The linearised field equations give outside a static source

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

The Geometry of Gravity, Ch Linearised gravity: weak fields, gravitomagnetism and the wave equation

Problem

The linearised field equations give h00=−2Φ/c2h_{00}=-2\Phi/c^2 outside a static source, with Φ=−GM/r~\Phi=-GM/\tilde r. In the Newtonian limit, g00=−(1+2Φ/c2)g_{00}=-(1+2\Phi/c^2) satisfies ∇2Φ=4πGρm\nabla^2\Phi=4\pi G\rho_{\rm m}, sourced by the slow-matter approximation T00≈ρT_{00}\approx\rho for the proper energy density ρ\rho. Also available is the gravitational redshift of a light signal travelling between two points of a weak, static field, Δν/ν≈(Φemit−Φobs)/c2\Delta\nu/\nu \approx (\Phi_{\rm emit}-\Phi_{\rm obs})/c^2 to first order. A GPS satellite moves on a circular orbit of radius r~orb=2.656×107\tilde r_{\rm orb}=2.656\times10^7 m; take the Earth's surface at r~⊕=6.371×106\tilde r_\oplus=6.371\times10^6 m, and GM⊕/c2=4.44GM_\oplus/c^2=4.44 mm. ((a)) Using h00=−2Φ/c2h_{00}=-2\Phi/c^2, compute h00h_{00} at the surface and at the orbit radius. ((b)) A clock signal is emitted by the satellite and received on the ground. Express the fractional frequency shift Δν/ν\Delta\nu/\nu of the received signal, relative to the emitted frequency, in terms of the two values of h00h_{00} found in part (a), and evaluate it. ((c)) State briefly why this result contains no physics beyond the Newtonian limit derived from T00≈ρT_{00}\approx\rho and ∇2Φ=4πGρm\nabla^2\Phi=4\pi G\rho_{\rm m}: what new content, if any, does the linearised metric perturbation add at this order?

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