# Does stability of relativistic dissipative fluid dynamics imply causality?

Shi Pua,c Tomoi Koideb Dirk H. Rischkea,b aInstitut für Theoretische Physik,
Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am
Main, Germany  bFrankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1,
D-60438 Frankfurt am Main, Germany  cDepartment of Modern Physics, University
of Science and Technology of China, Hefei 230026, P.R. China

###### Abstract

We investigate the causality and stability of relativistic dissipative fluid
dynamics in the absence of conserved charges. We perform a linear stability
analysis in the rest frame of the fluid and find that the equations of
relativistic dissipative fluid dynamics are always stable. We then perform a
linear stability analysis in a Lorentz-boosted frame. Provided that the ratio
of the relaxation time for the shear stress tensor, $\tau_{\pi}$, to the sound
attenuation length, $\Gamma_{s}=4\eta/3(\varepsilon+P)$, fulfills a certain
asymptotic causality condition, the equations of motion give rise to stable
solutions. Although the group velocity associated with perturbations may
exceed the velocity of light in a certain finite range of wavenumbers, we
demonstrate that this does not violate causality, as long as the asymptotic
causality condition is fulfilled. Finally, we compute the characteristic
velocities and show that they remain below the velocity of light if the ratio
$\tau_{\pi}/\Gamma_{s}$ fulfills the asymptotic causality condition.

##  I Introduction

Data from the Relativistic Heavy-Ion Collider (RHIC) on the collective flow of
matter in nucleus-nucleus collisions have delivered a surprising result: the
elliptic flow coefficient $v_{2}$ is sufficiently large Adams:2003zg ;
Adams:2003am ; Sorensen:2003kp ; Adler:2002pu  to be compatible with
calculations performed in the framework of ideal fluid dynamics review . This
has given rise to the notion that “RHIC physicists serve up the perfect
liquid” press ; Gyulassy:2004zy ; Shuryak:2004cy .

Of course, no real liquid can have zero viscosity: for all weakly coupled
theories, i.e., theories with well-defined quasi-particles, in the dilute
limit there is a lower bound which one can derive from the uncertainty
principle Danielewicz:1984ww : the ratio of shear viscosity to entropy density
$\eta/s\gtrsim 1/12$. For certain strongly coupled theories without
quasiparticles, there is also a lower bound which can be obtained from the
AdS/CFT conjecture Kovtun:2004de , $\eta/s\geq 1/(4\pi)$, i.e., surprisingly
close to the bound for dilute, weakly coupled systems.

In order to see whether the shear viscosity of the hot and dense matter
created in nuclear collisions at RHIC is close to the lower bound, one has to
perform calculations in the framework of relativistic dissipative fluid
dynamics. This program has only been recently initiated, but has already led
to an enormous activity in the literature muronga ; roma ; luzum1 ; luzum2 ;
Song:2007ux ; Song:2007fn ; chau ; du ; pasi ; pratt ; bha ; mol ; Betz:2008me
; gue ; dkkm1 ; dkkm2 ; dkkm3 ; dkkm4 ; dkkm5 ; knk .

Fluid dynamics is an effective theory for the long-wavelength, small-frequency
modes of a given theory. In order to see this, let us introduce three length
scales: (a) a microscopic length scale, $\ell_{\rm micro}$. In all theories,
at sufficiently large temperatures this length scale can be defined as the
thermal wavelength $\lambda_{\rm th}\sim 1/T$. In weakly coupled theories with
well-defined quasi-particles, this can be interpreted as the interparticle
distance. (b) A mesoscopic length scale, $\ell_{\rm meso}$. In weakly coupled
theories and in the dilute limit, this can be identified with the mean-free
path of particles between collisions. In strongly coupled theories, such a
scale is not known and should be identified with $\ell_{\rm micro}$. (c) A
macroscopic length scale, $\ell_{\rm macro}$. This is the scale over which the
conserved densities (e.g. the charge density, $n$, or the energy density,
$\varepsilon$) of the theory vary. Thus, $\ell_{\rm
macro}^{-1}\sim|\partial\varepsilon|/\varepsilon$, i.e., $\ell_{\rm
macro}^{-1}$ is proportional to the gradients of the conserved quantities.

We now define the quantity $K\equiv\ell_{\rm meso}/\ell_{\rm macro}$. For
dilute systems, this quantity is identical to the so-called Knudsen number. If
$K$ is sufficiently small, fluid dynamics as an effective theory can be
derived in a controlled way as a power series in terms $K$. Since
$K\sim\ell_{\rm macro}^{-1}$, this series expansion is equivalent to a
gradient expansion.

To zeroth order in $K$, one obtains the equations of ideal fluid dynamics. To
first order in $K$, one obtains the Navier-Stokes (NS) equations. So-called
second-order theories contain terms of second order in $K$. Examples for the
latter are the Burnett equations samojeden , the Israel-Stewart equations for
relativistic dissipative fluid dynamics is , the memory function theory dkkm1
; dkkm4 , extended thermodynamics jou ; dkkm4 , and others else . The main
difference between first and second-order theories is the velocity of signal
propagation. The relativistic NS equations allow for infinite signal
propagation speeds and are therefore acausal. On the other hand, all second-
order theories are considered to be causal in the sense that all signal
velocities are smaller than the speed of light, provided that the parameters
of the theory are suitably chosen.

The stability and causality of fluid-dynamical theories are usually studied
around a hydrostatic state (i.e., for vanishing macroscopic flow velocity)
which is in thermodynamical equilibrium. However, if a theory is stable around
a hydrostatic state, it does not necessarily imply that it is stable in a
state of nonzero flow velocity. Following this idea, the stability and
causality of first and second-order fluid dynamics for a state with nonzero
background flow velocity (mathematically realized by a Lorentz boost) were
studied for the case of nonzero bulk viscosity, but for vanishing shear stress
and heat flow in Ref. dkkm3 . There it was found that causality and stability
are intimately related: for all parameters considered, the theory becomes
unstable if and only if there is a mode which propagates faster than the speed
of light.

In this paper, we extend this analysis to the case of nonvanishing shear
viscosity in second-order theories of relativistic dissipative fluid dynamics.
A similar analysis for a hydrostatic background has already been done by
Hiscock, Lindblom, and Olson his ; his2 , but they discussed exclusively the
low- and high-wavenumber limits his2 . As we shall show in this paper, their
analysis missed a divergence of the group velocity of a shear mode at
intermediate wavenumbers. This anomalous behavior is generic, i.e., it cannot
be removed by tuning the parameters of the theory, e.g., the relaxation time
for the shear stress tensor, $\tau_{\pi}$, and the shear viscosity, $\eta$.
However, if the ratio $\tau_{\pi}/\Gamma_{s}$, where
$\Gamma_{s}=2(D-2)\eta/[(D-1)(\varepsilon+P)]$ is the sound attenuation length
in $D$ space-time dimensions, is chosen such that the large-momentum limit of
the group velocity associated with the perturbation remains below the velocity
of light (the so-called asymptotic causality condition), one can ensure that
the divergence is restricted to a finite range of momenta. It will be
demonstrated that in this case, the causality of the theory is not
compromised. On the other hand, second-order fluid dynamics is always stable
in the rest frame of the fluid, even if we use a parameter set which violates
the asymptotic causality condition.

We also study the causality and stability for a state with nonzero background
flow velocity, i.e., in a Lorentz-boosted frame. We find that the divergence
of the group velocity is removed. However, depending on the boost velocity the
group velocity of either the shear or the sound mode may still exceed the
speed of light in a certain range of wavenumbers. Nevertheless, provided that
the ratio $\tau_{\pi}/\Gamma_{s}$ fulfills the asymptotic causality condition,
we can show that the equations are stable. In contrast to the analysis in the
rest frame, however, they become unstable if the asymptotic causality
condition is violated. We shall demonstrate that if the asymptotic causality
condition is fulfilled, the causality of the theory as a whole is not
compromised. In this sense, causality and stability are intimately related.

So far, the discussion was limited to the fluid-dynamical equations in the
linear approximation. Therefore, we expect the results to be valid for all
versions of second-order theories presently discussed in the literature, since
they differ only by nonlinear terms. We also compute the characteristic
velocities for the so-called simplified IS equations Song:2007ux  without
linearizing these equations. Our analysis strongly indicates that the
characteristic velocities remain below the velocity of light if the ratio
$\tau_{\pi}/\Gamma_{s}$ is chosen such that the asymptotic causality condition
is fulfilled.

The asymptotic causality condition implies that, for a given
$\Gamma_{s}\sim\eta$, $\tau_{\pi}$ must not be arbitrarily small. This
explains why relativistic NS theory is acausal, because there
$\tau_{\pi}\rightarrow 0$, while $\eta$ is non-zero. It also implies that
second-order theories are not per se causal; they can violate causality (and
become unstable) if a too small value for $\tau_{\pi}$ is chosen. The
statement that second-order theories automatically cure the shortcomings of NS
theory is therefore not true.

