A star of uniform mass density (the mean density of the Earth

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · Differential geometry of curved spacetime

The Geometry of Gravity, Ch The covariant derivative, parallel transport and geodesics

Problem

A star of uniform mass density ρ=5.51×103 kg m−3\rho = 5.51\times10^3\ \mathrm{kg\,m^{-3}} (the mean density of the Earth, used here as a numerical stand-in) has, inside its surface, the Newtonian potential Φ(r)=2πGρ3r2\Phi(r) = \dfrac{2\pi G\rho}{3}r^2 (the constant of integration dropped, as it drops out of −∇Φ-\nabla\Phi), where rr is the distance from the centre. A test particle moves slowly along a straight radial line through the centre, in the weak, static field for which the metric component is g00=−(1+2Φc2)g_{00} = -\left(1+\dfrac{2\Phi}{c^2}\right) and the spatial metric is flat to the order needed. ((a)) Starting from the geodesic equation and its slow-motion, weak-field, static-field reduction, show that r¨=−ω2r\ddot r = -\omega^2 r for some constant ω\omega, and give ω2\omega^2 in terms of GG and ρ\rho. ((b)) Compute ω\omega numerically for the given ρ\rho, and hence the period Torb=2π/ωT_{\rm orb} = 2\pi/\omega of the resulting oscillation.

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