A small moon is modelled as a two-sphere of radius m with metric

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 small moon is modelled as a two-sphere of radius a=5.0×103a = 5.0\times10^3 m 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=∂θ\xi = \partial_\theta. ((a)) Compute the component Lξgϕϕ\mathcal{L}_\xi g_{\phi\phi} of the Lie derivative of the metric, evaluated at colatitude θ=30∘\theta = 30^\circ, and decide whether ξ\xi satisfies Killing's equation. ((b)) State whether ξ=∂θ\xi=\partial_\theta appears among the sphere's three independent Killing vectors. ((c)) Say what this implies for the existence of a conserved quantity ξμuμ\xi_\mu u^\mu along a geodesic on the moon.

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