A foundational question in relativistic fluid mechanics concerns the properties of the hydrodynamic gradient expansion at large orders. We establish the precise conditions under which this gradient expansion diverges for a broad class of microscopic theories admitting a relativistic hydrodynamic limit, in the linear regime. Our result does not rely on highly symmetric fluid flows utilized by previous studies of heavy-ion collisions and cosmology. The hydrodynamic gradient expansion diverges whenever energy density or velocity fields have support in momentum space exceeding a critical momentum, and converges otherwise. This critical momentum is an intrinsic property of the microscopic theory and is set by branch point singularities of hydrodynamic dispersion relations.

10 pages, 2 figures; v2: results unchanged, reorganized and expanded presentation with new figures and new appendix on purely temporal gradient expansion, matches published version