In this chapter we derive a system of equations for “relativistic fluid streamlines” in terms of the divergence of a thermodynamic mass-energy current and directed pressure gradient scalar field. One of the equations leads to interesting thermodynamic inequalities involving the behaviour of the “specific entropy scalar” along the relativistic flow lines. Conditions are also derived for the flow to be “locally adiabatic” or “locally isentropic”. The special case (relevant to certain cosmological models) is the “cold collisionless gas” or “dust” describing timelike geodesic flows.