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

Abstract

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: ${U}_{\ensuremath{\nu}}={\ensuremath{\mu}}^{\ensuremath{-}1}({\ensuremath{\varphi}}_{,\ensuremath{\nu}}+\ensuremath{\alpha}{\ensuremath{\beta}}_{,\ensuremath{\nu}}+\ensuremath{\theta}{S}_{,\ensuremath{\nu}})$. 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: $\mathcal{L}={(\ensuremath{-}g)}^{\frac{1}{2}}(R+16\ensuremath{\pi}p)$, 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.