General Relativity: An Introduction for Physicists · Problem 8.2

The gravitational field equations — Hobson, Efstathiou & Lasenby

Physics · perfect-fluid-equations-of-motion

Problem

Show that, for any fluid, uν∇μuν=0.u_\nu \nabla_\mu u^\nu = 0. Hence show that a perfect fluid in a gravitational field must satisfy the equations ∇μ(ρuμ)+pc2∇μuμ=0,\nabla_\mu(\rho u^\mu) + \frac{p}{c^2}\nabla_\mu u^\mu = 0, (ρ+pc2)uμ∇μuν=(gμν−uμuνc2)∇μp.\left(\rho + \frac{p}{c^2}\right) u^\mu \nabla_\mu u^\nu = \left(g^{\mu\nu} - \frac{u^\mu u^\nu}{c^2}\right)\nabla_\mu p. Obtain the equation of motion for the worldline xμ(τ)x^\mu(\tau) of a particle in a perfect fluid with pressure, and hence show that the particle is 'pushed off' geodesics by the pressure gradient.

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