Correspondence between Israel-Stewart and first-order casual and stable hydrodynamics for the boost-invariant massive case with zero baryon density

Exact correspondence between Israel-Stewart theory and first-order causal and stable hydrodynamics is established for the boost-invariant massive case with zero baryon density and the same constant relaxation times used in the shear and bulk sectors. Explicit expressions for the temperature dependent regulators are given for the case of a relativistic massive gas. The stability and causality conditions known in the first-order approach are applied, and one finds that one of them is violated in this case.

Figure 1

Temperature dependence of the regulators ${ϵ}_1$, ${ϵ}_2$, ${\pi }_1$, and ${\pi }_2$ obtained for the equation of state given by Eqs. (33) with $m=m_{\pi }=140\mathrm{MeV}$. The results are scaled by the values corresponding to the $m\to 0$ limit Eqs. (32).