A hypothetical uniform medium

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 hypothetical uniform medium, not identified with the cosmological constant, has proper energy density ρ=2.0×10−9 J m−3\rho=2.0\times10^{-9}\,\mathrm{J\,m^{-3}} and isotropic pressure p=−ρp=-\rho, at rest, so T00=ρT_{00}=\rho and Tij=pδij=−ρδijT_{ij}=p\delta_{ij}=-\rho\delta_{ij} in the local rest frame. Take Λ=0\Lambda=0: the medium's own stress-energy, not a separate constant, supplies the p=−ρp=-\rho behaviour. ((a)) Derive the Newtonian-limit field equation sourced by this medium and show that its sign is opposite to that of ordinary matter of the same energy density. Evaluate ∇2Φ\nabla^2\Phi numerically, using ρm=ρ/c2\rho_{\rm m}=\rho/c^2. ((b)) Show that for p=−ρp=-\rho the stress-energy tensor is Tμν=−ρgμνT_{\mu\nu}=-\rho g_{\mu\nu}, and use this to show that ∇μTμν=0\nabla_\mu T^{\mu\nu}=0 holds exactly, the medium's own gravity included. Then explain why a uniform, static medium with any other relation between pp and ρ\rho would violate this condition wherever its own field ∇Φ\nabla\Phi is non-zero. ((c)) Explain why the split of the ten field equations into four constraints and six evolution equations does not depend on the equation of state p(ρ)p(\rho) of the source at all.

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