ones reported in Ref. [@Hiscock:1987zz; @Osada:2011gx].

The results indicate that second-order dissipative hydrodynamics is stable in the Eckart frame if the transport coefficients satisfy the condition (\[eq:twkw\]). It can be immediately seen that the first-order theory is unstable in the Eckart frame by taking the limit of vanishing relaxation time $\tau_W\to 0$.

It is important to note that the space-like projection of the energy-momentum conservation law leads to $$\begin{aligned}
(e+P)Du^\mu &=& \nabla^\mu P
- W^\mu \nabla_\nu u^\nu \nonumber \\
&-& W^\nu \nabla_\nu u^\mu - \Delta^{\mu \nu} DW_\nu , \label{eq:emcperp}\end{aligned}$$ which is also used to convert the thermodynamic forces (\[eq:thermoforce\]). The higher order terms in the identity is important even in the stability analyses of the first-order theory because if one neglects the correction by truncation and use it to remove the acceleration term in the energy dissipation current, the equation can become seemingly “stable" at the first order. This is because the relaxation term-like correction originating from the last term in Eq. (\[eq:emcperp\]) is effectively introduced by the procedure at the second order even though it is not apparent. The prefactor before this effective relaxation term is $\kappa_W/(e+P)T$, which is the minimum value of the relaxation time required for hydrodynamic 