visc2}}_{\mu\nu}$. Moreover, we can also
perform a fit to the solution of ideal hydrodynamics using the ideal
dispersion relation, and obtain the analytical difference
$\left(T^{\text{visc1}}_{\mu\nu}-T^{\text{ideal}}_{\mu\nu}\right)_{\text{analytical}}$
that we plot in dotted grey, obtaining also a good estimate.

To be precise, we perform the aforementioned fits using data from some time
$t>0$ with $t\overline{T}\simeq 10$, and not from $t=0$. The reason is that we
observe that the fit is better if we avoid the first instants of the evolution
since there is a very small quasinormal mode that relaxes quickly and is not
captured by (3.7). It would be interesting to study in detail this quasinormal
mode, but we leave it for future work. In principle $\tilde{A}$ and
$\tilde{B}$ can be obtained from the initial data analytically, as the initial
data is also analytical. However, the fit that we obtain in doing so is not so
good precisely because of the presence of the aforementioned quasinormal mode.

Using the analytical solution we can now provide an answer to the question:
even if the $a_{1}$, $a_{2}$ terms are not small compared to the shear term,
is it possible that solutions obtained in different frames capture the same
physics up to first order? In other words: can Criterion B be violated while
Criterion C be satisfied? The answer is yes, and we have an example in which
this is under control: In the dispersion relation (3.6) the constant $a_{2}$
enters at $O(k^{4})$ (and $a_{1}$ enters at $O(k^{6})$); thus, if we consider
a frame with large value of $a_{2}$, the corresponding $a_{2}$ term in the
stress tensor becomes large compared to the shear term but the analytical
solution is barely modified because it only receives corrections at $O(k^{4})$
and hence the evolution of the stress tensor in different frames will be very
similar.

Even if we could consider these solutions as counterexamples to the statement
that Criterion A implies Criterion B, they are obtained by choosing
unreasonably large values of $a_{1}$ and $a_{2}$. If we restrict
$\\{a_{1},a_{2}\\}$ to values that are close to saturating the inequalities
(2.9), then we can exclude these counterexamples. Below we provide further
arguments in favor of this choice.

####  3.1.2 A solution marginally in the effective field theory regime

Up to this point we have considered solutions that are in the regime of
hydrodynamics by construction. In the experimental description of the QGP
created in heavy-ion collisions, when the hydrodynamics codes are initialized,
gradients might be large so it is unclear whether the QGP should be well-
described by (viscous) hydrodynamics. Motivated by this picture, we now
explore a system that is on the verge of the hydrodynamics regime. We consider
another pair of simulations, also in the frames $\\{a_{1},a_{2}\\}=\\{5,5\\}$
and $\\{a_{1},a_{2}\\}=\\{10,10\\}$, and initial data (3.3) but now with
momentum a factor of ten larger, $k/\overline{T}\simeq 1.84$. In Fig. 3
(right) we show the relevant quantities of these two evolutions at $x=0$, and
in Fig. 4 (right) we depict the relevant quantities to evaluate the different
criteria A, B and C in the whole spacetime.

We observe that Criterion A is violated, that is, the system is slightly
outside the regime of hydrodynamics since the ratio of the $L_{1}$ norms of
the second order and first order terms is $18\%$. Furthermore, the ratio
between the $L_{1}$ norm of the $a_{1}$ term and the shear term is $15\%$, so
Criterion B is also violated. However, Criterion C is satisfied, that is, the
two solutions obtained in different frames are close to each other since the
ratio between the $L_{1}$ norms of
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ and
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ is $5.3\%$. This indicates that,
according to Criterion C, the invariance of the physics up to first order
under frame redefinitions is robust, even if the system is marginally in the
effective field theory regime.

The fact that Criterion B is satisfied or not in a situation in where the
system is close to the boundary of the regime of hydrodynamics may depend on
the specific values of $\\{a_{1},a_{2}\\}$. If for a pair of values
$\\{a_{1},a_{2}\\}$ the system is about to violate Criterion B, then choosing
slightly larger values $\\{a_{1},a_{2}\\}$ will trivially lead to a violation
of this criterion (assuming that the threshold value for acceptance is kept
the same). Thus, the fact that Criterion B is satisfied or not in these
limiting cases depends on the values of $\\{a_{1},a_{2}\\}$ chosen. Therefore,
to avoid ambiguities, it is preferable to work with values $\\{a_{1},a_{2}\\}$
that are close to saturate the bound (2.9). On the other hand, Criterion C is
still robust in these circumstances.

One would also like to understand how this system hydrodynamizes. That is, how
long it takes for the system to become well described by hydrodynamics
according to Criterion A. The answer is that it seems that this system does
not hydrodynamize in times comparable to $\overline{T}$ or even much larger.
Linear physics indicates that the solution will remain away from the effective
field theory regime, as second order terms decay at the same rate as first
order terms, with a ratio of amplitudes that is constant in time and thus does
not decrease. It could be that non-linear effects change the picture at much
later times.

####  3.1.3 The choice of frame

We now comment on the choice of frame in practical applications. Above we have
discussed in detailed examples that show that physics up to first order in the
gradient expansion is independent of the chosen hydrodynamic frames
$\\{a_{1},a_{2}\\}$ as long as the system is within the regime of
hydrodynamics. So, in principle, one could use any values of
$\\{a_{1},a_{2}\\}$ that obey the hyperbolicity bounds (2.9) and obtain
equivalent physical descriptions up to first order. However, working with
large values of $\\{a_{1},a_{2}\\}$ is not practical. The first reason is that
the characteristic velocities of the PDEs depend on the values of
$\\{a_{1},a_{2}\\}$, and for larger values of $\\{a_{1},a_{2}\\}$ these
velocities are smaller. Then, if there are velocities in the system that are
larger than the characteristic velocities the code will crash; we provide
further comments in the study of shockwaves below. The second reason is that
larger values of $\\{a_{1},a_{2}\\}$ will set more restrictive bounds on the
size of corresponding terms, which may result in rather artificial violations
of Criterion B. For example, in the case studied in Fig. 6 (right), if we used
$\\{a_{1},a_{2}\\}=\\{100,100\\}$, instead of $\\{a_{1},a_{2}\\}=\\{10,10\\}$,
we would obtain that the difference is already $O(1)$ between the two
simulations. If we have done the same in (Left) we would still be well in the
regime of hydrodynamics. Thus, this example illustrates the fact that using
larger values of $\\{a_{1},a_{2}\\}$ may set a more restrictive bounds for the
validity of Criterion B.

Recall that $\\{a_{1},a_{2}\\}$ have to satisfy the hyperbolicity bounds
(2.9). Thus, to avoid artificially large effects from changing frames it is
convenient to work values of $\\{a_{1},a_{2}\\}$ that are close to saturating
(2.9). In fact, the equality region in (2.9) was coined ‘sharply causal
region’, and it is interesting because the characteristics have exactly the
speed of light and thus one can in principle evolve shocks of arbitrarily
large amplitudes [30, 31]. Hence, for these reasons it is convenient to work
in the sharply causal frames; it is a sweet spot satisfying several
interesting conditions. One such example would be:
$\\{a_{1},a_{2}\\}=\\{\frac{25}{4},\frac{25}{7}\\}$. A typical value used in
this paper is $\\{a_{1},a_{2}\\}=\\{5,5\\}$ which is not far from this sharply
causal region, and while it is quite arbitrary, it is equally valid.