This paper is organized as follows. In Sec. II, we discuss the causality and
stability of the linearized second-order fluid-dynamical equations in the
local rest frame. We also extend this analysis to nonzero bulk viscosity and
show that the divergence of the group velocity still exists in this case. In
Sec. III, this discussion is generalized to a Lorentz-boosted frame. We
discuss Lorentz boosts both in and orthogonal to the direction of propagation
of the perturbation. It will be demonstrated that superluminal group
velocities will not compromise the causality of the theory as long as the
asymptotic causality condition is fulfilled. In Sec. IV, we compute the
characteristic velocities in the nonlinear case. A summary of our results
concludes this work in Sec. V. An Appendix contains details of our
calculations in Sec. IV. The metric tensor is $g^{\mu\nu}={\rm
diag}(+,-,-,-)$; our units are $\hbar=c=k_{B}=1$.

##  II Stability in the rest frame

As mentioned in the Introduction, there are several approaches to formulate a
second-order theory of relativistic dissipative fluids is ; dkkm1 ; dkkm3 ;
dkkm4 ; jou ; else . These approaches differ only by nonlinear (second-order)
terms. However, since we shall apply a linear stability analysis in the
following, these differences vanish and all approaches lead to the same set of
linearized fluid-dynamical equations. In this work, we do not consider any
conserved charges and thus are left with energy-momentum conservation,

$\partial_{\mu}T^{\mu\nu}=0\;,$ (1)

where

$T^{\mu\nu}=\varepsilon\,u^{\mu}u^{\nu}-(P+\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu}$
(2)

is the energy-momentum tensor. Here, $\varepsilon$ and $P$ are the energy
density and the pressure, while $u^{\mu}$, $\Pi$, and $\pi^{\mu\nu}$ are the
fluid velocity, the bulk viscous pressure, and the shear stress tensor,
respectively. We also introduced the projection operator

$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}\;,$ (3)

which projects onto the $(D-1)$-dimensional subspace orthogonal to the fluid
velocity. We compute in the Landau frame LL , where there is no energy flow in
the local rest frame.

In second-order theories of relativistic dissipative fluid dynamics, the bulk
viscous pressure and the shear stress tensor are determined from evolution
equations. In $D$ space-time dimensions ($D\geq 3$), these equations are given
by

$\displaystyle\tau_{\Pi}\,\frac{d}{d\tau}\Pi+\Pi$ $\displaystyle=$
$\displaystyle-\zeta\,\partial_{\mu}u^{\mu}\;,$ (4a)
$\displaystyle\tau_{\pi}\,P^{\mu\nu\alpha\beta}\,\frac{d}{d\tau}\pi_{\alpha\beta}+\pi^{\mu\nu}$
$\displaystyle=$ $\displaystyle
2\eta\,P^{\mu\nu\alpha\beta}\,\partial_{\alpha}u_{\beta}\;;$ (4b)

possible other second-order terms Betz:2008me  can be neglected for the
purpose of a linear stability analysis. In Eqs. (4), the comoving derivative
is denoted by $u^{\mu}\partial_{\mu}\equiv d/d\tau$. The relaxation times for
the bulk viscous pressure and the shear stress tensor are denoted by
$\tau_{\Pi}$ and $\tau_{\pi}$, respectively. The coefficients $\zeta,\,\eta$
are the bulk and shear viscosities, respectively. We also introduced the
symmetric rank-four projection operator

$P^{\mu\nu\alpha\beta}=\frac{1}{2}\left(\Delta^{\mu\alpha}\Delta^{\nu\beta}+\Delta^{\nu\alpha}\Delta^{\mu\beta}\right)-\frac{1}{D-1}\,\Delta^{\mu\nu}\Delta^{\alpha\beta}\;.$
(5)

The shear stress tensor is traceless $\pi^{\mu}{}_{\mu}=0$ and orthogonal to
the fluid velocity $u_{\mu}\pi^{\mu\nu}=0$.

The stability and causality of a relativistic dissipative fluid with bulk
viscous pressure only have been investigated in Ref. dkkm3 . Thus, for the
sake of simplicity, we shall first ignore the effects from bulk viscous
pressure and discuss the properties of the fluid-dynamical equations of motion
including only shear viscosity. The interplay between shear and bulk viscosity
will be discussed afterwards.

###  II.1 Shear viscosity only

For convenience, we introduce the following parameterization:

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle as\;,$ (6a)
$\displaystyle\tau_{\pi}$ $\displaystyle=$
$\displaystyle\frac{\eta}{\varepsilon+P}\,b=\frac{ab}{T}\;,$ (6b)

where $s$ and $T$ are the entropy density and the temperature, respectively.
From the second equation we obtain $\tau_{\pi}(\varepsilon+P)/\eta=b$. The
parametrization (6) is motivated by the leading-order results for the causal
shear viscosity coefficient and the relaxation time obtained in Ref. knk
where the relation $\tau_{\pi}=\eta/P$ was found. For a massless ideal gas
equation of state, $\varepsilon=(D-1)P$, this result is reproduced by choosing
$b=D$.

In this section, we discuss the stability of second-order relativistic fluid
dynamics in the local rest frame. Following Ref. his ; dkkm3 , let us
introduce a perturbation $\sim e^{i\omega t-ikx}$ around the hydrostatic
equilibrium state,

$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\varepsilon_{0}+\delta\varepsilon\,e^{i\omega t-ikx}\;,$ (7a)
$\displaystyle\pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle\pi^{\mu\nu}_{0}+\delta\pi^{\mu\nu}\,e^{i\omega t-ikx}\;,$ (7b)
$\displaystyle u^{\mu}$ $\displaystyle=$ $\displaystyle u^{\mu}_{0}+\delta
u^{\mu}\,e^{i\omega t-ikx}\;,$ (7c)

where $\varepsilon_{0}={\rm const.}$, $\pi^{\mu\nu}_{0}=0$, and
$u^{\mu}_{0}=(1,0,0,\ldots)$, respectively. In the linear approximation, the
velocity perturbation has no zeroth component,

$\delta u^{\mu}=(0,\delta u^{1},\delta u^{2},\ldots,\delta u^{D-1})\;,$ (8)

because $u^{\mu}u_{\mu}=1$. Moreover, in the local rest frame,
$\delta\pi^{0\nu}\equiv 0$ on account of the orthogonality condition
$u_{\mu}\pi^{\mu\nu}=0$. Since $\pi^{\mu\nu}$ is traceless,
$\delta\pi^{(D-1)(D-1)}$ is not an independent variable. Taking all of this
into account, the linearized fluid-dynamical equations can be written as

$AX=0\;,$ (9)

where

$\displaystyle X$ $\displaystyle=$ $\displaystyle(\delta\varepsilon,\delta
u^{1},\delta\pi^{11},\delta u^{2},\delta\pi^{12},\ldots,\delta
u^{D-1},\delta\pi^{1(D-1)},$
$\displaystyle\;\;\delta\pi^{22},\delta\pi^{33},\ldots,\delta\pi^{(D-2)(D-2)},\delta\pi^{23},\delta\pi^{24},\ldots,\delta\pi^{2(D-1)},\delta\pi^{34},\ldots,\delta\pi^{(D-2)(D-1)})^{T}\;.$

The matrix $A$ is expressed as

$A=\left(\begin{array}[]{cccc}T&0&0&0\\\ 0&B&0&0\\\ G&0&C&0\\\
0&0&0&E\end{array}\right)\;,$ (10)

with

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}i\omega&f_{1}&0\\\
-ikc_{s}^{2}&f_{2}&-ik\\\ 0&\Gamma&f\end{array}\right)\;,$ (11d)
$\displaystyle B$ $\displaystyle=$ $\displaystyle{\rm
diag}(B_{0},\ldots,B_{0})_{(D-2)\times(D-2)}\;,\;\;B_{0}=\left(\begin{array}[]{cc}f_{2}&-ik\\\
\Gamma_{1}&f\end{array}\right)\;,$ (11g) $\displaystyle G$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\Gamma_{2}&0\\\ &\ldots&\\\
0&\Gamma_{2}&0\end{array}\right)_{(D-3)\times 3}\;,$ (11k) $\displaystyle C$
$\displaystyle=$ $\displaystyle{\rm diag}(f,\ldots,f)_{(D-3)\times(D-3)}\;,$
(11l) $\displaystyle E$ $\displaystyle=$ $\displaystyle{\rm
diag}(f,\ldots,f)_{\frac{1}{2}(D-2)(D-3)\times\frac{1}{2}(D-2)(D-3)}\;,$ (11m)

where $c_{s}=\sqrt{\partial P/\partial\varepsilon}$ is the velocity of sound.
Here, we introduced the abbreviations

$\displaystyle f$ $\displaystyle=$ $\displaystyle
i\omega\,\tau_{\pi}+1\;,\qquad\;f_{1}=-ik\,(\varepsilon+P)\;,$ $\displaystyle
f_{2}$ $\displaystyle=$ $\displaystyle
i\omega\,(\varepsilon+P)\;,\qquad\Gamma=-ik\,\frac{2(D-2)}{D-1}\,\eta\;,$
$\displaystyle\Gamma_{1}$ $\displaystyle=$ $\displaystyle-
ik\,\eta\;,\qquad\qquad\Gamma_{2}=ik\,\frac{2}{D-1}\,\eta\;.$

For nontrivial solutions of Eq. (9), the determinant of the matrix $A$ should
vanish. This leads to the following conditions for the dispersion relations
$\omega(k)$:

$\displaystyle f$ $\displaystyle=$ $\displaystyle 0\;,$ (12a)
$\displaystyle\det B=\left(\det B_{0}\right)^{D-2}$ $\displaystyle=$
$\displaystyle 0\;,$ (12b) $\displaystyle\det
T=\det\left(\begin{array}[]{ccc}i\omega&f_{1}&0\\\ -ik\,c_{s}^{2}&f_{2}&-ik\\\
0&\Gamma&f\end{array}\right)$ $\displaystyle=$ $\displaystyle 0\;.$ (12f)

Equation (12a) gives a purely imaginary frequency

$\omega=\frac{i}{\tau_{\pi}}\;,$ (13)

which corresponds to a nonpropagating mode. The degeneracy of this mode is
$(D-3)[1+(D-2)/2]$.

Equation (12b) leads to a complex frequency,

$\omega=\frac{1}{2\tau_{\pi}}\left(i\pm\sqrt{\frac{4\,\eta\,\tau_{\pi}}{\varepsilon+P}\,k^{2}-1}\right)\;,$
(14)

corresponding to two propagating modes, if $k$ is larger than the critical
wavenumber

$k_{c}=\sqrt{\frac{\varepsilon+P}{4\,\eta\,\tau_{\pi}}}\equiv\frac{\sqrt{b}}{2\,\tau_{\pi}}\;.$
(15)

Following Ref. baier , we shall call these modes shear modes. There are in
total $2(D-2)$ shear modes.

Equation (12f) gives the same dispersion relation as Eq. (16) of Ref. dkkm3 ,
after replacing $2(D-2)\eta/(D-1)$ with $\zeta_{0}$. Introducing the sound
attenuation length in $D$ space-time dimensions

$\Gamma_{s}\equiv\frac{2(D-2)}{D-1}\,\frac{\eta}{\varepsilon+P}\equiv\frac{2(D-2)}{D-1}\,\frac{\tau_{\pi}}{b}\;,$
(16)

the analytic solution in the limit of small wavenumber $k$ is

$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{i}{\tau_{\pi}}\;,\\\
\displaystyle\pm\,k\,c_{s}+i\,\frac{\Gamma_{s}}{2}\,k^{2}\;,\end{array}\right.$
(17)

while for large wavenumber we obtain

$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{i}{\tau_{\pi}}\,\left[1+\frac{\Gamma_{s}}{\tau_{\pi}c_{s}^{2}}\right]^{-1}\;,\\\\[8.5359pt]
\displaystyle\pm\,k\,c_{s}\sqrt{1+\frac{\Gamma_{s}}{\tau_{\pi}c_{s}^{2}}}+\frac{i}{2\tau_{\pi}}\,\left[1+\frac{\tau_{\pi}c_{s}^{2}}{\Gamma_{s}}\right]^{-1}\;.\end{array}\right.$
(18)

This corresponds to another nonpropagating mode and two propagating modes
which we call sound modes in accordance with Ref. baier . All imaginary parts
are positive and therefore the nonpropagating, as well as the shear and sound
modes are stable around the hydrostatic equilibrium state. This fact is
already known from the study of Hiscock and Lindblom his .

In order to discuss the issue of causality, we follow Ref. his ; dkkm3  and
study the group velocity defined as

$v_{g}=\frac{\partial{\rm Re}\,\omega}{\partial k}\;.$ (19)

For the two nonpropagating modes, ${\rm Re}\,\omega=0$. Consequently, in order
to discuss causality, we have to consider the behavior of the imaginary part
dkkm3 . Let us digress for the moment and consider the diffusion equation with
diffusion constant $D_{0}$. There is a nonpropagating mode with dispersion
relation $\omega=iD_{0}k^{2}$. Moreover, it is known that the diffusion
equation is acausal. Therefore, we conjecture that a $k^{2}$ dependence of any
nonpropagating mode can be considered a sign of acausality. In our case, the
nonpropagating modes are either independent of $k$, or have a weak $k$
dependence (cf. Fig. 1). According to our conjecture, we conclude that the
nonpropagating modes do not violate causality.

Figure 1: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the sound modes (full lines) and the
nonpropagating mode (dashed line) obtained from Eq. (12f). The parameters are
$a=\frac{1}{4\pi}\,,\;b=6\,,\;c_{s}^{2}=\frac{1}{3}$ for the 3+1-dimensional
case, $D=4$. Figure 2: The group velocity (22) for
$a=1/(4\pi)\,,\;D=4\,,\;c_{s}^{2}=\frac{1}{3}$, and $b=6$ (full line), $b=2$
(dashed line), as well as $b=1.5$ (dotted line).

The dispersion relations resulting from Eq. (12f) are shown in Fig. 1, and the
corresponding group velocity resulting from Eq. (19) in Fig.2. The group
velocity has a maximum for a finite value of $k/T$ and approaches its
asymptotic value ($k\rightarrow\infty$) from above. For small values of $b$,
it may thus happen that the group velocity becomes superluminal. Nevertheless,
in Sec. III.3 we shall show that only the asymptotic value determines whether
the theory as a whole is causal or not. The asymptotic value of the group
velocity is

$v_{g,{\rm sound}}^{\rm as}=\lim_{k\rightarrow\infty}\frac{\partial
Re\,\omega}{\partial
k}=c_{s}\,\sqrt{1+\frac{\Gamma_{s}}{\tau_{\pi}c_{s}^{2}}}\;.$ (20)

Consequently, for the asymptotic group velocity of sound waves to be less than
the speed of light, $\tau_{\pi}$ and $\Gamma_{s}$ should satisfy the
following, so-called asymptotic causality condition:

$\frac{\Gamma_{s}}{\tau_{\pi}}\leq
1-c_{s}^{2}\;\;\Longleftrightarrow\;\;\frac{1}{b}\equiv\frac{\eta}{\tau_{\pi}(\varepsilon+P)}\leq\frac{D-1}{2(D-2)}(1-c_{s}^{2})\;.$
(21)

This is similar to the causality condition for the group velocity in the case
of bulk viscosity, Eq. (21) of Ref. dkkm3 . For conformal fluids, where
$c_{s}^{2}=1/(D-1)$, the condition (21) simplifies to
$\Gamma_{s}\leq(D-2)\tau_{\pi}/(D-1)$ or, equivalently, $b\geq 2$. For
example, for the values of $\eta$ and $\tau_{\pi}$ deduced from the AdS/CFT
correspondence baier ; Heller:2007qt ; push , $\eta=s/(4\pi)$,
$\tau_{\pi}=(2-\ln 2)/(2\pi T)$, the condition (21) is always satisfied
because $b=2(2-\ln 2)\simeq 2.614>2$.

Figure 3: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the shear modes obtained from Eq. (12b). The
parameters are $a=\frac{1}{4\pi}\,,\;b=6\,,\;c_{s}^{2}=\frac{1}{3}$ for the
3+1-dimensional case, $D=4$. Figure 4: The group velocity (22) for
$D=4\,,\;b=6\,,\;c_{s}^{2}=\frac{1}{3}$, and $a=1/(4\pi)$ (full line), $a=1/4$
(dashed line), as well as $a=1$ (dotted line).

The dispersion relations for the shear modes resulting from Eq. (12b) change
their behavior from nonpropagating to propagating at the critical wavenumber
(15), as shown in Fig. 3. It should be noted that a similar behavior is
observed in the case of bulk viscosity, cf. Fig. 1 in Ref. dkkm3 . For
wavenumbers larger than $k_{c}$, the (modulus of the) group velocity of the
propagating mode is

$v_{g}=v_{g,{\rm shear}}^{\rm as}\,\frac{k/k_{c}}{\sqrt{(k/k_{c})^{2}-1}}\;,$
(22)

where

$v_{g,{\rm shear}}^{\rm
as}\equiv\frac{1}{\sqrt{2\tau_{\pi}k_{c}}}\equiv\sqrt{\frac{\eta}{\tau_{\pi}(\varepsilon+P)}}\equiv\frac{1}{\sqrt{b}}$
(23)

is the asymptotic value of $v_{g}$ in the large-wavenumber limit. If the
asymptotic causality condition (21) is satisfied, $v_{g,{\rm shear}}^{\rm
as}\leq\sqrt{(D-1)(1-c_{s}^{2})/2(D-2)}$. This is smaller than 1 for any value
of $c_{s}$ and $D\geq 3$. However, near the critical wavenumber $k_{c}$ the
group velocity diverges, as shown in Fig. 4. From the definitions of $k_{c}$,
Eq. (15), and the parameters $a,b$, Eqs. (6), we observe that
$k_{c}/T=(2a\sqrt{b})^{-1}$. The $1/a$-scaling of $k_{c}/T$ for fixed $b$ can
be nicely observed in Fig. 4.

In Sec. III.3 we shall show that the apparent violation of causality of the
group velocity does not cause the theory as a whole to become acausal. The
important issue is whether the asymptotic causality condition is fulfilled. If
yes, the theory is causal.

Figure 5: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the sound modes obtained from Eq. (12f). The
parameters are $a=\frac{1}{4\pi}\,,\;b=1\,,\;c_{s}^{2}=\frac{1}{3}$ for the
3+1-dimensional case, $D=4$.

