Earth rotates and is not perfectly spherical, so, strictly speaking, the Schwarzschild metric does not describe spacetime above Earth's surface. But Earth rotates slowly and the Schwarzschild metric is a good approximation for purposes of analyzing the Global Positioning System.
Apply this equation twice, first to the orbiting satellite clock and second to a clock fixed at Earth's equator and rotating as Earth turns. Both the Earth clock and the satellite clock travel at constant radius around Earth's center.
So dr = 0 for each clock. Divide the Schwarzschild metric through by the square of the far-away time df2 to obtain, for either clock,
Here dx is the wristwatch time between ticks of either clock and v-r dty/dt is the tangential velocity along the circular path of the same clock as measured by the bookkeeper using far-away time measurement. Write down equation  first for the satellite, using r = Satellite v = ^satellite and dx = ¿ftsatellite between ticks of the satellite clock, second for the Earth clock, using r = rEarth, v - i>Earth ^d time dx = df Earth between ticks of the Earth clock, all these for the same time lapse dt on the far-away clock. Divide corresponding sides of these two equations to obtain the squared ratio of time lapses recorded on the satellite and earth clocks:
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