Stationary Clocks

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 [2] 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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