Figure 6: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the shear modes obtained from Eq. (12b). The
parameters are $a=\frac{1}{4\pi}\,,\;b=1\,,\;c_{s}^{2}=\frac{1}{3}$ for the
3+1-dimensional case, $D=4$.

We remark that, in the local rest frame, the stability of the system of fluid-
dynamical equations is not affected if we choose a parameter set which
violates the asymptotic causality condition (21), for instance a conformal
fluid in $D=4$ dimensions and $b=1$. This is demonstrated for the sound modes
in Fig. 5, and for the shear modes in Fig. 6.

###  II.2 Competition of bulk and shear

The question we would like to answer in this section is whether the problem of
the divergent group velocity can be removed by adding bulk viscosity to the
discussion. For the sake of simplicity, we consider only the 2+1-dimensional
case, i.e., $D=3$. Similarly to Eqs. (6), we introduce the parametrization

$\zeta=a_{1}s\;,\qquad\tau_{\Pi}=\frac{\zeta}{\varepsilon+P}\,b_{1}\;.$ (24)

As before, the equations of motion (4) have to be linearized, yielding Eq.
(9), where now

$X=(\delta\varepsilon,\delta u^{x},\delta\pi^{xx},\delta
u^{y},\delta\pi^{xy},\delta\Pi)^{T}\;,$ (25)

and

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}i\omega&-ik\,(\varepsilon+P)&0&0&0&0\\\
-ik\,c_{s}^{2}&i\omega\,(\varepsilon+P)&-ik&0&0&-ik\\\
0&-ik\,\eta&i\omega\,\tau_{\pi}+1&0&0&0\\\
0&0&0&i\omega\,(\varepsilon+P)&-ik&0\\\
0&0&0&-ik\,\eta&i\omega\,\tau_{\pi}+1&0\\\
0&-ik\,\zeta&0&0&0&i\omega\,\tau_{\Pi}+1\end{array}\right)\;.$ (32)

Then, the dispersion relations are given by solving the following equations:

$\displaystyle k^{2}\eta+i\omega\,(1+i\omega\,\tau_{\pi})(\varepsilon+P)$
$\displaystyle=$ $\displaystyle 0\;,$ (33a) $\displaystyle i\omega
k^{2}\,(1+i\omega\,\tau_{\Pi})\,\eta+(1+i\omega\,\tau_{\pi})\left[i\omega
k^{2}\,\zeta+(1+i\omega\,\tau_{\Pi})(\varepsilon+P)(c_{s}^{2}k^{2}-\omega^{2})\right]$
$\displaystyle=$ $\displaystyle 0\;.$ (33b)

The dispersion relations resulting from sound and bulk viscous modes, Eq.
(33b), are

$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{T}{2aa_{1}(b+b_{1}+bb_{1}c_{s}^{2})}\left\\{\frac{}{}ia(1+bc_{s}^{2})+ia_{1}(1+b_{1}c_{s}^{2})\right.\\\
\qquad\left.\pm\left[4aa_{1}c_{s}^{2}(b+b_{1}+bb_{1}c_{s}^{2})-(a+a_{1}+abc_{s}^{2}+a_{1}b_{1}c_{s}^{2})^{2}\right]^{1/2}\right\\}\;,\\\
\displaystyle\pm
k\sqrt{\frac{1}{b}+\frac{1}{b_{1}}+c_{s}^{2}}+\frac{i\,T}{2(b+b_{1}+bb_{1}c_{s}^{2})}\left(\frac{b}{a_{1}b_{1}}+\frac{b_{1}}{ab}\right)\;,\end{array}\right.$
(34)

for large $k$, and

$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{i}{\tau_{\pi}}\;,\\\\[8.5359pt]
\displaystyle\frac{i}{\tau_{\Pi}}\;,\\\ \pm c_{s}^{2}k\;,\end{array}\right.$
(35)

for small $k$.

Thus the asymptotic causality condition reads

$\frac{1}{b_{1}}+\frac{1}{b}\equiv\frac{\zeta}{\tau_{\Pi}(\varepsilon+P)}+\frac{\eta}{\tau_{\pi}(\varepsilon+P)}\leq
1-c^{2}_{s}\;.$ (36)

On the other hand, the equation for the shear modes, Eq. (33a), is the same as
Eq. (12b) and hence the corresponding group velocity again shows a divergence.
Thus, the inclusion of bulk viscosity does not solve the problem of the
divergent group velocity.

##  III Stability in Lorentz-boosted frame

The discussion of causality and stability in the case of nonzero bulk
viscosity in a Lorentz-boosted frame in Ref. dkkm3  has shown that causality
and stability are intimately related. Relativistic dissipative fluid dynamics
becomes unstable if the group velocity exceeds the speed of light. If this is
still true in the case of nonzero shear viscosity, the divergence of the group
velocity found in the rest frame may induce an instability in a moving frame.
In order to investigate this question, we consider the stability of the
hydrostatic state observed from a Lorentz-boosted frame, following Ref. dkkm3
. In this section, we restrict our investigations to the case $D=4$.

We consider a frame moving with a velocity $\vec{V}$ with respect to the
hydrostatic state. Then, the total fluid velocity $u^{\prime\;\mu}$ is given
by

$u^{\prime\;\mu}=\left(\begin{array}[]{cc}\gamma_{V}&V\gamma_{V}\vec{n}^{T}\\\
V\gamma_{V}\vec{n}&\gamma_{V}P_{\parallel}+Q_{\perp}\end{array}\right)u^{\mu},$
(37)

where $\gamma_{V}=1/\sqrt{1-V^{2}}$, $P_{\parallel}=\vec{n}\vec{n}^{T}$, and
$Q_{\perp}=1-P_{\parallel}$, with $\vec{n}=\vec{V}/|\vec{V}|$. We consider the
two cases where the direction of the Lorentz boost is parallel and where it is
perpendicular to the direction of propagation of the perturbation; the latter
we take to be the $x$ direction.

###  III.1 Boost along the $x$ direction

The perturbation of the fluid velocity is given by

$u^{\prime\;\mu}=u^{\prime\;\mu}_{0}+\delta u^{\prime\;\mu}\;e^{i\omega
t-ikx}\;,$ (38)

where

$\displaystyle u^{\prime\;\mu}_{0}$ $\displaystyle=$
$\displaystyle\gamma_{V}(1,V,0,0)\;,$ (39a) $\displaystyle\delta
u^{\prime\;\mu}$ $\displaystyle=$ $\displaystyle(V\gamma_{V}\delta
u^{x},\gamma_{V}\delta u^{x},\delta u^{y},\delta u^{z})\;,$ (39b)

where $\delta u^{\mu}$ is the velocity perturbation in the local rest frame.
The linearized fluid-dynamical equations are again given by Eq. (9), with

$X=(\delta\varepsilon,\delta u^{x},\delta\pi^{xx},\delta
u^{y},\delta\pi^{xy},\delta
u^{z},\delta\pi^{xz},\delta\pi^{yy},\delta\pi^{yz})^{T}\;,$ (40)

and

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}T_{1}&0&0&0\\\ 0&B_{1}&0&0\\\
G_{1}&0&C_{1}&0\\\ 0&0&0&E_{1}\end{array}\right)\;.$ (45)

The submatrices are given by

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\gamma_{V}^{2}\left(\begin{array}[]{ccc}i\omega(1+V^{2}c_{s}^{2})-ikV(1+c_{s}^{2})&\;\;i[2\omega
V-k(1+V^{2})](\varepsilon+P)&\;\;i\gamma_{V}^{-2}V(\omega V-k)\\\ i\omega
V(1+c_{s}^{2})-ik(V^{2}+c_{s}^{2})&\;\;i\left[\omega(1+V^{2})-2kV\right](\varepsilon+P)&\;\;i\gamma_{V}^{-2}(\omega
V-k)\\\ 0&\frac{4}{3}i\eta\gamma_{V}(\omega
V-k)&\;\;\gamma_{V}^{-2}F\end{array}\right)\;,$ (46d) $\displaystyle B_{1}$
$\displaystyle=$ $\displaystyle{\rm diag}(B_{01},B_{01})\;,\qquad
B_{01}=\left(\begin{array}[]{cc}i\gamma_{V}(\omega-
kV)(\varepsilon+P)&\;\;i(\omega V-k)\\\ i\eta\gamma_{V}^{2}(\omega
V-k)&\;\;F\end{array}\right)\;,$ (46h) $\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{}{}0\qquad-\frac{2}{3}i\eta\gamma_{V}(\omega
V-k)\qquad 0\right)\;,$ (46i) $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle E_{1}=F\;.$ (46j) Here we abbreviated $F=i\gamma_{V}(\omega-
kV)\tau_{\pi}+1\;.$ (46k)

Obviously,

$\displaystyle{\rm det}A={\rm det}T_{1}\times{\rm det}B_{1}\times F^{2}\;.$
(47)

From $F^{2}=0$, we only obtain two trivial propagating modes

$\omega=\frac{i}{\gamma_{V}\tau_{\pi}}+kV\;.$ (48)

The group velocity is $v_{g}=V$, which implies that these modes correspond to
the nonpropagating modes in the LRF.

