Perfect Fluids in General Relativity: Velocity Potentials and a Variational Principle

The equations of hydrodynamics for a perfect fluid in general relativity are cast in Eulerian form, with the four-velocity being expressed in terms of six velocity potentials: ${U}_{{ν}}={{μ}}^{{-}1}({{φ}}_{,{ν}}+{α}{{β}}_{,{ν}}+{θ}{S}_{,{ν}})$. Each of the velocity potentials has its own "equation of motion." These equations furnish a description of hydrodynamics that is equivalent to the usual equations based on the divergence of the stress-energy tensor. The velocity-potential description leads to a variational principle whose Lagrangian density is especially simple: L=(-g)1/2(R+16πp), where R is the scalar curvature of spacetime and p is the pressure of the fluid. Variation of the action with respect to the metric tensor yields Einstein's field equations for a perfect fluid. Variation with respect to the velocity potentials reproduces the Eulerian equations of motion.