####  3.1.4 Changing frame in the initial data

In the previous analyses we performed simulations in two different frames
$\\{a_{1},a_{2}\\}=\\{5,5\\}$ and $\\{a_{1},a_{2}\\}=\\{10,10\\}$, using the
same initial data (3.3) in both cases. This was consistent because for that
particular choice of initial data the effect of changing frames (2.7) is
exactly vanishing. In more realistic applications, e.g., to describe
experimental data in the context of heavy-ion collisions, one will be given
initial data for the stress tensor, and not directly initial data for the
evolution variables $\\{\epsilon,u_{x}\\}$ in the chosen causal frame. Thus,
in practice, one has to obtain the initial data for the evolution variables in
the working causal frame from a given stress tensor. Assuming that the initial
stress tensor in the regime of validity of effective field theory, one
possible way to proceed is the following: start from initial data for the
stress tensor and diagonalize it to obtain $\\{\epsilon,u_{x}\\}$ in the
Landau frame and then change frame using (2.7).

In generic circumstances the initial data will not satisfy exactly the ideal
hydrodynamics equations and the change of frame expressions will be non-
trivial at that initial time. However, if there are reasons to believe that
the system is well described by hydrodynamics, then the ideal hydrodynamics
equations should be nearly satisfied and corrections should appear in a
hierarchy of ever decreasing terms. For this reason, considering initial data
so that the time derivatives $\\{\partial_{t}\epsilon,\partial_{t}u_{x}\\}$
are chosen to satisfy the ideal equations seems to be a good approximation.
Moreover, this choice allowed us to analyse the effect of changing frames in
the evolutions without having to address the question of changing frame in the
initial data. Other choices, such as vanishing initial time derivatives of the
evolution variables , tend to result in configurations for which the ideal
hydrodynamics equations are far from being satisfied at $t=0$, which could
result in significant differences in the initial data in different causal
frames. In this section we address the effect of changing frame in the initial
data.

Let us be more precise about our procedure to change frames in the initial
data. To change frames one uses the field redefinitions (2.7), where the
constants $\\{a_{1},a_{2}\\}$ in this expression specify the new causal frame.
Note that this expression assumes that the system is in the regime of
applicability of hydrodynamics since it ignores second (and higher) order
derivatives. Therefore, to change from frame $\alpha$ to frame $\beta$, the
constants in (2.7) are
$\\{a_{1}^{\alpha}-a_{1}^{\beta},a_{2}^{\alpha}-a_{2}^{\beta}\\}$, with the
quantities on the right hand side in frame $\alpha$ and the ones on the left
hand side in frame $\beta$. In particular, the change of frame from the Landau
frame to the causal frame $\\{a_{1},a_{2}\\}$ is given by expression (2.7)
with the constants given by $\\{-a_{1},-a_{2}\\}$ and the quantities on the
right hand side in the Landau frame. One could consider other procedures to
change frames at the level of the initial data and they would differ from the
one described here by second and higher order gradients. Of course, if the
system is not in the hydrodynamic regime, then different procedures will
result in significantly different initial conditions.

Recall that to solve the first order viscous hydrodynamics equations we need
to provide initial data for the time derivatives of the fundamental variables
$\\{\partial_{t}\epsilon,\partial_{t}u_{x}\\}$. We now comment on possible
ways to consistently compute these time derivatives when changing the frame of
the initial data. First, note that the time derivatives are of first order in
gradients and hence they should be the same in all frames, up to higher order
terms that should be ignored. In practice, if we happen to know the
thermodynamic variables over a time interval151515This would be the case if
the experimental data is given as a time series, or we have computed the
solution using some other scheme such as MIS. then we can change frame in the
entire interval and compute the time derivatives in the new frame variables.
This is what we will do in the examples below. Alternatively, if one only has
data at $t=0$, one can compute the time derivative of expression (2.7) and
either ignore the second time derivatives or impose the equations of motion at
$t=0$ in the original frame to compute them.

In order to explore the effect of changing frame in the initial data we
consider two different exercises. First, we consider a numerical solution in a
given frame (see Fig. 3) at $t>0$, change frames and thus obtain initial data
to evolve the equations in the new frame. For concreteness consider the
solution in Fig. 1, which is in frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$, at time
$t\overline{T}\simeq 17.1$. We change to frame $\\{a_{1},a_{2}\\}=\\{10,10\\}$
using (2.7) and use this as initial data for an evolution in the new frame. In
Fig. 5 (left) we show the original evolution in frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$ at $x=0$, in solid lines, and the new evolution
in frame $\\{a_{1},a_{2}\\}=\\{10,10\\}$, in dashed lines; the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ is depicted in solid green. To
carry out these simulations we computed the time derivatives for the initial
data in the new causal frame $\\{a_{1},a_{2}\\}=\\{10,10\\}$; using instead
the time derivatives computed in the original frame does not make a noticeable
difference, as one might have expected. For comparison, we have also performed
an evolution of the ideal hydrodynamics equations using initial data at
$t\overline{T}\simeq 17.1$ from the solution in frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$ (using data in the frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$ does not make a difference). Notice that
although the ideal hydrodynamics equations are not exactly satisfied at
$t\overline{T}\simeq 17.1$, the error terms are small, as they should be since
the system is well in the regime of hydrodynamics by construction. The
difference $T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in is shown in solid
grey.

From these results we conclude that the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in the two evolutions in
different frames due to the change of frame in the initial data, see Fig. 5
(left), is comparable to the difference introduced solely upon time evolution
in different frames studied in Fig. 3 (left) where, recall, the change of
frame was vanishing at $t=0$. Moreover, we conclude that criteria A, B and C
are satisfied at a similar level in this situation. We repeated this exercise
for initial data at some other times $t\overline{T}\simeq 8.5,25.6$, obtaining
similar conclusions.

Motivated by the physics of the QGP, we now study a case which is marginally
in the regime of hydrodynamics by performing a similar exercise as the
previous one but with momentum ten times larger, $k/\overline{T}\simeq 1.84$.
In Fig. 5 (right) we show the result of using initial data at time
$t\overline{T}\simeq 1.71$. Again, we conclude that the change of frame in the
initial data introduces a difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ which is comparable to the
difference generated solely upon time evolution studied in the example in Fig.
3 (right). We find that Criteria A and B are violated at a similar level as in
the situation in Fig. 3 (right), and Criterion C is still satisfied at a
similar level. We repeated the same exercise at times $t\overline{T}\simeq
0.85,2.56$ obtaining similar conclusions.

