It should be emphasized that the transmitted navigation
messages provide the user only with a function from which the
satellite position can be calculated
in the ECEF
as a function of the transmission time. Usually, the satellite
transmission times
are unequal, so the coordinate system in which the satellite
positions are specified changes orientation from one measurement
to the next. Therefore, to implement Eqs. (1
), the receiver must generally perform a different rotation for
each measurement made, into some common inertial frame, so that
Eqs. (1
) apply. After solving the propagation delay equations, a final
rotation must usually be performed into the ECEF to determine the
receiver's position. This can become exceedingly complicated and
confusing. A technical note [10] discusses these issues in considerable detail.
Although the ECEF frame is of primary interest for navigation,
many physical processes (such as electromagnetic wave
propagation) are simpler to describe in an inertial reference
frame. Certainly, inertial reference frames are needed to express
Eqs. (1), whereas it would lead to serious error to assert Eqs. (1
) in the ECEF frame. A ``Conventional Inertial Frame'' is
frequently discussed, whose origin coincides with earth's center
of mass, which is in free fall with the earth in the
gravitational fields of other solar system bodies, and whose
z
-axis coincides with the angular momentum axis of earth at the
epoch J2000.0. Such a local inertial frame may be related by a
transformation of coordinates to the so-called International
Celestial Reference Frame (ICRF), an inertial frame defined by
the coordinates of about 500 stellar radio sources. The center of
this reference frame is the barycenter of the solar system.
In the ECEF frame used in the GPS, the unit of time is the SI second as realized by the clock ensemble of the U.S. Naval Observatory, and the unit of length is the SI meter. This is important in the GPS because it means that local observations using GPS are insensitive to effects on the scales of length and time measurements due to other solar system bodies, that are time-dependent.
Let us therefore consider the simplest instance of a transformation from an inertial frame, in which the space-time is Minkowskian, to a rotating frame of reference. Thus, ignoring gravitational potentials for the moment, the metric in an inertial frame in cylindrical coordinates is
and the transformation to a coordinate system
rotating at the uniform angular rate
is
This results in the following well-known metric (Langevin metric) in the rotating frame:
where the abbreviated expression
for the square of the coordinate distance has been used.
The time transformation
t
=
t
' in Eqs. (3) is deceivingly simple. It means that in the rotating frame the
time variable
t
' is really determined in the underlying inertial frame. It is an
example of coordinate time. A similar concept is used in the
GPS.
Now consider a process in which observers in the rotating
frame attempt to use Einstein synchronization (that is, the
principle of the constancy of the speed of light) to establish a
network of synchronized clocks. Light travels along a null
worldline, so we may set
in Eq. (4
). Also, it is sufficient for this discussion to keep only terms
of first order in the small parameter
. Then
and solving for (cdt ') yields
The quantity
is just the infinitesimal area
in the rotating coordinate system swept out by a vector from the
rotation axis to the light pulse, and projected onto a plane
parallel to the equatorial plane. Thus, the total time required
for light to traverse some path is
Observers fixed on the earth, who were unaware of earth
rotation, would use just
for synchronizing their clock network. Observers at rest in the
underlying inertial frame would say that this leads to
significant path-dependent inconsistencies, which are
proportional to the projected area encompassed by the path.
Consider, for example, a synchronization process that follows
earth's equator in the eastwards direction. For earth,
and the equatorial radius is
6,378,137 m, so the area is
. Thus, the last term in Eq. (7
) is
From the underlying inertial frame, this can be regarded as
the additional travel time required by light to catch up to the
moving reference point. Simple-minded use of Einstein
synchronization in the rotating frame gives only
, and thus leads to a significant error. Traversing the equator
once eastward, the last clock in the synchronization path would
lag the first clock by 207.4 ns. Traversing the equator once
westward, the last clock in the synchronization path would lead
the first clock by 207.4 ns.
In an inertial frame a portable clock can be used to
disseminate time. The clock must be moved so slowly that changes
in the moving clock's rate due to time dilation, relative to a
reference clock at rest on earth's surface, are extremely small.
On the other hand, observers in a rotating frame who attempt
this, find that the proper time elapsed on the portable clock is
affected by earth's rotation rate. Factoring Eq. (4), the proper time increment
on the moving clock is given by
For a slowly moving clock,
, so the last term in brackets in Eq. (9
) can be neglected. Also, keeping only first order terms in the
small quantity
yields
which leads to
This should be compared with Eq. (7). Path-dependent discrepancies in the rotating frame are thus
inescapable whether one uses light or portable clocks to
disseminate time, while synchronization in the underlying
inertial frame using either process is self-consistent.
Eqs. (7) and (11
) can be reinterpreted as a means of realizing coordinate time
t
'=
t
in the rotating frame, if after performing a synchronization
process appropriate corrections of the form +
are applied. It is remarkable how many different ways this can
be viewed. For example, from the inertial frame it appears that
the reference clock from which the synchronization process starts
is moving, requiring light to traverse a different path than it
appears to traverse in the rotating frame. The Sagnac effect can
be regarded as arising from the relativity of simultaneity in a
Lorentz transformation to a sequence of local inertial frames
co-moving with points on the rotating earth. It can also be
regarded as the difference between proper times of a slowly
moving portable clock and a Master reference clock fixed on
earth's surface.
This was recognized in the early 1980s by the Consultative Committee for the Definition of the Second and the International Radio Consultative Committee who formally adopted procedures incorporating such corrections for the comparison of time standards located far apart on earth's surface. For the GPS it means that synchronization of the entire system of ground-based and orbiting atomic clocks is performed in the local inertial frame, or ECI coordinate system [6].
GPS can be used to compare times on two earth-fixed clocks when a single satellite is in view from both locations. This is the ``common-view'' method of comparison of Primary standards, whose locations on earth's surface are usually known very accurately in advance from ground-based surveys. Signals from a single GPS satellite in common view of receivers at the two locations provide enough information to determine the time difference between the two local clocks. The Sagnac effect is very important in making such comparisons, as it can amount to hundreds of nanoseconds, depending on the geometry. In 1984 GPS satellites 3, 4, 6, and 8 were used in simultaneous common view between three pairs of earth timing centers, to accomplish closure in performing an around-the-world Sagnac experiment. The centers were the National Bureau of Standards (NBS) in Boulder, CO, Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig, West Germany, and Tokyo Astronomical Observatory (TAO). The size of the Sagnac correction varied from 240 to 350 ns. Enough data were collected to perform 90 independent circumnavigations. The actual mean value of the residual obtained after adding the three pairs of time differences was 5 ns, which was less than 2 percent of the magnitude of the calculated total Sagnac effect [4].
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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 |