The causal diamond of relativistic hydrodynamics

Abstract

The convergence of hydrodynamics is commonly investigated by Taylor-expanding the gapless dispersion relations $\omega(k)$ around $k=0$. We show that this diagnostic can fail arbitrarily close to the hydrodynamic limit: sound modes can collide at arbitrarily small wave number, rendering $\omega(k)$ nonanalytic, while the relationship between densities and fluxes remains analytic there. We instead formulate the gradient expansion off-shell (i.e. in the presence of external forces), directly at the level of the constitutive relations. For causal theories with relaxational spectra, we establish convergence for $|i\omega|+w|ik|<1/\tau_g$, where $w$ is the maximal information speed and $1/\tau_g$ the nonhydrodynamic gap. On the naturally Lorentzian $\{i\omega,ik\}$ plane, this region is a causal diamond, whose boundary marks the first possible encounter with nonhydrodynamic modes. Beyond it, analytic continuation reveals the nonhydrodynamic spectrum through singularities of the resummed constitutive relations. Thus, hydrodynamics is aware of the degrees of freedom beyond hydrodynamics, but this information becomes fully accessible only off-shell.