A satellite skims the surface of a spherical asteroid of radius m

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Symmetry: Lie derivatives, Killing vectors and conserved quantities

Problem

A satellite skims the surface of a spherical asteroid of radius a=6.0×106a = 6.0\times10^6 m, so that its motion is confined to a two-sphere with metric ds2=a2(dθ2+sin⁡2θ dϕ2)ds^2 = a^2(d\theta^2 + \sin^2\theta\,d\phi^2). Consider the vector field ξ=∂ϕ\xi = \partial_\phi. ((a)) Compute Lξgμν\mathcal{L}_\xi g_{\mu\nu} for this metric and decide whether ξ\xi satisfies Killing's equation. ((b)) State which one of the two-sphere's three independent Killing vectors ξ\xi is. ((c)) At the point where the satellite crosses colatitude θ=60∘\theta = 60^\circ, its trajectory has dϕ/dτ=1.0×10−3 s−1d\phi/d\tau = 1.0\times10^{-3}\ \mathrm{s^{-1}}. Compute the numerical value of the conserved quantity ξ\xi produces along its geodesic.

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