PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Einstein's field equations
The Geometry of Gravity, Ch Curvature: the Riemann tensor, geodesic deviation and the Einstein tensor
Problem
Two satellites orbit the Earth at r=7000 km and are, for an instant, at rest relative to each other, separated by ℓ=10.0 m along the radial direction. Suppose we are told that in the vacuum outside the Earth the Ricci tensor vanishes, Rμν=0. A colleague argues: 'Since spacetime is Ricci-flat outside the Earth, it is flat there, so there can be no relative acceleration between the two satellites; the radial tidal stretching described by the Newtonian tidal field must be a purely Newtonian artefact that disappears once general relativity is used properly.' Using GM⊕/c2=4.44 mm and c=3.00×108ms−1, evaluate this claim by computing the actual relative radial acceleration between the two satellites from Rr0r0=−2GM⊕/(r3c2), and state precisely what is wrong with the reasoning.