100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Relativity in the Global Positioning System 285Summary. In the present calculation, the effect of earth’s quadrupolemoment on the Keplerian orbit was accounted for. It was not necessaryto compute the orbit eccentricity. This approximate treatment of the orbitmakes no attempt to consider perturbations that are non-gravitational innature – e.g., solar radiation pressure. As a general conclusion, the fractionalfrequency shift can be estimated to very good accuracy from theexpression for the “factory frequency offset,”10. Secondary Relativistic Effectsδff = +3GM Eδa2c 2 a 2 . (57)There are several additional significant relativistic effects that must be consideredat the level of accuracy of a few cm (which corresponds to 100picoseconds of delay). Many investigators are modeling systematic effectsdown to the millimeter level so these effects, which currently are not sufficientlylarge to affect navigation, may have to be considered in the future.Signal Propagation Delay. The Shapiro signal propagation delaymay be easily derived in the standard way from the metric, Eq. (18), whichincorporates the choice of coordinate time rate expressed by the presenceof the term in Φ 0 /c 2 . Setting ds 2 = 0 and solving for the increment of coordinatetime along the path increment dσ = √ dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2givesdt = 1 [1 − 2Vc c 2 + Φ ]0c 2 dσ . (58)The time delay is sufficiently small that quadrupole contributions can beneglected. Integrating along the straight line path a distance l between thetransmitter and receiver gives for the time delay∆t delay = Φ 0 lc 2 c + 2GM Ec 3 ln[r1 + r 2 + lr 1 + r 2 − l], (59)where r 1 and r 2 are the distances of transmitter and receiver from earth’scenter. The second term is the usual expression for the Shapiro time delay.It is modified for GPS by a term of opposite sign (Φ 0 is negative), dueto the choice of coordinate time rate. This tends to cancel the logarithmterm. The net effect for a satellite to earth link is less than 2 cm and formost purposes can be neglected. One must keep in mind, however, that inthe main term, l/c, l is a coordinate distance and further small relativisticcorrections are required to convert it to a proper distance.