Special Relativity and General Relativity
Einstein’s special relativity is based on two postulates. The first one is called the principle of relativity, i.e., "No inertial system is preferred. The equations expressing the laws of physics have the same form in all inertial systems." The second one is called the principle of the constancy of the speed of light, i.e., "The speed of light is a universal constant independent of the state of motion of the source. Any light ray moves in the inertial system of coordinates with constant velocity, c, whether the ray is emitted by a stationary or by a moving source." Of course, the speed of light refers to velocity in a vacuum (Ashby and Spilker 1996).
Consider two inertial coordinate systems S’ and S in Fig. 5.6, where the x’-axis and x-axis coincide. Two origins are placed at point A and B, respectively. Origin A of system S’ moves with a constant velocity v along the x-axis toward B. The distance between A and B viewed in system S is Ax. The mirror surface is parallel to the x-axis and is faced to the x-axis. The perpendicular distance of the mirror to the x-axis is AL. Suppose a light flash is emitted from A and the reflected light is receipted at B by the moving system S’. Then the transmitting time of the light measured in system S is
According to our assumption, one gets
Substituting this into Eq. 5.122,
can be obtained by
Because of Einstein’s postulates, the speed of light, c, is the same in two systems. Therefore,
is the light flash transmitting time viewed in the moving system S’, i.e., 
and
Fig. 5.6.
Light transmission viewed in two inertial frames
This indicates that the time interval viewed in the system at rest is shorter than the time interval viewed in the system, which is moving with velocity v.
Again, because of the constant c in the two systems, one may denote
and
where
are the lengths of the light transmitting paths viewed in the two systems. Multiplying c by Eq. 5.124 gets
This indicates that the length viewed in the moving system is lengthened.
Consider the relation of
where c is constant in both systems,
is wavelength and f is the related frequency. Because the wavelength
viewed in two systems is different, denoted by
the relationship of the frequencies f and f’, which are viewed in two systems, can be obtained by dividing c into Eq. 5.125:
This indicates that the frequency f’ viewed in the moving system is reduced to fwhen it is viewed by a resting system. Using mathematical expansions
for Eqs. 5.124, 5.125 and 5.126, we have
This is the formula of the special relativity effects caused by a constant motion of a moving inertial coordinate system viewed from a resting inertial coordinate system.
Einstein’s general relativity incorporates gravitation by virtue of the principle of equivalence. The mathematics of the general relativity is extremely complex. However, for treatment of the relativistic effects on GPS, only a simplified and small fraction of the theory is required. Note that the right-hand side of Eq. 5.129 is indeed the point-mass (or unit mass) kinetic energy
scaled by the speed of light
That is, the special relativity effects may be interpreted as the effects caused by kinetic energy due to motion. The analogous effects may also be caused by potential energy 
‘ due to the presence of the gravitation field U. Then
represents the relativistic relations in the case of the presence of a gravitational field U. Thus the total relativistic effects may be formulated as
The presence of a gravitational field indicates an acceleration of the frame S’ with respect to the system S in rest.
The special relativity effects of rotation may be similarly discussed. Details can be found, e.g., in Ashby and Spilker (1996).
Relativistic Effects on GPS
The inertial coordinate system at rest, its origin located at the centre of the Earth, is taken as reference to view all GPS related activities. Because of the large motion velocities and near circular orbits of the GPS satellite, the non-negligible gravitational potential difference between the satellite and the users, as well as the rotation of the Earth, the relativistic effects have to be taken into account. For convenience, we may imagine that the whole GPS process is viewed in an inertial reference at a point where the gravitational potential is the same as that of the geoid of the Earth. Taking the Earth’s rotational effects into account, the view point is equivalent to the point of the GPS user on the geoid of the rotating Earth.
