Reference Frames and the Sagnac Effect

2 Reference Frames and the Sagnac Effect

Almost all users of GPS are at fixed locations on the rotating earth, or else are moving very slowly over earth’s surface. This led to an early design decision to broadcast the satellite ephemerides in a model earth-centered, earth-fixed, reference frame (ECEF frame), in which the model earth rotates about a fixed axis with a defined rotation rate, $ωE = 7.2921151467 × 10 −5 rad s− 1$ . This reference frame is designated by the symbol WGS-84 (G873) [19 , 3 ]. For discussions of relativity, the particular choice of ECEF frame is immaterial. Also, the fact the the earth truly rotates about a slightly different axis with a variable rotation rate has little consequence for relativity and I shall not go into this here. I shall simply regard the ECEF frame of GPS as closely related to, or determined by, the International Terrestrial Reference Frame established by the International Bureau of Weights and Measures (BIPM) in Paris.

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 $tj$ 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 ${t′,r′,φ′,z ′}$ rotating at the uniform angular rate $ωE$ is

This results in the following well-known metric (Langevin metric) in the rotating frame:

where the abbreviated expression $(d σ′)2 = (dr′)2 + (r′dφ′)2 + (dz ′)2$ 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 $ds2 = 0$ in Eq. (4 ). Also, it is sufficient for this discussion to keep only terms of first order in the small parameter $′ ωEr ∕c$ . Then

and solving for $(cdt′)$ yields

The quantity $r′2dφ ′∕2$ is just the infinitesimal area $dA ′z$ 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 $∫ ′ dσ ∕c$ 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, $2ω ∕c2 = 1.6227 × 10−21 s m −2 E$ and the equatorial radius is $a1 =$ 6,378,137 m, so the area is $2 14 2 πa 1 = 1.27802 × 10 m$ . 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 $∫ ′ dσ ∕c$ , 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 $d τ$ on the moving clock is given by

For a slowly moving clock, $(dσ′∕cdt′)2 ≪ 1$ , so the last term in brackets in Eq. (9

) can be neglected. Also, keeping only first order terms in the small quantity $ωEr ′∕c$ 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 + $∫ ′ 2 2ωE pathdA z∕c$ 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 ].