A globular cluster

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Matter in curved spacetime: minimal coupling and relativistic hydrodynamics

Problem

A globular cluster, far from other matter, is modelled on scales large compared with the spacing of its stars, with the stars' random motions neglected so that one velocity describes them at each point, as a pressureless perfect fluid ("dust") with stress-energy tensor Tμν=ρuμuνT^{\mu\nu}=\rho u^\mu u^\nu, where ρ\rho is the rest-frame energy density of the smoothed-out star distribution and uμu^\mu its four-velocity field, uμuμ=−1u^\mu u_\mu=-1. ((a)) Starting from ∇μTμν=0\nabla_\mu T^{\mu\nu}=0 and using uμ∇νuμ=0u_\mu\nabla_\nu u^\mu=0, derive the two separate equations ∇μ(ρuμ)=0,uν∇νuμ=0.\nabla_\mu(\rho u^\mu)=0, \qquad u^\nu\nabla_\nu u^\mu = 0. ((b)) Explain in one or two sentences why the second equation means every star (each on its own worldline within the smoothed fluid) moves on a geodesic, even though ρ\rho is a property of the whole cluster and not of any single star. ((c)) A real cluster has a small internal pressure p≪ρp\ll\rho once enough stars are averaged, by their velocity dispersion, to define a fluid element. Write the corresponding relativistic Euler equation and identify, in words, the term absent from part (a) that now prevents a fluid element (as opposed to a single star) from being in free fall.

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