Extension of relativistic dissipative hydrodynamics to third order

Abstract

Following the procedure introduced by Israel and Stewart, we expand the entropy current up to the third order in the shear stress tensor ${\ensuremath{\pi}}^{\ensuremath{\alpha}\ensuremath{\beta}}$ and derive a novel third-order evolution equation for ${\ensuremath{\pi}}^{\ensuremath{\alpha}\ensuremath{\beta}}$. This equation is solved for the one-dimensional Bjorken boost-invariant expansion. The scaling solutions for various values of the shear viscosity to the entropy density ratio $\ensuremath{\eta}/s$ are shown to be in very good agreement with those obtained from kinetic transport calculations. For the pressure isotropy starting with 1 at ${\ensuremath{\tau}}_{0}=0.4$ fm/$c$, the third-order corrections to Israel-Stewart theory are approximately $10%$ for $\ensuremath{\eta}/s=0.2$ and more than a factor of 2 for $\ensuremath{\eta}/s=3$. We also estimate all higher-order corrections to Israel-Stewart theory and demonstrate their importance in describing highly viscous matters.