= 1$ and the mostly-negative Minkowski metric $g^{\mu \nu} = \mathrm{diag}(+,-,-,-)$ is used in this paper.

Relativistic hydrodynamics in Landau and Eckart Frames {#sec2}
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The ideal hydrodynamic flow is uniquely determined since the local fluxes of the energy and the charge densities are in the same direction, *i.e.*, the directions of the eigenvector of the energy-momentum tensor and the conserved current match as $T^{\mu \nu} u_\nu = e u^\mu$ and $N^\mu = n u^\mu$. Here $e$ is the energy density and $n$ is the conserved charge density. On the other hand, the presence of the vector dissipative currents lead to the separation of the two local fluxes in relativistic systems. The Landau frame is chosen in the direction of the total energy flux so the dissipation of energy does not appear explicitly, $T^{\mu \nu} u^L_\nu = e_L u_L^\mu$. The Eckart frame is the choice of flow where the total conserved charge flux is diffusion-less $N^\mu = n_E u_E^\mu$. Here the subscripts $L$ and $E$ represent the Landau and the Eckart frames, respectively. The energy-momentum tensor, the conserved charge current, and the entropy current $s^\mu$ are assumed to be invariant under frame transformations [@Israel:1979wp].

The tensor decompositions read $$\begin{aligned}
T^{\mu 