Thus, we can conclude that the change of frame performed in this way in the
initial data does not change the picture regarding the applicability of
Criteria A, B and C in these solutions: they are satisfied or violated at a
similar level as in the situation of Fig. 3.

Figure 5:  (Left) Solid lines correspond to the same evolution as in Fig. 3
(left) in frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$. We include in dashed lines and
same color coding the result of using the data at $t\overline{T}\simeq 17.1$
(time indicated by a vertical dashed line) to perform a change of frame to
$\\{a_{1},a_{2}\\}=\\{10,10\\}$ and use it as initial data for an evolution in
frame $\\{a_{1},a_{2}\\}=\\{10,10\\}$. We show in solid green the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$. Moreover, we perform an
evolution of the ideal hydrodynamics equations and we show the difference
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey. (Right) Similar
exercise as in (left) for the evolution with momentum ten times larger
$k/\overline{T}\simeq 1.84$, to analyse a situation which is marginally in the
regime of hydrodynamics. The main conclusion from these plots is that changing
frame in the initial data according to the prescription of effective field
theory introduces a small difference comparable to the difference in the
evolutions in different frames analysed in Fig. 3.

We present now our second example. Recall that the initial data (3.3) was
chosen so that the ideal hydrodynamics equations are identically satisfied
(and hence effects of changing frames are identically zero). Here we want to
study a situation where initially the equations of ideal hydrodynamics are far
from being satisfied. To do so, we assume that initial data is in the Landau
frame and modify (3.3) by imposing $\partial_{t}u_{x}|_{t=0}=0$, that is:

$\displaystyle\epsilon|_{t=0}$
$\displaystyle=\overline{\mathcal{E}}\left(1+0.01\cos(kx)\right)\,\,,$ (3.8a)
$\displaystyle\partial_{t}\epsilon|_{t=0}$ $\displaystyle=0\,\,,$ (3.8b)
$\displaystyle u_{x}|_{t=0}$ $\displaystyle=0\,\,,$ (3.8c)
$\displaystyle\partial_{t}u_{x}|_{t=0}$ $\displaystyle=0\,\,.$ (3.8d)

Changing frames to our working causal frame $\\{a_{1},a_{2}\\}$ using the
effective field theory prescription (2.7) gives the following initial data:

$\displaystyle\epsilon|_{t=0}$
$\displaystyle=\overline{\mathcal{E}}\left(1+0.01\cos(kx)\right)\,\,,$ (3.9a)
$\displaystyle\partial_{t}\epsilon|_{t=0}$ $\displaystyle=0\,\,,$ (3.9b)
$\displaystyle u_{x}|_{t=0}$
$\displaystyle=\frac{3}{16}a_{2}\eta\frac{0.01k\sin(kx)}{\overline{\mathcal{E}}\left(1+0.01\cos(kx)\right)^{2}}\,\,,$
(3.9c) $\displaystyle\partial_{t}u_{x}|_{t=0}$ $\displaystyle=0\,\,.$ (3.9d)

One can consider the time derivatives in (3.9b) and (3.9d) in the Landau frame
or in the causal frame, as the difference is of higher order; here we have
chosen to keep them in the Landau frame. The fact that the time derivatives at
$t=0$ in (3.9) (or in (3.8)) vanish, necessarily imply that the ideal
equations are not satisfied and the error terms are not small. This might not
correspond to a physically relevant situation, where the system should be in
(or close to) the regime of applicability of hydrodynamics, but it allows to
assess the effect of the ideal equations not being satisfied in the initial
data in a situation in which this effect is large.

Let us consider initial data (3.9) in frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$; the
corresponding numerical evolution is shown in solid lines in Fig. 6 (left).
The same exercise in frame $\\{a_{1},a_{2}\\}=\\{10,10\\}$ is shown in dashed
lines, and the difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ is
depicted in solid green. In addition we compute the solution of the ideal
hydrodynamics equations with initial data (3.8a),(3.8c) and the corresponding
difference $T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ is shown in solid
grey.161616For the initial data of the ideal equations we have three natural
posibilities: use the data in the Landau frame (3.8a),(3.8c) or in any of the
two causal frames (3.9a),(3.9c). We computed the three cases to check that the
difference does not change the conclusions, even if it is not very small. In
Fig. 6 we plot only the first case.

Figure 6: (Left) Evolutions with initial data (3.9) and momentum
$k/\overline{T}\simeq 0.184$ in frames $\\{a_{1},a_{2}\\}=\\{5,5\\}$,
continuous lines, and $\\{a_{1},a_{2}\\}=\\{10,10\\}$, dashed lines, at $x=0$.
We show the difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in solid
green. We evolve the ideal equations with initial data (3.8a),(3.8c) and show
the result $T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey.
(Right) Same exercise as in (left) but with momentum ten times larger
$k/\overline{T}\simeq 1.84$. The conclusion is that changing frame in the
initial data introduces a difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ that is sustained along the
evolution.

The difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ introduced by the
initial data (3.9c) at $x=0$ can be seen in Fig. 6 (left), where the green
line intersects the vertical axis. Upon time evolution the amplitude of the
difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ does not vary
considerably. If we compare this example to the one in Fig. 3 (left) we find
that the initial data for which the ideal hydrodynamics equations are far from
being satisfied initially, introduces a difference which is much larger than
the difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ generated solely
via time evolution (with vanishing initial change of frame) by more than one
order of magnitude. On the other hand, in the example in Fig. 5 (left), in
which the initial is such that the ideal hydrodynamics equations are nearly
satisfied, the difference in the initial data is of the same order as the
difference generated upon time evolution, see Fig. 3 (left).

As we are using initial data for which the ideal hydrodynamics equations are
far from being satisfied one may suspect that at initial times Criterion A is
violated. However, Criterion A is satisfied at all times and in particular at
$t=0$. This is not just the effect of integrating over the spatial domain and
one period in time (which is the characteristic scale of the system), but even
locally in time the amplitudes of ideal, first order and second order terms
are each smaller than $10\%$ the previous one, even at initial times. We find
curious that in spite of the ideal equations are largely violated at $t=0$,
even initially there still is a hierarchy in the gradient expansion.

Criterion B is satisfied at all times. In this case we observe that at initial
times, and for a timescale much smaller than the period, the amplitude of the
$a_{1}$ term is even larger than the shear term, but it quickly decays. In
spite of this initial transient, integrating over a period in time the
conditions of Criterion B are satisfied.171717This example suggest that it
could be interesting to extend the definition of our criteria to consider
integration in different scales. For example, if we integrated in scales ten
times smaller than the period, criterion B would be violated at initial times.
The fact that the $a_{1}$ term is large at $t=0$ is a consequence of the
choice of the initial data.

Criterion C is satisfied at all times. However, if we compare locally in time
the amplitudes of the difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$
with $T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$, we find that initially it
is not smaller than a $10\%$ and this is restored at times
$t\overline{T}\simeq 4$.

