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
V
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 very a 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 International Bureau of Weights and
Measures (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
V
given by Eq. (13). This equation determines the radius
r
' 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 1/
r
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
. 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
r
' 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 [18,
9,
15]. 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
dt
. 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
t
'' by means of a constant rate change:
The correction is about seven parts in
(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
dt
' or
dt
'' 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
t
''. This is the correct way to express the fact that ideal clocks
at rest on the geoid provide all of our standard reference
clocks.
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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 |