  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.
