Action of perfect fluid

[smallphi]

The energy momentum tensor of perfect fluid is

T^{\alpha \beta} = \left( \rho + p \right) \, U^\alpha U^\beta - p \, g^{\alpha \beta}

It must be derived by varying the metric in the action of matter fields but I've never seen that action. Anyone knows it?

[smallphi]

However moved that question, it belongs to General Relativity section not to 'Homework and Coursework', geezus ....

[Chris Hillman]

However moved that question, it belongs to General Relativity section not to 'Homework and Coursework', geezus ....

I assume you are asking for a Lagrangian formulation of a complete thermodynamic description of a perfect fluid, i.e. with variables n (particle number density), rho (mass-energy density), p (pressure), T (temperature), s (entropy per particle), and \vec{U} (four-velocity of the fluid) as per MTW. If so, Schutz and Sorkin showed that any such formulation must force additional constraints. That is, there is no general formulation, but there are proposed action formulations for special cases. See for example gr-qc/9304026

[smallphi]

Yes, that's what I needed to see. Thanks.