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: $U_{\nu }={\mu }^{-1\mathrm{}}({\phi }_{,\nu }+\alpha {\beta }_{,\nu }+\theta S_{,\nu })$. 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: $\mathcal{L}={\mathrm{}(-g)\mathrm{}}^{\frac{1}{2}}(R+16\mathrm{}\pi p)$, 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.