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Journal Article

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


Schutz,  Bernard F.
Astrophysical Relativity, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;
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Schutz, B. F. (1970). Perfect Fluids in General Relativity: Velocity Potentials and a Variational Principle. Physical Review D, 2(12), 2762-2773. doi:10.1103/PhysRevD.2.2762.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-0F40-5
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 =µ-1( 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: L=(-g)1 / 2(R+16 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.