In Fig. 6 (right) we repeat the previous exercise using similar initial data
(3.9) but with momentum ten times larger, $k/\overline{T}\simeq 1.84$, to
consider a situation that is marginally in the regime of hydrodynamics. We
find that the difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ along
the evolution due to change of frame in the initial data is considerably
larger than the difference introduced solely upon time evolution (that is,
with vanishing change of frame at $t=0$) studied in Fig. 3 (right). Due to
this larger difference, in this case all criteria A, B and C are now violated
for the times shown in the plot.

###  3.2 Large amplitude Gaussian perturbation

In the previous subsection we studied solutions that were in the linear regime
by construction, even if we solved the full non-linear evolution equations. We
now extend these studies to solutions that are well in the non-linear regime.
For this purpose we consider a thermal homogeneous state with a large Gaussian
perturbation of the energy density as initial data:

$\displaystyle\epsilon|_{t=0}$
$\displaystyle=\overline{\mathcal{E}}\left(1+e^{-\frac{x^{2}}{2\sigma^{2}}}\right)\,\,,$
(3.10a) $\displaystyle\partial_{t}\epsilon|_{t=0}$ $\displaystyle=0\,\,,$
(3.10b) $\displaystyle u_{x}|_{t=0}$ $\displaystyle=0\,\,,$ (3.10c)
$\displaystyle\partial_{t}u_{x}|_{t=0}$
$\displaystyle=\frac{xe^{-\frac{x^{2}}{2\sigma^{2}}}}{4\sigma^{2}\,\left(1+e^{-\frac{x^{2}}{2\sigma^{2}}}\right)}\,\,.$
(3.10d)

By tuning the width $\sigma$ of the Gaussian we can have a system that is well
in the regime of hydrodynamics (large $\sigma$, i.e., small gradients), or a
system that is far from the regime of hydrodynamics (small $\sigma$, i.e.,
large gradients). As in (3.3), the time derivatives at $t=0$, eqs. (3.10c) and
(3.10d), are chosen such that the ideal hydrodynamics equations are initially
satisfied. As in previous subsection, $\overline{T}$ and
$\overline{{\mathcal{E}}}$ are the temperature and energy density of the
thermal state, related by
$\overline{\mathcal{E}}=\frac{3}{4}\pi^{4}\,\overline{T}^{4}$.

We first consider the evolution of the initial data (3.10) with width
$\sigma\overline{T}\simeq 1.45$ in the frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$.
This width is of the order of the microscopic scale and hence we expect that
the system is on the verge of the regime of applicability of hydrodynamics. In
Fig. 7 (top, left) we show the evolution of $T_{tt}$ component of the stress
tensor.

With the criteria defined in the previous subsection, we proceed with the
analysis of the solution. We start by considering Criterion A. We perform this
comparison for the $T_{xx}$ component of the stress tensor (2.8); in Fig. 7
(bottom, left) we depict this component at $x=0$, and in Fig. 8 (top, left) we
show the second order terms and $10\%$ of the shear term in $T_{xx}$ in the
whole spacetime. Note that since the initial data has vanishing velocity,
initially the shear term is zero but the presence of spatial gradients implies
that the second order terms are non-vanishing. Upon time evolution the system
quickly relaxes to the regime of hydrodynamics, with first order terms much
smaller than ideal terms, and second order terms smaller than shear term, thus
satisfying Criterion A. More precisely, we find that the $L_{1}$ norm of the
second order terms is larger than $10\%$ of the $L_{1}$ norm of the shear term
from $t\overline{T}=0$ to $t\overline{T}\simeq 1.57$. After this time, the
$L_{1}$ norm of the second order terms is smaller than $10\%$ of the $L_{1}$
norm of the shear term. Therefore, we conclude that Criterion A is violated at
initial times and restored at $t\overline{T}\simeq 1.57$; this may be
considered as the hydrodynamization time.

We may wonder why the system is initially away from the regime of
hydrodynamics if the gradients are not very large. This is a consequence of
the initial data: since the velocity is initially vanishing, the shear term is
exactly zero while second order terms are non-zero. So, even for very large
$\sigma$ and very small gradients, we have to wait for the system to evolve
for some time until first order terms become larger than second order ones.

We continue by studying Criterion B. In Fig. 7 (left, bottom) we plot the
$a_{1}$ term and the shear term at $x=0$. Moreover, in Fig. 8 (middle, left)
we plot the $a_{1}$ and $a_{2}$ terms together with $10\%$ of the shear term
in all spacetime. We find that the $a_{1}$, $a_{2}$ terms are smaller than
$10\%$ the shear term in most of the domain, except for small regions where
the shear term vanishes. However, computing the $L_{1}$ norms of the $a_{1}$
and $a_{2}$ terms and the shear term shows that Criterion B is satisfied at
all times.

In order to employ criterion C we perform another real-time evolution with
similar initial data but in frame $\\{a_{1},a_{2}\\}=\\{10,10\\}$. In Fig. 7
(bottom, left) we show the result of this evolution in dashed lines. Recall
that it is justified to use exactly the same initial data as the change of
frame is exactly vanishing at $t=0$, as it satisfies the ideal hydrodynamics
equations. We also perform an evolution using the equations of ideal
hydrodynamics with similar initial data, eqs. (3.11a), (3.11b). We plot the
difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in absolute value in
solid green in Fig. 7 (bottom, left), and the difference
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey. We also plot
these two quantities in the whole spacetime in Fig. 8 (bottom, left). We
observe that the first difference is smaller than $10\%$ the second difference
in almost all the domain, and by computing the corresponding $L_{1}$ norms we
conclude that Criterion C is satisfied.

In the previous subsection we learned that the ratio between the $a_{1}$ terms
in two evolutions in different frames
$\\{a_{1},a_{2}\\}=\\{5,5\\},\\{10,10\\}$ was about a factor of 4, according
to the approximate expression (3.5). Even if the approximate expression is not
applicable in these solutions, as they are well in the non-linear regime, we
still find a difference which is roughly a factor of 4.

Let us also comment that in Fig. 8, top and middle plots are for the case
$\\{a_{1},a_{2}\\}=\\{5,5\\}$. For the case $\\{a_{1},a_{2}\\}=\\{10,10\\}$,
the top plot would be basically similar with Criterion A satisfied or violated
in the same regions, but the middle plot would be qualitatively different: the
$a_{1}$, $a_{2}$ terms are much larger, roughly by a factor of 4 and 2
respectively. This makes that Criterion B is slightly violated in this case,
but it is restored at times $t\overline{T}\simeq 4.55$. This is expected as
larger values of $\\{a_{1},a_{2}\\}$ make the $a_{1}$, $a_{2}$ terms trivially
larger. Actually, the values $\\{a_{1},a_{2}\\}=\\{10,10\\}$ could be
considered large as they are far from saturating the inequalities (2.9); for
this reason we do not consider this as a counterexample to our main
conclusion.

