The core of a compact star 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

The core of a compact star is modelled, at a scale where a Newtonian-limit description applies, as a static perfect fluid at rest; at the point considered its proper energy density is ρ=6.0×1034 J m−3\rho=6.0\times10^{34}\,\mathrm{J\,m^{-3}} and its isotropic pressure p=2.0×1034 J m−3p=2.0\times10^{34}\,\mathrm{J\,m^{-3}}, both varying across the core as part (b) requires, so that T00=ρT_{00}=\rho and Tij=pδijT_{ij}=p\delta_{ij} in the local rest frame, with no bulk motion. Take Λ=0\Lambda=0. ((a)) Using the trace-reversed field equations and R00=∇2Φ/c2R_{00}=\nabla^2\Phi/c^2, show that the Newtonian-limit source of gravity for a perfect fluid is ρ+3p\rho+3p rather than ρ\rho alone, i.e. derive ∇2Φ=4πG(ρ+3p)/c2\nabla^2\Phi=4\pi G(\rho+3p)/c^2, using κ=8πG/c4\kappa=8\pi G/c^4 as already fixed from the dust case. Evaluate ∇2Φ\nabla^2\Phi for the core, and state what fraction of it is due to the pressure alone. ((b)) The spatial (ν=i\nu=i) part of ∇μTμν=0\nabla_\mu T^{\mu\nu}=0, in the Newtonian limit, reproduces the fluid's hydrostatic-equilibrium condition ∇p=−ρmg⃗\nabla p=-\rho_{\rm m}\vec g. Explain why this relation is not an extra assumption about the star but a necessary consequence of the field equations, and name the geometric identity responsible. ((c)) State how many of the ten field equations for this star are constraints and how many are evolution equations, and explain, without appeal to the star's particular equation of state, why this split can never depend on what TμνT_{\mu\nu} is.

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