4 The Realization of Coordinate Time

We are now able to address the real problem of clock synchronization within the GPS. In the remainder of this paper I shall drop the primes on ′′ t and just use the symbol t, with the understanding that unit of this time is referenced to UTC(USNO) on the rotating geoid, but with synchronization established in an underlying, locally inertial, reference frame. The metric Eq. (23View Equation) will henceforth be written

( ) ( ) 2 2(V-−-Φ0-) 2 2V- 2 2 2 2 2 2 − ds = − 1 + c2 (cdt) + 1 − c2 (dr + r d𝜃 + r sin 𝜃dϕ ). (24 )
The difference (V − Φ0 ) that appears in the first term of Eq. (24View Equation) arises because in the underlying earth-centered locally inertial (ECI) coordinate system in which Eq. (24View Equation) is expressed, the unit of time is determined by moving clocks in a spatially-dependent gravitational field.

It is obvious that Eq. (24View Equation) contains within it the well-known effects of time dilation (the apparent slowing of moving clocks) and frequency shifts due to gravitation. Due to these effects, which have an impact on the net elapsed proper time on an atomic clock, the proper time elapsing on the orbiting GPS clocks cannot be simply used to transfer time from one transmission event to another. Path-dependent effects must be accounted for.

On the other hand, according to General Relativity, the coordinate time variable t of Eq. (24View Equation) is valid in a coordinate patch large enough to cover the earth and the GPS satellite constellation. Eq. (24View Equation) is an approximate solution of the field equations near the earth, which include the gravitational fields due to earth’s mass distribution. In this local coordinate patch, the coordinate time is single-valued. (It is not unique, of course, because there is still gauge freedom, but Eq. (24View Equation) represents a fairly simple and reasonable choice of gauge.) Therefore, it is natural to propose that the coordinate time variable t of Eqs. (24View Equation) and (22View Equation) be used as a basis for synchronization in the neighborhood of the earth.

To see how this works for a slowly moving atomic clock, solve Eq. (24View Equation) for dt as follows. First factor out 2 (cdt) from all terms on the right-hand side:

[ ( ) 2 2 2 2 2 2] 2 2(V--−-Φ0-) 2V- dr--+-r-d𝜃--+-r-sin-𝜃d-ϕ- 2 − ds = − 1 + c2 − 1 − c2 (cdt)2 (cdt) . (25 )
I simplify by writing the velocity in the ECI coordinate system as
2 2 2 2 2 2 v2 = dr--+-r-d𝜃-+--r-sin-𝜃dϕ--. (26 ) dt2
Only terms of order c−2 need be kept, so the potential term modifying the velocity term can be dropped. Then, upon taking a square root, the proper time increment on the moving clock is approximately
[ (V − Φ ) v2 ] d τ = ds∕c = 1 + -----2-0- − --2 dt. (27 ) c 2c
Finally, solving for the increment of coordinate time and integrating along the path of the atomic clock,
∫ ∫ [ (V − Φ0) v2 ] dt = d τ 1 − -----2--- + --2 . (28 ) path path c 2c
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The relativistic effect on the clock, given in Eq. (27View Equation), is thus corrected by Eq. (28View Equation).

Suppose for a moment there were no gravitational fields. Then one could picture an underlying non-rotating reference frame, a local inertial frame, unattached to the spin of the earth, but with its origin at the center of the earth. In this non-rotating frame, a fictitious set of standard clocks is introduced, available anywhere, all of them being synchronized by the Einstein synchronization procedure, and running at agreed upon rates such that synchronization is maintained. These clocks read the coordinate time t. Next, one introduces the rotating earth with a set of standard clocks distributed around upon it, possibly roving around. One applies to each of the standard clocks a set of corrections based on the known positions and motions of the clocks, given by Eq. (28View Equation). This generates a “coordinate clock time” in the earth-fixed, rotating system. This time is such that at each instant the coordinate clock agrees with a fictitious atomic clock at rest in the local inertial frame, whose position coincides with the earth-based standard clock at that instant. Thus, coordinate time is equivalent to time that would be measured by standard clocks at rest in the local inertial frame [7].

When the gravitational field due to the earth is considered, the picture is only a little more complicated. There still exists a coordinate time that can be found by computing a correction for gravitational redshift, given by the first correction term in Eq. (28View Equation).

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