of thermodynamics implies that the entropy production is expressed in a quadratic form. It can be written in the Landau frame as $$\begin{aligned}
\partial_\mu s^\mu = - \frac{V_L^\mu V^L_\mu }{\kappa_V} \geq 0,\end{aligned}$$ and in the Eckart frame as $$\begin{aligned}
\partial_\mu s^\mu = - \frac{W_E^\mu W^E_\mu}{\kappa_W} \geq 0,\end{aligned}$$ with the mostly-minus metric. The first and the second order transport coefficients of the two frames are related by the identification of the entropy production: $$\begin{aligned}
\kappa_V &=& \kappa_W \bigg( \frac{n}{e+P} \bigg)^2, \label{kappaLE}\\
\tau_V &=& \tau_W - \frac{\kappa_W}{(e+P)T}, \label{tauLE} \\
\chi_V^a &=& \chi_W^a - \frac{\tau_W nT}{e+P}, \\
\chi_V^b &=& \chi_W^b + \tau_W T - \frac{\kappa_W}{e+P}, \\
\chi_V^c &=&  \chi_W^c + \frac{\kappa_W}{(e+P)T}, \\
\chi_V^d &=& \chi_W^d + \frac{\kappa_W}{(e+P)T}, \\
\chi_V^e &=& \chi_W^e . \label{chieLE}\end{aligned}$$ See Appendix \[sec:A\] for the derivation. Those relations indicate that the full second-order terms are necessary in addition to the conventional Israel-Stewart terms for understanding the frame dependence of relativistic dissipative hydrodynamics, because the vanishing second-order transport coefficients in one frame lead to non-vanishing ones in the other frame, except for $\chi_V^e$ and $\chi_W^e$.

CAUSALITY AND STABILITY OF SECOND-ORDER HYDRODYNAMICS {#sec3}
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In this section, causality and stability conditions of the relativistic full second-order hydrodynamic equations are investigated in the 