10 Secondary Relativistic Effects

There are several additional significant relativistic effects that must be considered at the level of accuracy of a few cm (which corresponds to 100 picoseconds of delay). Many investigators are modelling systematic effects down to the millimeter level, so these effects, which currently are not sufficiently large to affect navigation, may have to be considered in the future.

Signal propagation delay.  The Shapiro signal propagation delay may be easily derived in the standard way from the metric, Eq. (23View Equation), which incorporates the choice of coordinate time rate expressed by the presence of the term in Φ āˆ•c2 0. Setting ds2 = 0 and solving for the increment of coordinate time along the path increment āˆ˜ ------------------------- dσ = dr2 + r2dšœƒ2 + r2sin2 šœƒdĻ•2 gives

1[ 2V Φ0] dt = --1 − --2 + -2- dσ. (95 ) c c c
The time delay is sufficiently small that quadrupole contributions to the potential (and to Φ0) can be neglected. Integrating along the straight line path a distance l between the transmitter and receiver gives for the time delay
[ ] Φ0 l 2GME r1 + r2 + l Δtdelay = -2--+ ---3---ln ----------- , (96 ) c c c r1 + r2 − l
where r1 and r2 are the distances of transmitter and receiver from earth’s center. The second term is the usual expression for the Shapiro time delay. It is modified for GPS by a term of opposite sign (Φ0 is negative), due to the choice of coordinate time rate, which tends to cancel the logarithm term. The net effect for a satellite to earth link is less than 2 cm and for most purposes can be neglected. One must keep in mind, however, that in the main term lāˆ•c, l is a coordinate distance and further small relativistic corrections are required to convert it to a proper distance.

Effect on geodetic distance.  At the level of a few millimeters, spatial curvature effects should be considered. For example, using Eq. (23View Equation), the proper distance between a point at radius r1 and another point at radius r2 directly above the first is approximately

∫ r2 [ GME ] GME (r2 ) dr 1 + --2--- = r2 − r1 + --2---ln -- . (97 ) r1 c r c r1
The difference between proper distance and coordinate distance, and between the earth’s surface and the radius of GPS satellites, is approximately 4.43 ln(4.2) mm ≈ 6.3 mm. Effects of this order of magnitude would enter, for example, in the comparison of laser ranging to GPS satellites, with numerical calculations of satellite orbits based on relativistic equations of motion using coordinate times and coordinate distances.

Phase wrap-up.  Transmitted signals from GPS satellites are right circularly polarized and thus have negative helicity. For a receiver at a fixed location, the electric field vector rotates counterclockwise, when observed facing into the arriving signal. Let the angular frequency of the signal be ω in an inertial frame, and suppose the receiver spins rapidly with angular frequency Ω which is parallel to the propagation direction of the signal. The antenna and signal electric field vector rotate in opposite directions and thus the received frequency will be ω + Ω. In GPS literature this is described in terms of an accumulation of phase called “phase wrap-up”. This effect has been known for a long time [17202124], and has been experimentally measured with GPS receivers spinning at rotational rates as low as 8 cps. It is similar to an additional Doppler effect; it does not affect navigation if four signals are received simultaneously by the receiver as in Eqs. (1View Equation). This observed effect raises some interesting questions about transformations to rotating, spinning coordinate systems.

Effect of other solar system bodies.  One set of effects that has been “rediscovered” many times are the redshifts due to other solar system bodies. The Principle of Equivalence implies that sufficiently near the earth, there can be no linear terms in the effective gravitational potential due to other solar system bodies, because the earth and its satellites are in free fall in the fields of all these other bodies. The net effect locally can only come from tidal potentials, the third terms in the Taylor expansions of such potentials about the origin of the local freely falling frame of reference. Such tidal potentials from the sun, at a distance r from earth, are of order 2 3 GM āŠ™r āˆ•R where R is the earth-sun distance [8]. The gravitational frequency shift of GPS satellite clocks from such potentials is a few parts in 1016 and is currently neglected in the GPS.

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