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Journal Article

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

##### Fulltext (public)

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##### Supplementary Material (public)

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##### Citation

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

##### 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 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.