Note on a theorem of relativistic hydrodynamics

A well known theorem of relativistic hydrodynamics states that the streamlines of an isentropic perfect fluid are the future-pointing timelike (FPT) curves extremizing the integral J = integral(S1)(S2) fds, J, where f is the so-called index function and s the proper time on the world line of the fluid particle. The integral is taken over all possible FPT curves with regular representations x(i) = x(i)(s) joining the fixed end events E-1, E-2. The purpose of this note is to show that the streamlines of an adiabatic perfect fluid can likewise be regarded as extremizing curves of the functional J provided the class of admissible curves consists of those FPT curves satisfying the side condition u(i)partial derivative(i)S = 0, u(i) unit 4-velocity and S the specific proper entropy of the fluid, with the first end point fixed and the second being the end point variable.