## 3 GPS Coordinate Time and TAI

In the GPS, the time variable becomes a coordinate time in the rotating frame of the earth, which is realized by applying appropriate corrections while performing synchronization processes. Synchronization is thus performed in the underlying inertial frame in which self-consistency can be achieved.

With this understanding, I next need to describe the gravitational fields near the earth due to the earth’s mass itself. Assume for the moment that earth’s mass distribution is static, and that there exists a locally inertial, non-rotating, freely falling coordinate system with origin at the earth’s center of mass, and write an approximate solution of Einstein’s field equations in isotropic coordinates:

where are spherical polar coordinates and where is the Newtonian gravitational potential of the earth, given approximately by:
In Eq. (13), is the product of earth’s mass times the Newtonian gravitational constant, is earth’s quadrupole moment coefficient, and is earth’s equatorial radius. The angle is the polar angle measured downward from the axis of rotational symmetry; is the Legendre polynomial of degree 2. In using Eq. (12), it is an adequate approximation to retain only terms of first order in the small quantity . Higher multipole moment contributions to Eq. (13) have a very small effect for relativity in GPS.

One additional expression for the invariant interval is needed: the transformation of Eq. (12) to a rotating, ECEF coordinate system by means of transformations equivalent to Eqs. (3). The transformations for spherical polar coordinates are:

Upon performing the transformations, and retaining only terms of order , the scalar interval becomes:
To the order of the calculation, this result is a simple superposition of the metric, Eq. (12), with the corrections due to rotation expressed in Eq. (4). The metric tensor coefficient in the rotating frame is
where is the effective gravitational potential in the rotating frame, which includes the static gravitational potential of the earth, and a centripetal potential term.

The Earth’s geoid.  In Eqs. (12) and (15), the rate of coordinate time is determined by atomic clocks at rest at infinity. The rate of GPS coordinate time, however, is closely related to International Atomic Time (TAI), which is a time scale computed by the BIPM in Paris on the basis of inputs from hundreds of primary time standards, hydrogen masers, and other clocks from all over the world. In producing this time scale, corrections are applied to reduce the elapsed proper times on the contributing clocks to earth’s geoid, a surface of constant effective gravitational equipotential at mean sea level in the ECEF.

Universal Coordinated Time (UTC) is another time scale, which differs from TAI by a whole number of leap seconds. These leap seconds are inserted every so often into UTC so that UTC continues to correspond to time determined by earth’s rotation. Time standards organizations that contribute to TAI and UTC generally maintain their own time scales. For example, the time scale of the U.S. Naval Observatory, based on an ensemble of Hydrogen masers and Cs clocks, is denoted UTC(USNO). GPS time is steered so that, apart from the leap second differences, it stays within 100 ns UTC(USNO). Usually, this steering is so successful that the difference between GPS time and UTC(USNO) is less than about 40 ns. GPS equipment cannot tolerate leap seconds, as such sudden jumps in time would cause receivers to lose their lock on transmitted signals, and other undesirable transients would occur.

To account for the fact that reference clocks for the GPS are not at infinity, I shall consider the rates of atomic clocks at rest on the earth’s geoid. These clocks move because of the earth’s spin; also, they are at varying distances from the earth’s center of mass since the earth is slightly oblate. In order to proceed one needs a model expression for the shape of this surface, and a value for the effective gravitational potential on this surface in the rotating frame.

For this calculation, I use Eq. (15) in the ECEF. For a clock at rest on earth, Eq. (15) reduces to

with the potential given by Eq. (13). This equation determines the radius of the model geoid as a function of polar angle . The numerical value of can be determined at the equator where and . This gives
There are thus three distinct contributions to this effective potential: a simple contribution due to the earth’s mass; a more complicated contribution from the quadrupole potential, and a centripetal term due to the earth’s rotation. The main contribution to the gravitational potential arises from the mass of the earth; the centripetal potential correction is about 500 times smaller, and the quadrupole correction is about 2000 times smaller. These contributions have been divided by in the above equation since the time increment on an atomic clock at rest on the geoid can be easily expressed thereby. In recent resolutions of the International Astronomical Union [1], a “Terrestrial Time” scale (TT) has been defined by adopting the value . Eq. (18) agrees with this definition to within the accuracy needed for the GPS.

From Eq. (15), for clocks on the geoid,

Clocks at rest on the rotating geoid run slow compared to clocks at rest at infinity by about seven parts in 1010. Note that these effects sum to about 10,000 times larger than the fractional frequency stability of a high-performance Cesium clock. The shape of the geoid in this model can be obtained by setting and solving Eq. (16) for in terms of . The first few terms in a power series in the variable can be expressed as
This treatment of the gravitational field of the oblate earth is limited by the simple model of the gravitational field. Actually, what I have done is estimate the shape of the so-called “reference ellipsoid”, from which the actual geoid is conventionally measured.

Better models can be found in the literature of geophysics [18915]. The next term in the multipole expansion of the earth’s gravity field is about a thousand times smaller than the contribution from ; although the actual shape of the geoid can differ from Eq. (20) by as much as 100 meters, the effects of such terms on timing in the GPS are small. Incorporating up to 20 higher zonal harmonics in the calculation affects the value of only in the sixth significant figure.

Observers at rest on the geoid define the unit of time in terms of the proper rate of atomic clocks. In Eq. (19), is a constant. On the left side of Eq. (19), is the increment of proper time elapsed on a standard clock at rest, in terms of the elapsed coordinate time . Thus, the very useful result has emerged, that ideal clocks at rest on the geoid of the rotating earth all beat at the same rate. This is reasonable since the earth’s surface is a gravitational equipotential surface in the rotating frame. (It is true for the actual geoid whereas I have constructed a model.) Considering clocks at two different latitudes, the one further north will be closer to the earth’s center because of the flattening – it will therefore be more redshifted. However, it is also closer to the axis of rotation, and going more slowly, so it suffers less second-order Doppler shift. The earth’s oblateness gives rise to an important quadrupole correction. This combination of effects cancels exactly on the reference surface.

Since all clocks at rest on the geoid beat at the same rate, it is advantageous to exploit this fact to redefine the rate of coordinate time. In Eq. (12) the rate of coordinate time is defined by standard clocks at rest at infinity. I want instead to define the rate of coordinate time by standard clocks at rest on the surface of the earth. Therefore, I shall define a new coordinate time by means of a constant rate change:

The correction is about seven parts in 1010 (see Eq. (18)).

When this time scale change is made, the metric of Eq. (15) in the earth-fixed rotating frame becomes

where only terms of order have been retained. Whether I use or in the Sagnac cross term makes no difference since the Sagnac term is very small anyway. The same time scale change in the non-rotating ECI metric, Eq. (12), gives
Eqs. (22) and Eq. (23) imply that the proper time elapsed on clocks at rest on the geoid (where ) is identical with the coordinate time . This is the correct way to express the fact that ideal clocks at rest on the geoid provide all of our standard reference clocks.