Computation of Time Error in GPS Signals: Using Schwartzchild Time Dilation Equation

Abstract

Global Positioning System (GPS) is a satellite based navigation system that can be used to locate positions anywhere on earth and proper timing determination. The Global Positioning System (GPS) uses accurate, stable atomic clocks in satellites and on the ground to provide world-wide position and time determination. As a result of their motion and being in the earth gravitational field, these clocks are subject to one of the relativity effects known as time dilation and without carefully accounting for this effect, the system would not work. This research work discussed the concept and effect of special and general relativity on GPS navigation as well as using Karl Schwarzschild time dilation equation to estimate time error. Signals exchanged by atomic clocks at different altitudes are subject to general relativistic effects described using the Schwarzschild metric and neglecting these effects would make the GPS error-full. Applying the Kerl Schwarzchild time dilation equation in the obtained GPS data set, as measured at the orbit of the satellite, we found a difference between the proper time and the coordinate time whish gave us the dilated time within the range of 0.7-0.9 seconds. The dilated time for signal to travel from the orbit to the receiving station is what we regarded as Time Error. [PUBLICATION ABSTRACT]

Headnote

Abstract: Global Positioning System (GPS) is a satellite based navigation system that can be used to locate positions anywhere on earth and proper timing determination. The Global Positioning System (GPS) uses accurate, stable atomic clocks in satellites and on the ground to provide world-wide position and time determination. As a result of their motion and being in the earth gravitational field, these clocks are subject to one of the relativity effects known as time dilation and without carefully accounting for this effect, the system would not work. This research work discussed the concept and effect of special and general relativity on GPS navigation as well as using Karl Schwarzschild time dilation equation to estimate time error. Signals exchanged by atomic clocks at different altitudes are subject to general relativistic effects described using the Schwarzschild metric and neglecting these effects would make the GPS error-full. Applying the Kerl Schwarzchild time dilation equation in the obtained GPS data set, as measured at the orbit of the satellite, we found a difference between the proper time and the coordinate time whish gave us the dilated time within the range of 0.7-0.9 seconds. The dilated time for signal to travel from the orbit to the receiving station is what we regarded as Time Error.

Keywords: Global Positioning System (GPS), Kerl Schwarzchild Time Dilation Equation, Relativity Effect, Atomic Clocks, Time Dilation.

(ProQuest: ... denotes formulae omitted.)

1.0 Introduction

While best known as a system that makes it possible to determine one's location on the Earth, the Global Positioning System (GPS) is also an important contributor to precise time- and frequency-transfer systems (Ashby, 2002).

The Global Positioning System includes 24 satellites in orbit around the Earth with an orbital period of 12 sidereal hours. The 24 satellites are distributed unevenly in six orbital planes with an inclination of 55° with respect to the equatorial plane, so that from practically any point on the Earth four or more satellites are visible. Each satellite carries an operating atomic clock (along with several backup clocks) and emits timed signals that include a code telling its location. By analyzing signals from at least four of these satellites, a receiver on the surface of Earth with a built-in microprocessor can display the location of the receiver (latitude, longitude, and altitude). The accuracy of this timing estimate will be influenced both by the number of satellites being tracked by the GPS receiver as well as their geometry in the sky/atmosphere. Although the GPS has thousands of civilian users' worldwide and it's operated by the U. S. Military in Department of Defence (DOD). The GPS uses stable highly accurate atomic clocks in the satellites and on the Earth in order to provide position and time determination. Signals exchanged by atomic clocks at different altitudes are subject to general relativistic effects described using the Schwarzschild metric. Neglecting these effects would make the GPS useless.

3.0 Relativity Error: One of the Sources of GPS Error

Timing accuracy is limited by uncertainties in the GPS measurements; GPS is not a perfect system (Basic Land Navigation Journal, 2003). There are several different types of GPS errors that could result in GPS inaccuracy, but we are discussing just only relativity error among others.

3.1 Relativity Error

The GPS is one of the first operational systems, outside of particle accelerators, that has important effects from relativity. The reasons for this are threefold. The GPS satellites have a large velocity, there is a non-negligible gravitational potential difference between that of the satellites and that of the users (usually at or near the Earth's surface) and there are significant Earth rotation effects. These effects when coupled with the fact that GPS satellites carry precision atomic frequency standards, that pseudorange measurements are made to the centimetre level, relativistic effects can, indeed, be significant and must be taken into account and for a fixed user at sea level on Earth's surface, there are basically three primary consequences of relativity effects.

