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

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

Dec 1970

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970phrvd...2.2762s&link_type=abstract

Physical Review D, vol. 2, Issue 12, pp. 2762-2773

Physics

156

Scientific paper

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: Uν=μ- 1(φ,ν+αβ,ν+θS,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)12(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.

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