A dense core in a molecular cloud is modelled

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

The Geometry of Gravity, Ch The Einstein field equations and the Newtonian limit

Problem

A dense core in a molecular cloud is modelled, on the scales of interest, as a static cloud of uniform mass density ρm=2.0×10−16 kg m−3\rho_{\rm m} = 2.0\times10^{-16}\,\mathrm{kg\,m^{-3}}, at rest in the coordinates used, whose pressure holds it up but is negligible next to its energy density, so that in the source it acts as dust, with equivalent proper energy density ρ=ρmc2\rho = \rho_{\rm m}c^2. Take the cosmological constant to be zero. ((a)) Starting from the trace-reversed field equations Rμν=κ(Tμν−12Tgμν)R_{\mu\nu} = \kappa(T_{\mu\nu} - \tfrac12 Tg_{\mu\nu}) and the weak-field result R00=∇2Φ/c2R_{00} = \nabla^2\Phi/c^2, derive Poisson's equation for this dust and hence the value of κ\kappa. Evaluate ∇2Φ\nabla^2\Phi at the core. ((b)) Explain, starting from the contracted Bianchi identity, why ∇μTμν=0\nabla_\mu T^{\mu\nu} = 0 must hold for this dust as a consequence of the field equations, rather than being an extra assumption about the cloud. ((c)) The field equations Gμν=κTμνG^{\mu\nu} = \kappa T^{\mu\nu} are ten equations for the ten independent components of gμνg_{\mu\nu}. State how many of these are constraint equations containing no second time derivative, how many are genuine evolution equations, and explain what accounts for the mismatch with ten unknowns.

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