Signal propagation delay.
The Shapiro signal propagation delay may be easily derived in the
standard way from the metric, Eq. (23), which incorporates the choice of coordinate time rate
expressed by the presence of the term in
. Setting
and solving for the increment of coordinate time along the path
increment
gives
The time delay is sufficiently small that quadrupole
contributions to the potential (and to
) can be neglected. Integrating along the straight line path a
distance
l
between the transmitter and receiver gives for the time
delay
where
and
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 (
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
l
/
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. (23), the proper distance between a point at radius
and another point at radius
directly above the first is approximately
The difference between proper distance and coordinate
distance, and between the earth's surface and the radius of GPS
satellites, is approximately
. 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
in an inertial frame, and suppose the receiver spins rapidly
with angular frequency
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
. 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 [17,
20,
21,
24], 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. (1
). 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
where
R
is the earth-sun distance [8]. The gravitational frequency shift of GPS satellite clocks from
such potentials is a few parts in
and is currently neglected in the GPS.
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Relativity in the Global Positioning System
Neil Ashby http://www.livingreviews.org/lrr-2003-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |