Perfect fluids revisited: an action principle approach
We revisit the variational principle for relativistic perfect fluids in a manifestly covariant formulation based on differential forms, with particular attention to the boundary data required for a well-posed action principle. For timelike flows, the formulation reproduces the standard formulation of Schutz and Brown. We then examine the extension obtained by replacing proper time normalization of the velocity with a nullity constraint, with the he Bi{\v c}{á}k-Kucha{\v r} null dust action emerging as a special case. Requiring the complete action to be invariant under local rescalings of the null flow vector imposes the condition that the energy density depend only on the entropy per particle. For this class of null flows, the conventional equilibrium condition, \textit{i.e.}, that the energy density is convex, fails at non-zero temperatures, therefore limiting the interpretation of the fluid as an ordinary thermodynamic system.