A satellite moves in the Earth's equatorial plane

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry

The Geometry of Gravity, Ch Redshift, local inertial frames and the failure of flat spacetime

Problem

A satellite moves in the Earth's equatorial plane. Describe its position by polar coordinates (r,φ)(r,\varphi) centred on the Earth, so that v2=r˙2+r2φ˙2v^2=\dot r^2+r^2\dot\varphi^2, and take Φ(r)=−GM⊕/r\Phi(r)=-GM_\oplus/r with GM⊕=3.986×1014 m3 s−2GM_\oplus=3.986\times10^{14}\ \mathrm{m^3\,s^{-2}} and dτ/dt≈1+Φ/c2−v2/2c2d\tau/dt\approx1+\Phi/c^2-v^2/2c^2. ((a)) Apply δ∫dτ=0\delta\int d\tau=0 with the end events fixed, treating rr and φ\varphi as the two coordinates. Show that a circular orbit, with rr constant and φ˙=ω\dot\varphi=\omega constant, is a solution only if ω2r3=GM⊕\omega^2r^3=GM_\oplus, which is Kepler's third law. ((b)) A geostationary satellite has period T=86 164T=86\,164 s, one sidereal day. Find its orbital radius. ((c)) At one event on this orbit two clocks part. One rides the satellite. The other is held by rockets at that point, at rest relative to the distant stars. One period later the satellite comes round and the two clocks meet again. Which has recorded more proper time, and by how much? Explain why this does not contradict the principle that the freely falling worldline records the most proper time.

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