From ${\rm det}B_{1}=0$, we obtain

$[iT+ab\gamma_{V}(kV-\omega)](kV-\omega)+a\gamma_{V}(kV-\omega)^{2}T=0\;,$
(49)

corresponding to the shear modes. There are in total four modes satisfying
this relation. The solutions are given by

$\omega_{\pm}=\frac{1}{2a(b-V^{2})\gamma_{V}}\left[i\,T-2a(1-b)kV\gamma_{V}\pm\sqrt{-T^{2}+4iakTV\gamma_{V}^{-1}+4a^{2}bk^{2}\gamma_{V}^{-2}}\right]\;.$
(50)

On the other hand, the sound modes result from

$\displaystyle
c_{s}^{2}(\varepsilon+P)\left[\frac{}{}1-i\gamma_{V}\tau_{\pi}(kV-\omega)\right]\left\\{\frac{}{}k^{2}\left[\frac{}{}V^{2}+(V-1)^{2}V\gamma_{V}^{2}+1\right]\right.$
(51) $\displaystyle+$
$\displaystyle\left.2kV\omega\left[\frac{}{}(V-1)V\gamma_{V}^{2}-1\right]+V^{2}\omega^{2}-c_{s}^{-2}(\omega-
kV)^{2}\frac{}{}\right\\}$ $\displaystyle+$
$\displaystyle\frac{4}{3}i\gamma_{V}\eta(k-V\omega)^{2}\left\\{\frac{}{}kV\left[\frac{}{}c_{s}^{2}\gamma_{V}^{2}V(1-V)-1\right]+\omega\right\\}\qquad=0\;.$

In Fig. 7, the dependence of the group velocity on the wavenumber is shown for
various values of the boost velocity $V$. The left panel shows the behavior of
one of the shear modes and the right panel one of the sound modes. The
parameter set used here is $a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3}$,
which satisfies the asymptotic causality condition. We observe that the
divergence of the group velocity of the shear mode in the rest frame is
tempered by the Lorentz boost to result in a peak of finite height. However,
the group velocity may still exceed the speed of light in a certain range of
wavenumbers. As we increase the boost velocity, the peak height diminishes,
until the group velocity remains below the speed of light for all wavenumbers.
However, further increasing the boost velocity leads to an acausal group
velocity in the sound mode.

Figure 7: The group velocity calculated for one of the shear modes (left
panel) and one of the sound modes (right panel). We set
$a=1/(4\pi),b=6,c_{s}^{2}=1/3$. The solid line is for a boost velocity
$V=0.05$, the dashed line for $V=0.4$ and the dotted line for $V=0.99$,
respectively.

Although the group velocity of the shear or the sound mode may exceed the
speed of light, as long as the asymptotic causality condition is fulfilled,
the theory is still stable. This is demonstrated in the left panel of Fig. 8,
where the imaginary parts of the modes are shown for the parameter set
$a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3}$. We observe that all
imaginary parts are positive, indicating the stability of the theory.

In contrast to the rest frame, where the theory is stable even for parameters
which violate the asymptotic causality condition (21), this is no longer the
case in a Lorentz-boosted frame. In the right panel of Fig. 8, the imaginary
parts of the modes are calculated with the parameter set
$a=\frac{1}{4\pi},\,b=1,\,c_{s}^{2}=\frac{1}{3}$. Now one observes the
appearance of negative imaginary parts, indicating that the theory becomes
unstable.

Figure 8: The imaginary parts of the dispersion relations for a boost in $x$
direction with velocity $V=0.9$. The left panel shows the results for the
parameter set $a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3}$, which fulfills
the asymptotic causality condition, while the right panel is for
$a=\frac{1}{4\pi},\,b=1,\,c_{s}^{2}=\frac{1}{3}$, which violates this
condition. The dashed lines are for the shear modes, while the solid lines are
for the sound modes.

###  III.2 Boost along the $y$ direction

Now we consider a Lorentz boost along the $y$ direction. The perturbation of
the fluid velocity is given by

$u^{\prime\;\mu}=u^{\prime\;\mu}_{0}+\delta u^{\prime\;\mu}\;e^{i\omega
t-ikx}\;,$ (52)

where

$\displaystyle u^{\prime\;\mu}_{0}$ $\displaystyle=$
$\displaystyle\gamma_{V}(1,0,V,0)\;,$ (53a) $\displaystyle\delta
u^{\prime\;\mu}$ $\displaystyle=$ $\displaystyle(V\gamma_{V}\delta
u^{y},\delta u^{x},\gamma_{V}\delta u^{y},\delta u^{z})\;.$ (53b)

Similarly to the preceding discussion, the linearized fluid-dynamical
equations take the form (9), where the matrix $A$ is

$A=\left(\begin{array}[]{cccc}T_{2}&H_{1}&H_{2}&0\\\
H_{3}&B_{2}&H_{4}&H_{5}\\\ G_{2}&H_{6}&C_{2}&0\\\
0&H_{7}&0&E_{2}\end{array}\right)\;,$ (54)

with

$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}i\omega\gamma_{V}^{2}(1+c_{s}^{2}V^{2})&-ik\gamma_{V}(\varepsilon+P)&0\\\
-ikc_{s}^{2}&i\omega\gamma_{V}(\varepsilon+P)&-ik\\\
0&-\frac{4}{3}ik\eta&F_{1}\end{array}\right)\;,$ (55d) $\displaystyle H_{1}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}2i\omega
V(\varepsilon+P)\gamma_{V}^{2}&-ikV&0&0\\\ 0&i\omega V&0&0\\\
-\frac{2}{3}i\omega V\eta\gamma_{V}&0&0&0\end{array}\right)\;.$ (55h)
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}i\omega
V^{2}&0&0\end{array}\right)^{T}\;,$ (55j) $\displaystyle H_{3}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}i\omega
V\gamma_{V}^{2}(1+c_{s}^{2})&-ikV\gamma_{V}(\varepsilon+P)&0\\\ 0&i\omega
V\gamma_{V}^{2}\eta&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)\;,$ (55o)
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}i\omega\gamma_{V}^{2}(1+V^{2})(\varepsilon+P)&-ik&0&0\\\
-ik\gamma_{V}\eta&F_{1}&0&0\\\ 0&0&i\omega\gamma_{V}(\varepsilon+P)&-ik\\\
0&0&-ik\eta&F_{1}\end{array}\right)\;,$ (55t) $\displaystyle H_{4}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}i\omega
V&0&0&0\end{array}\right)^{T}\;,\qquad\qquad
H_{5}=\left(\begin{array}[]{cccc}0&0&i\omega V&0\end{array}\right)^{T}\;,$
(55w) $\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\frac{2}{3}ik\gamma_{V}^{2}\eta&0\end{array}\right)\;,\qquad\qquad
H_{6}=\left(\begin{array}[]{cccc}\frac{4}{3}i\omega
V\gamma_{V}^{3}\eta&0&0&0\end{array}\right)\;,$ (55z) $\displaystyle H_{7}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}0&0&i\omega
V\gamma_{V}^{2}\eta&0\end{array}\right)\;,\qquad\;\;\;C_{2}=E_{2}=F_{1}\;.$
(55ab)

Here we abbreviated

$F_{1}=i\omega\gamma_{V}\tau_{\pi}+1\;.$

The condition ${\rm det}A=0$ leads again to the following nine modes: three
nonpropagating modes, four shear modes and two sound modes.

The nonpropagating mode has almost the same form as that in the LRF,

$\omega=\frac{i}{\gamma_{V}\tau_{\pi}}\;.$ (56)

The shear modes are given by the solution of the following equation

$\displaystyle
k^{2}\eta+\gamma_{V}\omega\left[V^{2}\gamma_{V}\eta\omega+(\varepsilon+P)(i-\gamma_{V}\tau_{\pi}\omega)\right]=0\;,$
(57)

and the solutions are given by

$\omega_{\pm}=\frac{1}{2a(b-V^{2})\gamma_{V}}\left[i\,T\pm\sqrt{-T^{2}+4a^{2}bk^{2}-4a^{2}k^{2}V^{2}}\right]\;.$
(58)

We find that the critical wavenumber is now given by
$\tilde{k}_{c}=T/(2a\sqrt{b-V^{2}})$, below which the shear modes become
nonpropagating modes.

On the other hand, the sound modes and another nonpropagating mode result from

$\displaystyle
3c_{s}^{2}(\varepsilon+P)(-i+\gamma_{V}\tau_{\pi}\omega)(k^{2}+V^{2}\gamma_{V}^{2}\omega^{2})$
(59) $\displaystyle+$
$\displaystyle\gamma_{V}\omega\left\\{4k^{2}\eta+\gamma_{V}\omega\left[3i(\varepsilon+P)+4V^{2}\gamma_{V}\eta\omega-3(\varepsilon+P)\gamma_{V}\tau_{\pi}\omega\right]\right\\}=0\;.$

The real and imaginary parts of this dispersion relation are calculated with a
parameter set satisfying the asymptotic causality condition. The results are
shown in Fig. 9. One observes that the real parts are symmetric around
$\omega=0$. This symmetry is due to the fact that the direction of the Lorentz
boost is orthogonal to the direction of the perturbation. The critical wave
number $\tilde{k_{c}}$ where the shear mode changes from nonpropagating to
propagating mode can be clearly seen. The imaginary parts are seen to be
positive. We confirmed that the imaginary parts become negative if we use a
parameter set which violates the asymptotic causality condition.

