stability. The constitutive relation is qualitatively modified and thus cannot be regarded as a first-order theory.

The asymptotic forms of the propagating and the non-propagating modes at small $k$ are $$\begin{aligned}
\omega = \pm c_s k + i \frac{\kappa_W (c_s^2 d_2 - d_4 )}{2 c_s^2} k^2 ,\end{aligned}$$ and $$\begin{aligned}
\omega = \frac{i}{\tau_W - \kappa_W \beta/h} ,\end{aligned}$$ aside from $\omega = 0$. Those modes are causal and stable if the Routh-Hurwitz criteria are satisfied.

The transverse equations $$\begin{aligned}
\det(\mathcal{M}^E_{xy}) &=& \det(\mathcal{M}^E_{xz}) \nonumber \\
&=& i\omega [h+i\omega (\tau_V - \kappa_W \beta)] = 0, \label{eq:MExy}\end{aligned}$$ have the non-propagating solutions $\omega = i/(\tau_W - \kappa_W \beta/h)$ and $\omega = 0$. One can see that all the modes satisfy the stability and causality conditions if the relaxation time is sufficiently larger than the conductivity (\[eq:twkw\]).

Comparing the two frames, the characteristic equations in the Landau frame (\[eq:MLxx\]) and (\[eq:MLxy\]) and their solutions are equivalent to those in the Eckart frame (\[eq:MExx\]) and (\[eq:MExy\]) under the identification of the conductivities (\[kappaLE\]) and the relaxation times (\[tauLE\]) that follow from the matching of the entropy production. The relation of the relaxation times in the two frames implies that the Eckart stability condition on $\tau_W$ is closely related 