Frequency Effects
The fundamental frequency
of the GPS system is selected as 10.23 MHz. All clocks on the GPS satellites and GPS receivers operate based on this frequency. If all the GPS satellites are working simply on the frequency
then we will view a frequency f at our reference point, and f is not the same as
due to relativistic effects. In order to be able to view the fundamental frequency
the desired working frequency f ‘ of the GPS satellites can be computed using Eq. 5.131 by
where v is the velocity of the satellite and
is the difference of the Earth’s gravitational potential between the satellite and the geoid. The difference between the setting frequency f’ of satellite clock and the fundamental frequency
is called the offset in the satellite clock frequency. Such an offset of the relativistic effects has been implemented in the satellite clock settings, and therefore users do not need to consider this effect. The offset can be computed by using the mean velocity of the satellite and
where
is the gravitational constant of the Earth,
is the Earth radius (ca. 6 370 km), and H is the height of the satellite above the Earth (ca. 20200 km). The offset is approximately 0.00457 Hz; in other words, the satellite clock frequency is set to
For the receiver fixed on the earth’s surface, the frequency of the clock in the receiver is also affected by the relativistic effects. The effects can be represented analogously by Eq. 5.132, where
is the velocity of the receiver due to the rotation of the Earth. Such effects are corrected by the software of the receiver.
Path Range Effects
The general relativity effects of the signal transmitting from the GPS satellite to the receiver can be represented by the Holdridge (1967) model:
where
are the geocentric distances of the satellite j and station i, respectively,
is the distance between the satellite and the observing station,
has the units of meters and a maximum value of about 2 cm. It is notable that by computing the distance
the effect of the rotation of the Earth during the signal transmission has to be taken into account (if it is done in the Earth’s fixed system).
Earth’s Rotational Effects
All corrections related to the rotation of the Earth are called Sagnac corrections. The geocentric vector of the GPS satellite is denoted by
the geocentric vector of the receiver by
and the velocity vector of the receiver by
These are the vectors during GPS signal emission. Suppose the transmitting time between the signal emission from satellite and signal reception of receiver is
During the time of GPS signal transmission, the receiver has moved to position
Observing from the non-rotating frame, the distance of the signal transmission can be represented by
Therefore the transmitting path correction due to the rotation of the Earth can be presented as
This can be simplified as (Ashby and Spiler 1996)
The correction can reach up to 30 meters and must be taken into account.
If the signal transmitting time At has been solved through iteration of Eq. 5.134, then the Sagnac correction will automatically be taken into account.
This term of correction is also valid for the kinematic GPS receivers that are not fixed on the Earth’s surface. The velocity vector in Eq. 5.136 is
where the first term on the right-hand side is the velocity vector of the receiver due to the Earth’s rotation, and the second term
is the kinematic velocity vector of the receiver related to the Earth’s surface. A kinematic motion of 100 km h-1 related to the Earth’s surface can cause additional Sagnac effects up to 2 meters.
The Sagnac correction also has to be taken into account for low-Earth orbit (LEO) satellites (e.g., TOPEX, CHAMP and GRACE), which are equipped with GPS receivers onboard for satellite-satellite tracking (SST).
Relativistic Effects due to the Orbit Eccentricity
The theoretical formula of the clock correction of the satellite can be written as (Ashby and Spilker 1996)
where a is the semimaj or axis of the satellite orbit, e is the eccentricity of the orbit, E is the eccentric anomaly of the orbit,
is the gravitational constant of the Earth, and
is the clock correction due to the eccentricity of the orbit. The second term on the right-hand side is a constant that cannot be separated from the clock offset. This total correction has already been taken into account in the GPS orbits determination and is broadcasted in the navigation message by the parameters of the clock error polynomial. Therefore, this term of correction only needs to be considered in the satellite orbits determination.
Using the relation of
(cf. Kaula 1966), the Eq. 5.138 can be presented by the position (x,y, z) and velocity
of the satellite.
General Relativity Acceleration of the Satellite
The IERS standard correction for the acceleration of the Earth satellite is (McCarthy 1996)
where c is the speed of light, p is the gravitational constant of the Earth, r, v, and a are the geocentric satellite position, velocity and acceleration vectors, respectively.