Figure 9: The real and imaginary parts for the dispersion relations of the
shear modes (dashed lines) and sound modes (solid lines), for a Lorentz boost
in $y$ direction. We use
$a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3},\,V=0.9$ in the
3+1-dimensional case.

###  III.3 Causality of wave propagation

In the preceding discussion we have seen that the theory is stable if the
asymptotic causality condition is fulfilled. The reverse is in general not
true, as the discussion in the local rest frame has shown, since a stable
theory may also violate the asymptotic causality condition. However, the
discussion in the Lorentz-boosted frame has revealed that the stability of a
theory is contingent upon whether the asymptotic causality condition is
fulfilled.

In this section, we shall show that the causality of the theory as a whole is
guaranteed if the asymptotic stability condition is fulfilled. The group
velocity may become superluminal, or even diverge, as long as this apparent
violation of causality is restricted to a finite range of momenta. The
argument leading to this conclusion is analogous to that of Sommerfeld and
Brillouin in classical electrodynamics jackson ; bri . For instance, in the
case of anomalous dispersion the group velocity may become superluminal, but
the causality of the theory as a whole is not affected.

The change in a fluid-dynamical variable induced by a general perturbation is
given by

$\delta X(x,t)=\sum_{j}\int d\omega\,\widetilde{\delta
X}_{j}(\omega)\,e^{i\omega t-ik_{j}(\omega)x}\;,$ (60)

where $\delta X(x,t)$ stands for $\delta\varepsilon$, $\delta u^{\mu}$, and
$\delta\pi^{\mu\nu}$. The index $j$ denotes the different modes, i.e., the
shear modes, the sound modes etc. The function $k_{j}(\omega)$ is the inverted
dispersion relation $\omega_{j}(k)$ of the respective mode. The Fourier
components are given by

$\sum_{j}\widetilde{\delta
X}_{j}(\omega)=\frac{1}{2\pi}\int^{\infty}_{-\infty}dt\,\delta
X(0,t)\,e^{-i\omega t}\;.$ (61)

We assume that the incident wave has a well-defined front that reaches $x=0$
not before $t=0$. Thus $\delta X(0,t)=0$ for $t<0$. This condition on $\delta
X(0,t)$ ensures that $\sum_{j}\widetilde{\delta X}_{j}(\omega)$ is analytic in
the lower half of the complex $\omega$ plane jackson . On the other hand, in
Sec. II.1 we have found that the group velocity of the shear modes diverges
for certain values of $k$. These divergences correspond to singularities in
the complex $\omega$ plane. However, if the asymptotic causality condition is
fulfilled, the imaginary part of the dispersion relation is always positive,
i.e., the singularities only appear in the upper half of the complex $\omega$
plane. In this case, the system is also stable. On the other hand, if the
asymptotic causality condition is violated, the singularities may appear also
in the lower half-plane, i.e., for negative imaginary part of the dispersion
relation, and the system is unstable.

We shall now demonstrate that the divergences in the group velocity do not
violate causality as long as the asymptotic causality condition is satisfied,
i.e., as long as the asymptotic group velocity remains subluminal. To this
end, we compute Eq. (60) by contour integration in the complex $\omega$ plane.
To close the contour, we have to know the asymptotic behavior of the
dispersion relations. In our calculation, we found that the real part of the
dispersion relation at large $k$ is proportional to $k$ [see Eq. (18)], with a
coefficient which is the large-$k$ limit of the group velocity, i.e.,
$v_{gj}^{\rm as}$,

$\lim_{k\rightarrow\infty}{\rm Re}~{}\omega_{j}(k)=v_{gj}^{\rm as}\,k\;.$ (62)

Then, in the large-$k$ limit, the exponential becomes

$\exp[i\omega
t-ik_{j}(\omega)x]\rightarrow\exp\left[-i\,\frac{\omega}{v_{gj}^{\rm
as}}\,(x-v_{gj}^{\rm as}\,t)\right]\;.$ (63)

In the case $x>v_{gj}^{\rm as}\,t$, we have to close the integral contour in
the lower half plane. If the asymptotic causality condition is fulfilled,
there are no singularities in the lower half plane, and Eq. (60) vanishes. On
the other hand, the contour should be closed in the upper half plane if $x\leq
v_{gj}^{\rm as}\,t$. Then, because of the singularities, Eq. (60) may have a
nonzero value. However, as long as we choose a parameter set for which the
asymptotic group velocity $v_{gj}^{\rm as}$ is smaller than the speed of
light, i.e., for which the asymptotic causality condition is fulfilled, the
signal propagation does not violate causality, since the locations $x$ where
the disturbance has travelled lie within the cone given by $v_{gj}^{\rm as}$
which, in turn, lies within the lightcone, q.e.d.

To conclude this section, we have shown that the asymptotic causality
condition not only implies stability in a general (Lorentz-boosted) frame, but
also causality of the theory as a whole.

##  IV Characteristic velocities

So far, we have analyzed the causality and stability of relativistic
dissipative fluid dynamics with shear viscosity using a linear stability
analysis. However, there is another possibility to analyze causality, namely
by studying the characteristic velocities. For the sake of simplicity, we
consider the 2+1-dimensional case with shear viscosity only. The fluid-
dynamical equations can be written in the following form:

$\left(A_{ab}^{t}\partial_{t}+A_{ab}^{x}\partial_{x}+A_{ab}^{y}\partial_{y}\right)Y_{b}=B_{a}\;,$
(64)

where $Y_{b}^{T}=(\varepsilon,u^{x},u^{y},\pi^{xx},\pi^{xy})$ and
$B_{a}^{T}=(0,\;0,\;0,\;\pi^{xx},\;\pi^{xy})$. The expressions for the
components of $A$ are given in the Appendix. Then, as discussed in Ref. his ,
the characteristic velocities are defined as the roots of the following
equations,

$\displaystyle\det(v_{x}A^{t}-A^{x})$ $\displaystyle=$ $\displaystyle 0\;,$
(65a) $\displaystyle\det(v_{y}A^{t}-A^{y})$ $\displaystyle=$ $\displaystyle
0\;.$ (65b)

For the case of bulk viscosity, see Ref. dkkm3 .

For the sake of simplicity, we consider $u^{\mu}=(1,\;0,\;0)$ and
$\pi^{xx}=\pi^{xy}=0$. Then, the characteristic velocities are given by

$v_{x}=v_{y}=\left\\{\begin{array}[]{l}0\;,\\\
\displaystyle\pm\sqrt{\frac{1}{b}}\;,\\\
\displaystyle\pm\sqrt{\frac{1}{b}+c_{s}^{2}}\;.\end{array}\right.$ (66)

Interestingly, the second velocity is identical to the asymptotic group
velocity (23) for the shear modes and the third velocity is the same as the
asymptotic group velocity (20) for the sound modes (since $D=3$). As a matter
of fact, if the asymptotic causality condition (21) is satisfied, the velocity
(66) is smaller than the speed of light.

In Fig. 10, we show the $b$ dependence of one of the five characteristic
velocities. We set $u^{\mu}=(\sqrt{5}/2,\;1/2,\;0),\;\pi^{xx}=\pi^{xy}=0$, and
$c_{s}^{2}=1/2$. The velocity exhibits a divergence at small values of $b$,
and thus exceeds the speed of light. This divergence occurs also for at least
one other characteristic velocity. As far as we have checked numerically, in
order to satisfy causality, one should use a value of $b$ which is larger than
about 2. This condition is consistent with the asymptotic causality condition
(21).

Figure 10: One of the five characteristic velocities determined from the roots
of Eqs. (65). The left panel is for $v_{x}$ and the right panel is for
$v_{y}$. We set $u^{\mu}=(\sqrt{5}/2,\;1/2,\;0),\;\pi^{xx}=\pi^{xy}=0$, and
$c_{s}^{2}=1/2$.

##  V Concluding remarks

In this work, we have discussed the stability and causality of relativistic
dissipative fluid dynamics, based on a linear stability analysis around a
hydrostatic state. Following the usual argument, we calculated the group
velocity from the dispersion relation of the perturbation. We found that the
group velocity diverges at a critical wavenumber $k_{c}$. The appearance of
the divergence is independent of the dimensionality of space-time and can be
removed neither by tuning the parameters of the theory nor by adding bulk
viscosity to the discussion.

Nevertheless, in the rest frame of the background this acausal group velocity
does not cause the fluid to become unstable. Moreover, investigating causality
and stability in a Lorentz-boosted frame, we found that the fluid-dynamical
equations of motion are stable, if we choose parameters which satisfy a so-
called asymptotic causality condition. They become unstable if this condition
is violated. In this sense, the problems of acausality and instability are
still correlated even in the case of shear viscosity, as was already found for
the case of bulk viscosity dkkm3 .

We have then demonstrated that the causality of the theory as a whole is
guaranteed if the asymptotic causality condition is fulfilled. Therefore, a
superluminal group velocity in a finite range of momenta can cause the theory
neither to become acausal nor unstable. Finally, we studied the characteristic
velocities and found a violation of causality for small values of
$\tau_{\pi}(\varepsilon+P)/\eta$, but not for values which satisfy the
asymptotic causality condition.

