The difference
that appears in the first term of Eq. (24
) arises because in the underlying earth-centered locally
inertial (ECI) coordinate system in which Eq. (24
) is expressed, the unit of time is determined by moving clocks
in a spatially-dependent gravitational field.
It is obvious that Eq. (24) 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. (24) is valid in a coordinate patch large enough to cover the earth
and the GPS satellite constellation. Eq. (24
) 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. (24
) represents a fairly simple and reasonable choice of gauge.)
Therefore, it is natural to propose that the coordinate time
variable
t
of Eqs. (24
) and (22
) 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. (24) for
dt
as follows. First factor out
from all terms on the right-hand side:
I simplify by writing the velocity in the ECI coordinate system as
Only terms of order
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
Finally, solving for the increment of coordinate time and integrating along the path of the atomic clock,
The relativistic effect on the clock, given in Eq. (27), is thus corrected by Eq. (28
).
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. (28). 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. (28).
![]() |
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 |