Figure 7: (top, left): Spacetime evolution of the $T_{tt}$ component of the
stress tensor with initial data of a gaussian, large amplitude perturbation of
a homogeneous thermal state (3.10), width $\sigma\overline{T}\simeq 1.45$ and
frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$. In (bottom, left) we show the log plot of
the absolute value of the different contributions to the $T_{xx}$ component of
the stress tensor (2.8) at $x=0$ as a function of time: ideal term in solid
black, shear term in solid blue, $a_{1}$ term in solid red ($a_{2}$ term is
vanishing at $x=0$ by symmetry). We include the result of evaluating the
second order terms (2.10) in solid purple. We also include the data for
another run, with similar initial data, performed in a different frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$, in dashed lines. We plot the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in solid green and the
difference $T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey. See
Fig. 8 for plots in all spacetime domain. In the plots on the right we present
the result of performing the same exercise but with smaller width
$\sigma\overline{T}\simeq 0.73$ in the initial data, and it corresponds to a
system marginally in the regime of hydrodynamics.

$\displaystyle Criterion$$\displaystyle A$ 10%Shear Second order$\displaystyle
Criterion$$\displaystyle B$ 10%Shear $a_{1}$term $a_{2}$term$\displaystyle
Criterion$$\displaystyle C$
10%$\left(T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}\right)$
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$

Figure 8:  For the evolutions presented in Fig. 7 we show here relevant
quantities for criteria A, B and C in the spacetime domain. In the top panels
we plot a $10\%$ of the shear term in blue and second order terms in purple
(ideal terms are much larger than shear, so we do not plot them). In middle
panels we plot a $10\%$ of the shear term (blue) and $a_{1}$, $a_{2}$ terms
(red, orange). In bottom panel we plot a $10\%$ of
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey and
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in solid green. Left plots
correspond to the case $\sigma\overline{T}\simeq 1.45$, and right plots to
$\sigma\overline{T}\simeq 0.73$. All quantities are in absolute values. Top an
middle panels correspond to evolutions in frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$.

We proceed now with another simulation with initial data (3.10) and a smaller
width, $\sigma\overline{T}\simeq 0.73$, so that the system now has larger
gradients and it is slightly outside the regime of hydrodynamics. We plot the
result in Fig. 7 (top, right) for the $T_{tt}$ component of the stress tensor.
In Fig. 7 (bottom, right) we plot the different elements of the $T_{xx}$
component of the stress tensor in solid lines. This case shares an important
aspect with the QGP created in heavy-ion collisions, which is that initially
both systems are marginally in the regime of hydrodynamics. Therefore, even
though the two systems are very different, one may hope that the conclusions
obtained here might be relevant for the implementation of the first order
viscous hydrodynamic equations to describe the QGP in experiments.

The system violates Criterion A up to times $t\overline{T}\simeq 12.3$, when
it is restored; this is the hydrodynamization time. The initial Gaussian
explodes into two waves that travel in opposite directions, as we observe in
Fig. 7 (top, right). Initially, each of these waves has large gradients, which
are dissipated as they travel, resulting in smoother profiles; at times
$t\overline{T}\simeq 12.3$ they are smooth enough so that Criterion A is
satisfied (in the plots we show times only up to $t\overline{T}\simeq 6.84$
but we have run the simulations for longer). Relevant plots are Fig. 7
(bottom, right) and Fig. 8 (top, right).

Criterion B is slightly violated at initial times, and it is restored around
times $t\overline{T}\simeq 2.1$, and satisfied from this time on. Relevant
plots are Fig. 7 (bottom, right) and Fig. 8(middle, right).

In order to evaluate criterion C we perform another evolution in frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$ with similar initial data and also an
evolution using the ideal hydrodynamics equations. We plot the difference
$T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$ in absolute value in solid green
in Fig. 7 (bottom, right), and the difference
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey. We also plot
these quantities in the whole spacetime in Fig. 8 (bottom, right). We find
that criterion C is satisfied at all times. Thus, this is another example of a
system that is marginally in the regime of hydrodynamics (actually slightly
away according to our criterion A) and yet Criterion C is satisfied, which
indicates the robustness of the frame independence of the first order physics.
We have checked that Criterion C continues to be satisfied for values of
$\sigma$ all the way down to $\sigma\overline{T}\simeq 0.24$.

The main conclusions obtained from the analysis above are the following. We
observe that if Criterion A is satisfied, then criteria B and C are satisfied.
We find that even if Criterion A is slightly violated (and B is also
violated), Criterion C is still satisfied. This suggests that the frame
independence of the first order physics is robust and it applies slightly
beyond the hydrodynamics regime.

####  3.2.1 Changing frame in the initial data

We now explore the effect of changing frame in the initial data. To do so, we
perform a similar exercise as in Fig. 5; we start from the evolutions in the
frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$ presented in Fig. 7 and use the solution
at a timeslice $t>0$ to obtain initial data for the frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$ via (2.7). In Fig. 9 (left, dashed lines) we
present the same evolution as in Fig. 7 (left) but now performed in frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$, initialized with data from the frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$ at $t\overline{T}\simeq 1.71$. In Fig. 9 (right)
we present the same exercise for the case $\sigma\overline{T}\simeq 0.73$.

The conclusions are similar to those in the previous exercise described in
Fig. 5: The difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$
introduced by changing frame in the initial data, Fig. 9, is comparable to the
difference introduced by the time evolution, Fig. 7. Criteria A, B and C are
satisfied at a similar level as for the evolutions with vanishing initial
change of frame effects, Fig 7, with similar times for when each criterion is
violated or satisfied. Moreover, for completeness we repeated both exercise
for times $t\overline{T}\simeq 3.42,5.13$ obtaining similar conclusions.

Figure 9:  (Left) Solid lines correspond to the same evolution as in Fig. 7
(left), which is in frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$. We include in dashed
lines and same color coding the result of using data at $t\overline{T}\simeq
1.71$ (time indicated by a vertical dashed line) to perform a change of frame
to $\\{a_{1},a_{2}\\}=\\{10,10\\}$ and use it as initial data for an evolution
in frame $\\{a_{1},a_{2}\\}=\\{10,10\\}$. We show in solid green the
difference $T^{\text{visc1}}_{xx}-T^{\text{visc2}}_{xx}$. Moreover, we perform
an evolution of the ideal hydrodynamics equations and we show the difference
$T^{\text{visc1}}_{xx}-T^{\text{ideal}}_{xx}$ in solid grey. (Right): Similar
exercise as in left for the case $\sigma\overline{T}\simeq 0.73$, to analyse a
situation which is marginally in the regime of hydrodynamics. The main
conclusion from these plots is that changing frame in the initial data
according to the prescription of effective field theory introduces a small
difference comparable to the difference due to perform evolutions in different
frames analysed in Fig. 7 .