The asymptotic causality condition requires that the ratio
$\tau_{\pi}/\Gamma_{s}$ is sufficiently large, i.e., that the time scale
$\tau_{\pi}$ over which the shear viscous pressure relaxes towards its NS
value is not too small compared to the sound attenuation length
$\Gamma_{s}\sim\eta/(\varepsilon+P)\equiv\eta/(Ts)$. This is an important
finding for practitioners of fluid dynamics, who frequently consider
$\tau_{\pi}$ and the shear viscosity-to-entropy density ratio $\eta/s$ to be
independent from each other. We have demonstrated that this is not the case if
one wants the theory to remain causal. Therefore, second-order theories of
relativistic dissipative fluid dynamics are not automatically causal by
construction. Our findings also illuminate why NS theory violates causality
from a different perspective, because there $\tau_{\pi}\rightarrow 0$ while
$\eta$ remains non-zero.

## Acknowledgement

Shi Pu thanks Zhe Xu and Qun Wang for helpful discussions. We acknowledge G.
Moore and the referee for valuable comments concerning causality and the
divergence of the group velocity which have resulted in the discussion
presented in Sec. III.3. This work was (financially) supported by the
Helmholtz International Center for FAIR within the framework of the LOEWE
program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer
Exzellenz) launched by the State of Hesse.

##  Appendix A Matrix elements in Eq. (64)

The fluid-dynamical equations can be expressed in the form (64). Let us
parameterize the velocity of the fluid as
$u^{\mu}=(\cosh\theta,\sinh\theta\cos\phi,\sinh\theta\sin\phi)$. The matrix
elements of $A_{ab}^{x}$ are

$\displaystyle A^{x}_{11}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\cos\phi\;,$
$\displaystyle A^{x}_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{sech}^{3}\theta\left\\{\frac{}{}2\sinh^{2}\theta\left[(2w+\pi^{xx})\sin^{2}\phi+3w\cos^{2}\phi-\pi^{xy}\sin\phi\cos\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{4}\theta(\cos(2\phi)+3)+w+\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{x}_{13}$ $\displaystyle=$
$\displaystyle\text{sech}^{3}\theta\left\\{\frac{}{}\sinh^{2}\theta\cos\phi\left[(w-\pi^{xx})\sin\phi+\pi^{xy}\cos\phi\right]+w\sinh^{4}\theta\sin\phi\cos\phi+\pi^{xy}\right\\}\;,$
$\displaystyle A^{x}_{14}$ $\displaystyle=$
$\displaystyle\tanh\theta\cos\phi\;,$ $\displaystyle A^{x}_{15}$
$\displaystyle=$ $\displaystyle\tanh\theta\sin\phi\;,$ $\displaystyle
A^{x}_{21}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\cos^{2}\phi+c_{s}^{2}\;,$
$\displaystyle A^{x}_{22}$ $\displaystyle=$ $\displaystyle
2w\sinh\theta\cos\phi\;,$ $\displaystyle A^{x}_{24}$ $\displaystyle=$
$\displaystyle A^{x}_{35}=1\;,$ $\displaystyle A^{x}_{31}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\sin\phi\cos\phi\;,$
$\displaystyle A^{x}_{32}$ $\displaystyle=$ $\displaystyle
w\sinh\theta\sin\phi\;,$ $\displaystyle A^{x}_{33}$ $\displaystyle=$
$\displaystyle w\sinh\theta\cos\phi\;,$ $\displaystyle A^{x}_{42}$
$\displaystyle=$
$\displaystyle\text{sech}^{2}\theta\left\\{\frac{}{}\sinh^{4}\theta\cos^{2}\phi\left[\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{2}\theta\left[2(\eta-\tau_{\pi}\pi^{xx})\cos^{2}\phi+\eta\sin^{2}\phi\right]+\eta\frac{}{}\right\\}\;,$
$\displaystyle A^{x}_{43}$ $\displaystyle=$
$\displaystyle-2\tau_{\pi}\tanh^{2}\theta\cos^{2}\phi\left[\sinh^{2}\theta\cos\phi(\pi^{xy}\cos\phi-\pi^{xx}\sin\phi)+\pi^{xy}\right]\;,$
$\displaystyle A^{x}_{44}$ $\displaystyle=$ $\displaystyle
A^{x}_{55}=\tau_{\pi}\sinh\theta\cos\phi\;,$ $\displaystyle A^{x}_{52}$
$\displaystyle=$
$\displaystyle\frac{\tanh^{2}\theta\cos\phi}{2(\sinh^{2}\theta\cos^{2}\phi+1)}\left\\{\frac{}{}-2\sinh^{2}\theta\left(\pi^{xx}\sin^{3}\phi+2\pi^{xx}\sin\phi\cos^{2}\phi+\pi^{xy}\cos^{3}\phi\right)\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\sin^{2}(2\phi)(\pi^{xy}\cos\phi-2\pi^{xx}\sin\phi)-2\pi^{xx}\sin\phi-2\pi^{xy}\cos\phi\frac{}{}\right\\}\;,$
$\displaystyle A^{x}_{53}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{sech}^{2}\theta\left\\{\frac{}{}2\sinh^{4}\theta\cos^{2}\phi\left[\eta-\tau_{\pi}\pi^{xx}\cos(2\phi)+\tau_{\pi}\pi^{xx}-\tau_{\pi}\pi^{xy}\sin(2\phi)\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{2}\theta\left[(\eta+\tau_{\pi}\pi^{xx})\cos(2\phi)+3\eta+\tau_{\pi}\pi^{xx}-\tau_{\pi}\pi^{xy}\sin(2\phi)\right]+2\eta\frac{}{}\right\\}\;.$

The matrix elements of $A_{ab}^{t}$ are given by

$\displaystyle A^{t}_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(c_{s}^{2}+1\right)\cosh(2\theta)-c_{s}^{2}+1\right]\;,$
$\displaystyle A^{t}_{12}$ $\displaystyle=$
$\displaystyle\frac{2\sinh\theta}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\left\\{\frac{}{}\sinh^{2}\theta\cos\phi\left(2w\cos^{2}\phi+\pi^{xx}\sin^{2}\phi-\pi^{xy}\sin\phi\cos\phi\right)\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{4}\theta\cos^{5}\phi+(w+\pi^{xx})\cos\phi+\pi^{xy}\sin\phi\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{13}$ $\displaystyle=$ $\displaystyle
2\sinh\theta\left(w\sin\phi+\frac{\pi^{xy}\cos\phi-\pi^{xx}\sin\phi}{\sinh^{2}\theta\cos^{2}\phi+1}\right)\;,$
$\displaystyle A^{t}_{14}$ $\displaystyle=$
$\displaystyle\frac{\cos(2\phi)}{\text{csch}^{2}\theta+\cos^{2}\phi}\;,$
$\displaystyle A^{t}_{15}$ $\displaystyle=$
$\displaystyle\frac{\sin(2\phi)}{\text{csch}^{2}\theta+\cos^{2}\phi}\;,$
$\displaystyle A^{t}_{21}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\cos\phi\;,$
$\displaystyle A^{t}_{31}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\sin\phi\;,$
$\displaystyle A^{t}_{22}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{2}\left\\{\frac{}{}2\sinh^{2}\theta\left[(2w+\pi^{xx})\sin^{2}\phi+3w\cos^{2}\phi-\pi^{xy}\sin\phi\cos\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{4}\theta\left[\cos(2\phi)+3\right]+2w+2\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{23}$ $\displaystyle=$
$\displaystyle\text{sech}^{3}\theta\left\\{\sinh^{2}\theta\cos\phi\left[w\sinh^{2}\theta\sin\phi+(w-\pi^{xx})\sin\phi+\pi^{xy}\cos\phi\right]+\pi^{xy}\right\\}\;,$
$\displaystyle A^{t}_{24}$ $\displaystyle=$
$\displaystyle\tanh\theta\cos\phi\;,$ $\displaystyle A^{t}_{25}$
$\displaystyle=$ $\displaystyle\tanh\theta\sin\phi\;,$ $\displaystyle
A^{t}_{32}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\left\\{\frac{}{}\sinh^{2}\theta\left[(w+3\pi^{xx})\sin\phi\cos\phi+3\pi^{xy}\sin^{2}\phi+2\pi^{xy}\cos^{2}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\left[3(w+\pi^{xx})\sin\phi\cos^{3}\phi+(w+5\pi^{xx})\sin^{3}\phi\cos\phi+2\pi^{xy}\sin^{4}\phi+\pi^{xy}\cos^{4}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{16}\sinh^{6}\theta\left[10\sin(2\phi)+\sin(4\phi)\right]\left[(w-\pi^{xx})\cos(2\phi)+w+\pi^{xx}-\pi^{xy}\sin(2\phi)\right]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{8}\theta\sin\phi\cos^{5}\phi+\pi^{xy}\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{33}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{8\left(\sinh^{2}\theta\cos^{2}\phi+1\right)}\left\\{\frac{}{}\sinh^{4}\theta\left[4(w+2\pi^{xx})\cos(2\phi)+(\pi^{xx}-w)\cos(4\phi)+21w\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.9\pi^{xx}+10\pi^{xy}\sin(2\phi)+\pi^{xy}\sin(4\phi)\right]+4\sinh^{2}\theta\left[6w+2\pi^{xx}\cos(2\phi)-4\pi^{xx}\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.3\pi^{xy}\sin(2\phi)\right]-4w\sinh^{6}\theta\cos^{2}\phi\left[\cos(2\phi)-3\right]+8w-8\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{34}$ $\displaystyle=$
$\displaystyle-\frac{\tanh\theta\sin\phi\left(\sinh^{2}\theta\sin^{2}\phi+1\right)}{\sinh^{2}\theta\cos^{2}\phi+1}\;,$
$\displaystyle A^{t}_{35}$ $\displaystyle=$
$\displaystyle\frac{\tanh\theta\cos\phi}{2\sinh^{2}\theta\cos^{2}\phi+2}\left\\{\frac{}{}2-\sinh^{2}\theta[\cos(2\phi)-3]\right\\}\;,$
$\displaystyle A^{t}_{42}$ $\displaystyle=$
$\displaystyle\tanh\theta\cos\phi\left\\{\frac{}{}\sinh^{2}\theta\left\\{2\sin\phi\left[(\eta-\tau_{\pi}\pi^{xx})\sin\phi+\tau_{\pi}\pi^{xy}\cos\phi\right]+\eta\cos^{2}\phi\right\\}+\eta-2\tau_{\pi}\pi^{xx}\right\\}\;,$
$\displaystyle A^{t}_{43}$ $\displaystyle=$
$\displaystyle-\tanh\theta\left\\{\frac{}{}\sinh^{2}\theta\cos^{2}\phi\left[(\eta-2\tau_{\pi}\pi^{xx})\sin\phi+2\tau_{\pi}\pi^{xy}\cos\phi\right]+\eta\sin\phi+2\tau_{\pi}\pi^{xy}\cos\phi\right\\}\;,$
$\displaystyle A^{t}_{44}$ $\displaystyle=$ $\displaystyle
A^{t}_{55}=\tau_{\pi}\cosh\theta\;,$ $\displaystyle A^{t}_{52}$
$\displaystyle=$
$\displaystyle\frac{\tanh\theta}{4\sinh^{2}\theta\cos^{2}\phi+4}\left\\{\frac{}{}-2\sinh^{2}\theta\left\\{\sin\phi\left[-2\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)+3\tau_{\pi}\pi^{xx}\right]+2\tau_{\pi}\pi^{xy}\cos^{3}\phi\right\\}\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\sin^{2}(2\phi)\left[(\eta-2\tau_{\pi}\pi^{xx})\sin\phi+2\tau_{\pi}\pi^{xy}\cos\phi\right]+4(\eta-\tau_{\pi}\pi^{xx})\sin\phi-4\tau_{\pi}\pi^{xy}\cos\phi\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{53}$ $\displaystyle=$
$\displaystyle\tanh\theta\left\\{\frac{}{}\sinh^{2}\theta\left[\eta\cos^{3}\phi+\tau_{\pi}\pi^{xx}\sin\phi\sin(2\phi)-2\tau_{\pi}\pi^{xy}\sin\phi\cos^{2}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.(\eta+\tau_{\pi}\pi^{xx})\cos\phi-\tau_{\pi}\pi^{xy}\sin\phi\frac{}{}\right\\}\;.$

