A satellite in a circular orbit at altitude km above the Earth's surface (as for the…

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 in a circular orbit at altitude 400400 km above the Earth's surface (as for the International Space Station) carries a clock. Take the Earth's radius R⊕=6.371×106R_\oplus = 6.371\times10^{6} m and GM⊕/c2=4.44×10−3GM_\oplus/c^2 = 4.44\times10^{-3} m. Using the rate formula dτ/dt≈1+(Φsat−Φground)/c2−v2/2c2d\tau/dt \approx 1 + (\Phi_{\rm sat}-\Phi_{\rm ground})/c^2 - v^2/2c^2 for a circular orbit, estimate: ((a)) the fractional rate of the satellite clock relative to a ground clock at R⊕R_\oplus; ((b)) the accumulated time difference after one day, in microseconds, stating whether the satellite clock runs fast or slow.

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