A junior traveler, making her first trip by train from the United States into Mexico, sees the town of Zacatecas outside her window and reassures herself by the marginal note in the guidebook that this ancient silver-mining town is 1848 kilometers from San Diego, California, and 1506 kilometers from New Orleans, Louisiana. On a surface, two distances thus suffice to fix location. But in space it is three. Find those three distances, to each of three nearest satellites of the Global Positioning System, by finding the time taken by light or radio pulse to come from each satellite to us. Simple enough! Or simple as soon as we correct, as we must and as we have, for the clock rates at each end of the signal path. (1) General relativity predicts that both the relative altitudes and the relative speeds of satellite and

Earth clocks affect their relative rates. (2) The clock in the hand-held receiver on Earth is far less accurate than the atomic clock in each satellite, so the signal from a fourth satellite is employed to correct the Earth clock. With these corrections, we can use the Global Positioning System to locate ourselves anywhere on Earth with an uncertainty of only a few meters.

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