The matrix elements of $A_{ab}^{y}$ are

$\displaystyle A^{y}_{11}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\sin\phi\;,$
$\displaystyle A^{y}_{21}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\sin\phi\cos\phi\;,$
$\displaystyle A^{y}_{12}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\left\\{\frac{}{}\sinh^{2}\theta\left[(w+3\pi^{xx})\sin\phi\cos\phi+3\pi^{xy}\sin^{2}\phi+2\pi^{xy}\cos^{2}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\left[3(w+\pi^{xx})\sin\phi\cos^{3}\phi+(w+5\pi^{xx})\sin^{3}\phi\cos\phi+2\pi^{xy}\sin^{4}\phi+\pi^{xy}\cos^{4}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{16}\sinh^{6}\theta[10\sin(2\phi)+\sin(4\phi)][(w-\pi^{xx})\cos(2\phi)+w+\pi^{xx}-\pi^{xy}\sin(2\phi)]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{8}\theta\sin\phi\cos^{5}\phi+\pi^{xy}\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{13}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{8\left(\sinh^{2}\theta\cos^{2}\phi+1\right)}\left\\{\frac{}{}\sinh^{4}\theta[4(w+2\pi^{xx})\cos(2\phi)+(\pi^{xx}-w)\cos(4\phi)+21w\right.$
$\displaystyle-$
$\displaystyle\left.9\pi^{xx}+10\pi^{xy}\sin(2\phi)+\pi^{xy}\sin(4\phi)]+4\sinh^{2}\theta[6w+2\pi^{xx}\cos(2\phi)-4\pi^{xx}\right.$
$\displaystyle+$
$\displaystyle\left.3\pi^{xy}\sin(2\phi)]-4w\sinh^{6}\theta\cos^{2}\phi[\cos(2\phi)-3]+8w-8\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{14}$ $\displaystyle=$
$\displaystyle-\frac{\tanh\theta\sin\phi\left(\sinh^{2}\theta\sin^{2}\phi+1\right)}{\sinh^{2}\theta\cos^{2}\phi+1}\;,$
$\displaystyle A^{y}_{15}$ $\displaystyle=$
$\displaystyle\frac{\tanh\theta\cos\phi}{2\sinh^{2}\theta\cos^{2}\phi+2}\left\\{\frac{}{}2-\sinh^{2}\theta\left[\cos(2\phi)-3\right]\right\\}\;,$
$\displaystyle A^{y}_{22}$ $\displaystyle=$ $\displaystyle
w\sinh\theta\sin\phi\;,$ $\displaystyle A^{y}_{23}$ $\displaystyle=$
$\displaystyle w\sinh\theta\cos\phi\;,$ $\displaystyle A^{y}_{25}$
$\displaystyle=$ $\displaystyle 1\;,$ $\displaystyle A^{y}_{31}$
$\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\sin^{2}\phi+c_{s}^{2}\;,$
$\displaystyle A^{y}_{32}$ $\displaystyle=$
$\displaystyle\frac{2\sinh\theta\left[\sinh^{2}\theta\sin\phi\cos\phi(\pi^{xx}\sin\phi-\pi^{xy}\cos\phi)+\pi^{xx}\cos\phi+\pi^{xy}\sin\phi\right]}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\;,$
$\displaystyle A^{y}_{33}$ $\displaystyle=$ $\displaystyle
2\sinh\theta\left(w\sin\phi+\frac{\pi^{xy}\cos\phi-\pi^{xx}\sin\phi}{\sinh^{2}\theta\cos^{2}\phi+1}\right)\;,$
$\displaystyle A^{y}_{34}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}\theta\sin^{2}\phi+1}{\sinh^{2}\theta\cos^{2}\phi+1}\;,$
$\displaystyle A^{y}_{35}$ $\displaystyle=$
$\displaystyle\frac{\sin(2\phi)}{\text{csch}^{2}\theta+\cos^{2}\phi}\;,$
$\displaystyle A^{y}_{42}$ $\displaystyle=$
$\displaystyle\tanh^{2}\theta\sin\phi\cos\phi\left\\{\frac{}{}\sinh^{2}\theta[2\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)]+2\eta-2\tau_{\pi}\pi^{xx}\right\\}\;,$
$\displaystyle A^{y}_{43}$ $\displaystyle=$
$\displaystyle-\frac{\text{sech}^{2}\theta}{2}\left\\{\frac{}{}2\sinh^{4}\theta\cos^{2}\phi[\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{2}\theta\left\\{\eta[\cos(2\phi)+3]+2\tau_{\pi}\pi^{xy}\sin(2\phi)\right\\}+2\eta\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{44}$ $\displaystyle=$ $\displaystyle
A^{y}_{55}=\tau_{\pi}\sinh\theta\sin\phi\;,$ $\displaystyle A^{y}_{52}$
$\displaystyle=$
$\displaystyle\frac{\tanh^{2}\theta}{8(\sinh^{2}\theta\cos^{2}\phi+1)}\left\\{\frac{}{}\sinh^{2}\theta\left[(\tau_{\pi}\pi^{xx}-\eta)\cos(4\phi)+9\eta+4\tau_{\pi}\pi^{xx}\cos(2\phi)-5\tau_{\pi}\pi^{xx}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.8\tau_{\pi}\pi^{xy}\sin\phi\cos^{3}\phi\right]+2\sinh^{4}\theta\sin^{2}(2\phi)[\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)]\right.$
$\displaystyle+$
$\displaystyle\left.4[4\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}-\tau_{\pi}\pi^{xy}\sin(2\phi)]+8\eta\text{csch}^{2}\theta\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{53}$ $\displaystyle=$
$\displaystyle\tau_{\pi}\tanh^{2}\theta\sin\phi\left[\sinh^{2}\theta\sin(2\phi)(\pi^{xx}\sin\phi-\pi^{xy}\cos\phi)+\pi^{xx}\cos\phi-\pi^{xy}\sin\phi\right]\;,$

where we defined $w=\varepsilon+P$. All other elements vanish.

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