Bounds on transport from hydrodynamic stability

It was recently shown that the dispersion relations describing singularities of retarded two-point functions in causal quantum field theories always satisfy the fundamental inequality $\mathfrak{Im} \, \omega \leq |\mathfrak{Im} \, k|$, and that several rigorous bounds on transport coefficients follow directly from such inequality. Here, we prove that the same inequality, $\mathfrak{Im} \, \omega \leq |\mathfrak{Im} \, k|$, is a necessary condition for a fluid theory to be covariantly stable (i.e. stable in all frames of reference). Hence, the same bounds on transport that follow from causality in quantum field theory also emerge as stability conditions within relativistic hydrodynamics. This intimate connection between bounds from causality and bounds from stability stems from the fact that covariant stability is possible only in the presence of causality. As a quick application, we show that fluids with luminal speed of sound have vanishing viscosities.