###  3.3 Shockwaves

In this section we study the effects of using different causal frames in
shockwave solutions. Shockwaves are discontinuous solutions of the ideal
hydrodynamic equations, which are smoothed out once viscosity is included. The
notion of shockwave may sound like a strong disturbance and even if viscous
hydrodynamics provides a continuous profile, one could think that gradients
are still large, suggesting that the solution is not in the effective field
theory regime. However, this intuition is not correct. Shockwave solutions of
viscous hydrodynamics are in the regime of validity of effective field theory
as long as their amplitude is small [5].181818Small amplitude shockwaves admit
analytical solutions [5], in which the width is inversely proportional to the
amplitude of the shockwave, and thus if the amplitude is small, the width is
large and gradients are small. This picture is confirmed by numerical results,
in which we can consider shockwaves of finite amplitude and check up to which
amplitude Criterion A is still satisfied. In the examples below we confirm
that they satisfy Criterion A up to velocities of the order of $v=0.66$.

We start by considering a smoothed out version of the Riemann problem with
initial data given by

$\displaystyle\epsilon|_{t=0}$
$\displaystyle=\overline{\mathcal{E}}\left[\textstyle
1+\frac{\Sigma-1}{2}\left(\tanh(\frac{x-\Delta}{\tilde{\sigma}})-\tanh(\frac{x+\Delta}{\tilde{\sigma}})\right)\right]\,\,,$
(3.11a) $\displaystyle\partial_{t}\epsilon|_{t=0}$ $\displaystyle=0\,\,,$
(3.11b) $\displaystyle u_{x}|_{t=0}$ $\displaystyle=0\,\,,$ (3.11c)
$\displaystyle\partial_{t}u_{x}|_{t=0}$ $\displaystyle=0\,\,.$ (3.11d)

Since we impose periodic boundary conditions, we consider initial data that is
$x\leftrightarrow-x$ symmetric. In the following plots we only show the region
$x>0$.191919We do not impose that the ideal equations are satisfied at $t=0$
because we will focus on the shockwave formed at later times, so for this
purpose it is not relevant if the ideal equations are initially satisfied or
not. We perform a numerical evolution of the initial data (3.11) with
parameters $\overline{\mathcal{E}}=1$, $\Sigma=3$, $\Delta\overline{T}\simeq
20.5$, $\tilde{\sigma}\overline{T}\simeq 0.684$, on a spatial domain
$L\overline{T}\simeq 136.8$ and frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$. In this
setup the energy $\overline{\mathcal{E}}$ and temperature $\overline{T}$ are
the ones of the low energy homogeneous region of the initial data.

The initial sharp profile evolves into a rarefaction wave that propagates to
the left and a shockwave that propagates to the right, with a homogeneous
region in between. See Fig. 10 (top, left) for the profile of the $T_{xx}$
component of the stress tensor. This is the expected solution in relativistic
hydrodynamics [47]; studies of the Riemann problem in the case of first-order
viscous hydrodynamics were done in [31, 33, 34]. In Fig. 10 (top, right) we
show the profile of the $T_{xx}$ component of the stress tensor in the
shockwave solution at times $t\overline{T}=\\{20.5,34.2,47.9,61.5\\}$. In Fig.
10 (bottom, left) we show the same profiles shifted to have their mid height
at $x=0$ so that they appear superimposed. This plot shows how the profile of
the shockwave evolves in time and it asymptotes to a final configuration at
late times. Also, the velocity of the shockwave asymptotes to a terminal
velocity $v_{\text{shockwave}}\simeq 0.66$.

We can compare the asymptotic solution that we obtain from our numerical
evolutions with the solutions of the ODEs that follow from considering the
system in the local rest frame of the shockwave and assuming stationarity
[31]. Under these conditions, the conservation equation for the stress tensor
$\partial_{x}T^{x\mu}=0$ can be easily integrated to obtain $T^{xt}=C_{1}$ and
$T^{xx}=C_{2}$, where $C_{1}$ and $C_{2}$ are two real constants. Solving for
$\\{\epsilon^{\prime},v^{\prime}\\}$ gives

$\displaystyle\epsilon^{\prime}$
$\displaystyle={\frac{-4\epsilon\sqrt{1-v^{2}}\left((a_{1}-4)C_{1}+((3a_{2}+4a_{1}-4)C_{2}-(4+a_{2})\epsilon)v-3(a_{1}+2a_{2})C_{1}v^{2}+3a_{2}(C_{2}+\epsilon)v^{3}\right)}{9a_{1}a_{2}\eta(v-c_{1})(v-c_{2})(v-c_{3})(v-c_{4})}}\,\,,$
(3.12a) $\displaystyle v^{\prime}$
$\displaystyle={\frac{-(1-v^{2})^{\frac{3}{2}}\left(a_{2}(\epsilon-3C_{2})+3(a_{1}+2a_{2})C_{1}v-3((4a_{1}+a_{2})C_{2}+a_{2}\epsilon)v^{2}+9a_{1}C_{1}v^{3}\right)}{9a_{1}a_{2}\eta(v-c_{1})(v-c_{2})(v-c_{3})(v-c_{4})}}\,\,,$
(3.12b)

where $c_{i}$, $i=1,2,3,4$ are the characteristic velocities,

$c_{i}=\pm\sqrt{\frac{a_{1}(2+a_{2})\pm
2\sqrt{a_{1}(a_{1}+a_{1}a_{2}+a_{2}^{2})}}{3a_{1}a_{2}}}\,.$ (3.13)

Boosting the solutions of these ODEs so that the fluid in front of the shock
is at rest, we obtain a profile for the shockwave that can be compared with
our asymptotic solutions, dashed black curve in Fig. 10 (bottom, left). This
comparison confirms that indeed the late time asymptotic profile coincides
with the solution of the ODEs. The latter are particularly interesting and
useful because they are computationally cheaper (and more accurate) to obtain.
For this reason, in the rest of this section we will use the solutions to the
ODEs to study the effect of using different frames in shockwaves.

The two asymptotic regions in a shockwave solution as well as its velocity are
related to each other by the stress tensor conservation, which gives the
Rankine-Hugoniot conditions. Viscous hydrodynamics provides a smooth
transition between the two asymptotic states, but the latter and the velocity
of the shockwave do not depend on using the viscous theory. Thus, the change
of frames will only affect the specific shape of the profile but not the
asymptotic regions nor the velocity of the shock. In the following we study
the profile of $T_{xx}$ in a shockwave solution in a Lorentz frame such that
the fluid in front of the shock is at rest; in the local rest frame of the
shockwave this analysis would be trivial as $T_{xx}$ is a constant.

Figure 10:  (Top, left) Profile of the $T_{xx}$ component of the stress tensor
at $t\overline{T}=0$ corresponding to the initial data (3.11) and at later
times when the system has evolved into a rarefaction wave propagating to the
left, a shockwave propagating to the right and a homogeneous region in
between. (Top, right) Profile of the $T_{xx}$ component of the stress tensor
of the shockwave at times $t\overline{T}=\\{20.5,34.2,47.9,61.5\\}$. (Bottom,
left) Same profiles as in (top, right) shifted to have their midheight at
$x=0$. We also include in dashed black the solution obtained by solving the
ODEs. (Bottom, right) Different contributions to the $T_{xx}$ component of the
stress tensor of the shockwave at late times. We also include second order
terms as in previous sections. Criteria A and B are satisfied for this
solution.

