  We show that it is possible to define a timelike future-directed information
current within relativistic first-order hydrodynamics. This constitutes the
first step toward a covariantly stable and causal formulation of first-order
fluctuating hydrodynamics based on thermodynamic principles. We provide several
explicit examples of first-order theories with an information current, covering
many physical phenomena, ranging from electric conduction to viscosity and
elasticity. We use these information currents to compute the corresponding
equal-time correlation functions, and we find that the physically relevant
(equal-time) correlators do not depend on the choice of the hydrodynamic frame
as long as the frame leads to causal and stable dynamics. In the example of
chiral hydrodynamics, we find that circularly polarized shear waves have
different probabilities of being excited depending on their handedness,
generating net helicity in chiral fluids.
