A satellite moves on a circular orbit of areal radius km around a non-rotating…

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 moves on a circular orbit of areal radius rs=2.6571×104r_s=2.6571\times10^4 km around a non-rotating idealised Earth, described by the weak-field metric ds2=−(1+2Φ/c2)c2dt2+dr2+r2dΩ2ds^2=-(1+2\Phi/c^2)c^2dt^2+dr^2+r^2d\Omega^2, Φ(r)=−GM⊕/r\Phi(r)=-GM_\oplus/r. A ground clock sits static at Earth's surface, R⊕=6371R_\oplus=6371 km, also idealised as non-rotating. ((a)) Using the conserved quantities generated by the Killing vectors ∂t\partial_t and ∂ϕ\partial_\phi along the satellite's circular geodesic, together with the geodesic equation, recover the Newtonian circular-orbit speed vs2=GM⊕/rsv_s^2=GM_\oplus/r_s. ((b)) Combine the gravitational time-dilation factor for each clock with the special-relativistic time dilation of the moving satellite to find the fractional rate difference (dτsat/dτground)−1(d\tau_{\rm sat}/d\tau_{\rm ground})-1, to leading order in Φ/c2\Phi/c^2 and vs2/c2v_s^2/c^2. ((c)) Evaluate this numerically using GM⊕/c2=4.44GM_\oplus/c^2=4.44 mm, and express it as a number of microseconds gained or lost by the satellite clock per day.

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