In Fig. 10 (bottom, right) we show the different contributions of the gradient
expansion to the $T_{xx}$ component of the stress tensor. By computing the
norms, we find that Criterion A is only marginally satisfied since the ratio
of the norms of the second order and the shear terms is $9\%$. Therefore,
according to this criterion, the solution is marginally within the effective
field theory regime. In addition, the ratio of the norms of the $a_{1}$ term
($a_{2}$ term) and the norm of the shear term is $3.9\%$ ($7.3\%$). Hence,
Criterion B is also satisfied.

In order to study Criterion C we construct a shockwave solution in the frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$ with the same asymptotics, see Fig. 11 (left)
in solid grey for the $T_{xx}$ component of the stress tensor, together with
the solution in frame $\\{a_{1},a_{2}\\}=\\{5,5\\}$ in dashed black. We
observe that both profiles are very similar to each other, see the inset for a
detailed view of the difference. We note that Criterion C as previously
defined is not applicable in shockwave solutions because in ideal
hydrodynamics the shocks are discontinuous. In this case we use the following
alternative criterion: we consider that the criterion is satisfied if the
maximum difference between the two profiles divided by the difference between
the asymptotic states is smaller than $5\%$.202020As before, this particular
choice of threshold is arbitrary. In our solution this ratio is $0.48\%$, and
hence this criterion would be satisfied.

References [30, 31, 33, 34] provide evidence that the first-order viscous
hydrodynamics equations admit shockwave solutions as long as their velocity
profile does not cross any characteristic velocity. We have verified that this
is indeed the case for the chosen frames,
$\\{a_{1},a_{2}\\}=\\{5,5\\},\,\\{10,10\\}$, for which the characteristic
velocities are always larger than the upstream velocity. We also verified that
for frames with characteristic velocities that cross the velocity profile of
the shockwave (in the local rest frame), the real-time evolutions crash and
the ODEs do not seem to admit smooth shockwave solutions, in agreement with
the results in [30, 31, 33, 34].

Sharply causal frames, that is, frames saturating the inequalities (2.9), are
specially interesting as they allow to capture arbitrarily large shocks while
respecting causality. In Fig. 11 (right) we repeat the same exercise as above
but in this case with two frames in the sharply causal region, namely
$\\{a_{1},a_{2}\\}=\\{4,4\\}$ and
$\\{a_{1},a_{2}\\}=\\{\frac{25}{4},\frac{25}{7}\\}$. We find that for these
solutions Criteria A, B and C are satisfied. In this case the ratio of the
maximum difference and the differences of the asymptotic states is $0.08\%$;
this ratio is smaller than above as the two frames here are ‘close’ to each
other. It would be interesting to have a precise notion of the separation
between two frames in the space of frames (with, for example, some metric),
which could be used in the definition of Criterion C. Otherwise two solutions
can trivially be very close to each other just because the parameters defining
the corresponding frames are very close to each other, even if such solutions
are not in the effective field theory regime. We leave this for future work.

Figure 11: Comparison of the $T_{xx}$ component of the stress tensor of
shockwave solutions with similar asymptotic regions and different frames. For
these comparisons we use solutions obtained by solving the ODEs and boosting
the solution such that the fluid in front of the shock is at rest. Solutions
are shifted to have their midheight at $x=0$. The inset corresponds to the
difference of the two solutions. (Left) The black dashed line corresponds to
the same solution as in Fig. 10 (bottom, left) which is in frame
$\\{a_{1},a_{2}\\}=\\{5,5\\}$, and we compare it with the solution with frame
$\\{a_{1},a_{2}\\}=\\{10,10\\}$, in solid grey. The ratio of the maximum
difference to the amplitude of the shockwave is $0.48\%$. (Right) Similar
comparison as in (left) and with similar asymptotics, but in this case we use
two sharply causal frames $\\{a_{1},a_{2}\\}=\\{4,4\\}$ and
$\\{a_{1},a_{2}\\}=\\{\frac{25}{4},\frac{25}{7}\\}$. In this case the ratio is
$0.08\%$.

We now consider stronger shocks. We use the ODEs to obtain shockwave solutions
with velocity $v_{shockwave}=0.85$ in the frames $\\{a_{1},a_{2}\\}=\\{4,4\\}$
and $\\{a_{1},a_{2}\\}=\\{\frac{25}{4},\frac{25}{7}\\}$. In Fig. 12 (left) we
display the corresponding $T_{xx}$ component of the stress tensor, after
performing a boost to a Lorentz frame in which the fluid in front of the shock
is at rest and has energy $\overline{\mathcal{E}}=1$. We find that criteria A
and B are violated whilst Criterion C is satisfied. This indicates that the
system is not in the effective field theory regime and yet the two profiles
are still very close to each other, with a ratio of the maximum deviation to
the amplitude of the shockwave of $1.4\%$. We note that the $a_{1}$, $a_{2}$
terms are not much smaller than the shear term, so in Fig. 12 (right, dashed
black) we plot the full first order term $T_{xx}^{(1)}$ in the stress tensor.

We conclude that also in shockwave solutions criterion C is robust even if the
system is exiting the effective field theory regime. On the other hand, for
sufficiently large velocities Criterion C is also violated. For example, for
solutions with $v_{shockwave}=0.9999$ in frames $\\{a_{1},a_{2}\\}=\\{4,4\\}$
and $\\{a_{1},a_{2}\\}=\\{\frac{25}{4},\frac{25}{7}\\}$ we find that the ratio
of the maximum difference to the amplitude of the shockwave is $41\%$.

Figure 12: (Left) Shockwave solutions with velocity $v_{shockwave}=0.85$ in
frames $\\{a_{1},a_{2}\\}=\\{4,4\\}$, dashed black, and
$\\{a_{1},a_{2}\\}=\\{\frac{25}{4},\frac{25}{7}\\}$, solid grey. Solutions are
shifted to have their midheight at $x=0$. The inset shows the difference of
the two profiles. (Right) Different contributions to the $T_{xx}$ component of
the stress tensor for the solution in frame $\\{a_{1},a_{2}\\}=\\{4,4\\}$. We
include the value of the full first order term $T_{xx}^{(1)}$ in black dashed,
as in this case the $a_{1}$, $a_{2}$ terms are not much smaller than the shear
term. Criteria A and B are violated and criterion C satisfied. The ratio of
the maximum deviation to the amplitude of the shockwave is $1.4\%$.

##  4 Discussion

In this paper we studied real-time numerical evolutions of the relativistic
first-order viscous hydrodynamics equations for a conformal fluid. Our main
goal was to explore the consequences of using different hydrodynamic frames
(or field redefinitions) within the causal set of frames (2.9) in practical
situations. By performing evolutions with different sets of initial data and
by considering specific quantitative criteria, we have made precise and
provided evidence for the statement that the physics up to first order remains
invariant under field redefinitions as long as the system is within the
effective field theory regime.

