At a point inside a static

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

At a point inside a static, weak-field ball of radiation-dominated fluid, the fluid is at rest and has the perfect-fluid stress-energy tensor Tμν=(ρ+p)uμuν/c2+pgμνT^{\mu\nu} = (\rho+p)u^\mu u^\nu/c^2 + pg^{\mu\nu}, where ρ\rho is the local proper energy density and pp the local pressure, with the radiation equation of state p=ρ/3p = \rho/3. Both are values at that point; neither is uniform across the ball. ((a)) Show that T00=ρT_{00} = \rho, T=−ρ+3pT = -\rho+3p, and hence T00−12Tg00=12(ρ+3p)T_{00} - \tfrac12 Tg_{00} = \tfrac12(\rho+3p). ((b)) Using the trace-reversed field equations Rμν=κ(Tμν−12Tgμν)R_{\mu\nu} = \kappa\left(T_{\mu\nu} - \tfrac12 Tg_{\mu\nu}\right) with κ=8πG/c4\kappa = 8\pi G/c^4, together with the relation R00=∇2Φ/c2R_{00} = \nabla^2\Phi/c^2 for the weak, static tidal field, show that ∇2Φ=4πG(ρm+3p/c2)\nabla^2\Phi = 4\pi G(\rho_{\rm m} + 3p/c^2) with ρm=ρ/c2\rho_{\rm m} = \rho/c^2, and give the numerical factor by which this radiation's source exceeds that of dust (p=0p=0) of the same ρm\rho_{\rm m}. ((c)) For a region of mass density ρm=4.0×1017 kg m−3\rho_{\rm m} = 4.0\times10^{17}\ \mathrm{kg\,m^{-3}}, evaluate ∇2Φ\nabla^2\Phi numerically. ((d)) Explain, without introducing any extra postulate, why ∇μTμν=0\nabla_\mu T^{\mu\nu} = 0 must hold for this fluid, and say what this equation becomes in the Newtonian limit, referring to the continuity and momentum equations for an ideal fluid.

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