$\omega = i/\tau_V$. The causality and stability conditions are trivially satisfied. Those results of the longitudinal and the transverse modes indicate that the second-order diffusive hydrodynamics is causal and stable in the Landau frame.

Eckart Frame
------------

In the Eckart frame, the energy-momentum tensor and the conserved charge current are expressed as $$\begin{aligned}
\delta T^{\mu \nu} &=& (e+P) (\delta u^\mu u^\nu + u^\mu \delta u^\nu) + \delta e u^\mu u^\nu \nonumber \\
&-& \delta P g^{\mu \nu} + \delta W^\mu u^\nu + \delta W^\nu u^\mu, \\
\delta N^\mu &=& n \delta u^\mu + \delta n u^\mu ,\end{aligned}$$ and the energy dissipation current as $$\begin{aligned}
\delta W^\mu &=& - \kappa_W \beta_\mathrm{eq} D \delta u^\mu - \kappa_W \nabla^\mu \delta \beta - \tau_W \Delta^{\mu \nu} D \delta W_\nu . \nonumber \\\end{aligned}$$ The perturbed equations of motion are $$\begin{aligned}
\mathcal{M}^E_{xx}
\begin{pmatrix}
\delta e\\
\delta n\\
\delta u^x \\
\delta W^x \\
\end{pmatrix}
&=&0 , \end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^E_{xy}
\begin{pmatrix}
\delta u^y \\
\delta W^y \\
\end{pmatrix}
=0 , \
\mathcal{M}^E_{xz}
\begin{pmatrix}
\delta u^z \\
\delta W^z \\
\end{pmatrix}
=0 , \end{aligned}$$ where $$\begin{aligned}
\mathcal{M}^E_{xx} = \begin{pmatrix}
i\omega&0&-ik h&-ik\\
-ik \frac{\partial P}{\partial e}|_n&-ik \frac{\partial P}{\partial n}|_e&i\omega h&i\omega \\
0&i\omega&-ikn&0\\
ik\kappa_W \frac{\partial \beta}{\partial e}|_n&ik\kappa_W \frac{\partial \beta}{\partial 