Our studies are motivated by the future implementation of the relativistic
first-order viscous hydrodynamics equations to model physical systems of
interest such as the QGP created in heavy-ion collision experiments or
astrophysical systems like neutron star mergers. An important step towards
this goal is to understand to which extent the physics is independent of the
choice of hydrodynamic frame. Providing specific quantitative criteria for
this assessment is the main motivation for this paper.

We considered the following sets of initial data. First, we considered a small
amplitude sinusoidal perturbation of a homogeneous thermal state. Such a well-
controlled system allows to define the criteria that we then use in more non-
trivial settings. We continued by considering large amplitude Gaussian
perturbations of a homogeneous thermal state and finally shockwave solutions.

We proposed three criteria, denoted by A, B and C, that assess different
aspects of our solutions. The detailed definitions of these criteria can be
found in Section 3.1, and we summarize them here. They involve measuring the
‘size’ of certain terms in the gradient expansion of the stress tensor in our
solutions at a certain time $t$. To do so we will use the $L_{1}$ norm
computed over the spatial domain and times $\\{t-\tilde{t},t+\tilde{t}\\}$,
where $\tilde{t}$ is a characteristic timescale of the problem. Here we use
the $L_{1}$ norm for convenience, but other norms are possible and should give
similar results. In practical applications one may need to be quantitative
about what ‘much smaller’ means; here we use $10\%$, but this threshold value
is arbitrary and in practice it may depend on the initial conditions and/or
the details of the fluid under consideration. In future work we will make this
precise in the case of holographic fluids, where we have control over the
microscopic theory.

Criterion A. We measure the sizes of the ideal and first order terms on our
solutions by evaluating the corresponding expressions in (2.8). In addition,
for each solution, we also measure the size of the second order terms by using
the second line of (2.10). In this way we compare the ideal, first and second
order terms in the gradient expansion and we check if each term is much
smaller than the previous one. This is the standard criterion in truncated
effective field theories. Our criterion A is: the system is in the effective
field theory regime at a given time $t$ if, for all components of the stress
tensor, the $L_{1}$ norm of the shear term is much smaller than the $L_{1}$
norm the ideal term, and in turn the $L_{1}$ norm of the second order terms is
much smaller than the $L_{1}$ norm of the shear term.

Criterion B. We measure the size of the $a_{1}$, $a_{2}$ terms in (2.8) and
compare them with the size of the shear term. The motivation for this
criterion is as follows. The change of frame terms (2.7) are proportional to
the lowest order equation of motion, that is, the equations of ideal
hydrodynamics (2.5). Thus, by using the equations of first-order viscous
hydrodynamics these terms are, on shell, equivalent to second order terms.
Then, if the system is in the effective field theory regime, the frame
dependent terms in (2.8) should be much smaller than the shear term, which is
frame independent. With this motivation, we propose the following criterion:
the physics to first order is independent of the chosen causal frame at a
given time $t$ if, for the spatial components of the stress tensor, the
$L_{1}$ norms of the $a_{1}$ and $a_{2}$ terms are much smaller than the
$L_{1}$ norm of the shear term.

Criterion C. We consider two evolutions in different (and sufficiently
separated, at least in parameter space) causal frames, and a third evolution
using ideal hydrodynamics; we compare the difference between the two viscous
evolutions with the difference between one viscous evolution and the ideal
evolution. In this way we compare the effect of changing frame with the effect
of including first order viscous terms in the hydrodynamics description. Our
specific criterion is: the physics to first order is independent on the chosen
hydrodynamic frame at a given time $t$ if the $L_{1}$ norm of the difference
between the stress tensors obtained from two evolutions performed in different
causal frames, $T^{\text{visc1}}_{\mu\nu}-T^{\text{visc2}}_{\mu\nu}$, is much
smaller than the $L_{1}$ norm of the difference between the stress tensors
obtained with ideal hydrodynamics and one of the viscous evolutions,
$T^{\text{visc}}_{\mu\nu}-T^{\text{ideal}}_{\mu\nu}$.

The main conclusion of this paper is that in all solutions that we have
studied, if Criterion A is satisfied, then criteria B and C are are also
satisfied; this is a quantitative and precise version of the statement that if
the system is in the effective field theory regime, then the physics to first
order is independent of the arbitrarily chosen frame.

Thus, we confirm that it is consistent to truncate the hydrodynamic gradient
expansion to include only the viscous terms. This is a non-trivial result
since each of these three criteria assesses a different aspect of what it
means for a solution to be ‘in the effective field theory regime’. One might
have thought that a given solution is indeed well described by viscous
hydrodynamics only of the three criteria are simultaneously satisfied. We have
shown that this is not the case and Criterion A is sufficient to imply to
other two.

From the main conclusion and from a practical perspective, one could think of
the terms $a_{1}$, $a_{2}$ as mere regulators of the equations (that make them
well posed) and work as if we were in the usual Landau frame, as the choice of
frame difference is a higher order effect. However, this is only applicable
when the system is well within the effective field theory regime, and one must
be careful with this statement if it is not clear that the system is in that
regime.

One possibility that could happen is that Criterion B is satisfied and the
terms $a_{1}$, $a_{2}$ are small, but the effects of choosing different frames
accumulate over sufficiently long times and they eventually become large. That
is, can Criterion B be satisfied and C violated? In our solutions we have
always observed that if Criterion B is satisfied, then Criterion C is also
satisfied.

Regarding Criterion B, even if in most cases we have found that when Criterion
A is satisfied then Criterion B is also satisfied, we have found some
counterexamples; for example see discussion after eq. (3.7). However, these
counterexamples only happen when the values of $\\{a_{1},a_{2}\\}$ are
unreasonable large. If we restrict the values of $\\{a_{1},a_{2}\\}$ to be
close to saturating the inequalities (2.9), then these counterexamples can be
ruled out. Moreover, in these solutions we found that Criterion C is
satisfied, even if the $a_{1}$, $a_{2}$ terms are not small compared to shear
term.

In the actual description of the QGP created in heavy-ion collision
experiments, when the hydrodynamic codes are initialized, there are situations
in which gradients are large. This motivates us to consider solutions in which
the system is marginally in the effective field theory regime, that is,
Criterion A is almost violated or slightly violated.212121See Figs. 3(right),
5(right), 6(right), 7(right) and 12(top). We want to test if the physics to
first order is independent of the arbitrarily chosen frame even under these
circumstances. A generic conclusion from our studies is that even if the
system is not well within the regime of hydrodynamics, that is, Criterion A is
slightly violated, the change of frame still does not significantly affect the
viscous evolutions compared to the ideal evolution, that is, Criterion C is
still satisfied; we also observe that Criterion B is also slightly violated.
Therefore, we conclude that the physics to first order is robust against
changes of frame according to Criterion C, even in situations where the system
is on the boundary of the regime of applicability of effective field theory.

We emphasize that it is convenient in practice to choose hydrodynamic frames