Landau and the Eckart frames. A plane wave perturbation $\delta Q = \delta \bar{Q} e^{i(\omega t - kx)}$ is considered for a macroscopic variable $Q$ around global equilibrium where $u^\mu = (1,0,0,0)$. The perturbed equations of motion are used to analyze the hydrodynamic modes [@Hiscock:1985zz; @Hiscock:1987zz].

Landau Frame
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In the Landau frame, the perturbed energy-momentum tensor and the conserved charge current are $$\begin{aligned}
\delta T^{\mu \nu} &=& (e+P) (\delta u^\mu u^\nu + u^\mu \delta u^\nu) \nonumber \\
&+& \delta e u^\mu u^\nu - \delta P g^{\mu \nu}, \\
\delta N^\mu &=& n \delta u^\mu + \delta n u^\mu + \delta V^\mu ,\end{aligned}$$ which follow the conservation law and the constitutive relation $$\begin{aligned}
\delta V^\mu &=& \kappa_V \nabla^\mu \delta \alpha - \tau_V \Delta ^{\mu \nu} D \delta V_\nu ,\end{aligned}$$ where $\alpha = \mu/T$. The longitudinal and the transverse modes relevant to the diffusion are given by $$\begin{aligned}
\mathcal{M}^L_{xx}
\begin{pmatrix}
\delta e\\
\delta n\\
\delta u^x \\
\delta V^x \\
\end{pmatrix}
&=&0 , \end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^L_{xy}
\begin{pmatrix}
\delta u^y \\
\delta V^y \\
\end{pmatrix}
=0 , \
\mathcal{M}^L_{xz}
\begin{pmatrix}
\delta u^z \\
\delta V^z \\
\end{pmatrix}
=0 , \end{aligned}$$ where $$\begin{aligned}
\mathcal{M}^L_{xx} = \begin{pmatrix}
i\omega&0&-ikh&0\\
-ik \frac{\partial P}{\partial e}|_n&-ik\frac{\partial P}{\partial 