4.0 Relativistic Effects on GPS

This section gives brief information on the three effects of relativity. In the normal life we are quite unaware of the omnipresence of the theory of relativity. However it has an influence on many processes, among them is the proper functioning of the GPS system. As we already learnt, the time is a relevant factor in GPS navigation and must be accurate to 20-30 nanoseconds to ensure the necessary accuracy (NOAA, 2011). In order for time accuracy to be certain, a correction by General Relativity was provided. According to the theory of relativity, due to their constant movement and height relative to the Earth-cantered, non-rotating approximately inertial reference frame, the clocks on the satellites are affected by their speed. Special relativity predicts that the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground clocks, thereby resulting to a delay of about 7 ps/day. The effect of gravitational frequency shift on the GPS due to general relativity is that a clock closer to a massive object will be slower than a clock farther away. Applied to the GPS, the receivers are much closer to Earth than the satellites, causing the GPS satellite clocks to be faster. GPS satellites are affected by relativity in seven different ways but only three are common ones: Gravitational redshift (blueshift) effect, Time dilation effect (second-order Doppler effect) and Sagnac effect.

4.1 Time Dilation & Gravitational Effect

Albert Einstein's Special and General Theories of relativity predicted that a clock in orbit around the earth would appear to run faster than a clock on its surface, this is due to their greater speed and the weaker gravity around them, the clocks in the GPS satellites do appear to run faster than the clocks in GPS receivers. There are actually two parts to the effects. These are: Gravitational redshift (blueshift) effect & Time dilation effect (second-order Doppler effect). The first effect considered stems from a Lorentz transformation for inertial reference frames. This effect from special relativity is derived from the postulate that the speed of light is constant in all inertial reference frames [Lorentz et. al., 1923], The correction accounts for Time Dilation.

4.1.1 Relative Velocity Time Dilation

When two observers are in relative uniform motion and uninfluenced by any gravitational mass, the point of view of each will be that the other's (moving) clock is ticking at a slower rate than the local clock. The faster the relative velocity, the greater the magnitude of time dilation. This case is sometimes called special relativistic time dilation.

4.1.2 Gravitational Time Dilation

Velocity and Gravity each slow down time as they increase. Velocity increase slows down time, whereas at low gravity field the time speeds up. The velocity time dilation (explained above) is making a bigger difference, and slowing down time. The (time-speeding up) effects of low-gravity would not cancel out these (time-slowing down) effects of velocity unless the GPS satellite orbited much farther from earth. A very few scientist has been able to work on relativistic effect of space machines like Satellite, GPS, Shuttle etc. The Schwarzschild metric constitute an important achievement in relativistic theory, since it is one of the few exact solutions of the field equation. It has provided a foundation for much of the present day understanding of gravitation, as well as the basis for many of the experimental verification for GR theory that have been devised. In this section we shall briefly review some experimental work which is particularly relevant for the relativistic effects on moving frame (mostly in clocks on GPS satellites, flown atomic clocks, other satellites).

5.0 Karl Schwarzchild Equation

According to Einstein's equivalence principle, the influence of gravitation on phenomena in a local reference frame that is at rest in the field is equivalent to the influence of the accelerated motion of a local reference frame in which phenomena are described in the absence of gravitation. The first known non-trivial solution to Einstein equations, the solution is due to the astronomer Karl Schwarzschild and in his honour the solution is referred to as the Schwarzschild solution for empty space. In consequence of the equivalence principle, the metric in the above-mentioned frame will have the form:

t/s2 = c2 t/f'2 - dx'2 - r2 (t/#2 + sin2 θ?φ^ ) (5.i.o)

where the local spatial coordinate x' in the radial direction and the local time coordinate / ' arc related to r, / by the formulae:

... (5.1.1)

... (5.1.2)

where ν is the velocity that a radially accelerated frame in the absence of gravitational field must have in order to be equivalent to the stationary frame in question for a given value of the coordinate r. After the appropriate substitutions, we get a metric of the form:

... (5.1.3)

Let us assume that ν is given by the Newtonian formula for free fall with the initial velocity zero "at infinity", i.e., it is equal to the escape velocity in Newton's gravitation theory:

... (5.1.4)

Where G is the gravitational constant and Mis the mass of the source of the gravitational field.

Finally, by substituting (5.1.4) to (5.1.3), we get the form of the Schwarzschild metric:

... (5.1.5)

f(r )has been calculated and it's found out to be ... (5.1.6)

Where ... (5.1.7)

Substituting equations (5.1.6) and (5.1.7) into (5.1.5) we obtain:

... (5.1.8)

Great importance is attached to the spherical symmetric metric because most of the large gravitating masses of the physical world are spherically symmetric.

The requirement of spherical symmetry can be expressed by writing the metric to be determined in the form:

ds = gQQ(r)dt^ + g\\ (r)dr^ - + sin^ θdφ^ ) (5.1.9)

Here the metric coefficients §22 and §33 have been chosen to be equal to - 1? Γ Sill θ respectively. Consider perturbation as a background of the flat space in a gravitational field and also assume that the gravitational field does not alter the symmetrical disposition of θ and Φ co-ordinates. The retention of the factor Γ in ?22 and B33 imposes a condition upon the ratio of the co-ordinate distance from origin of an object subtending an angle to the width of the object. Putting §oo(f) in a form to show its relationship with the metric in the limiting condition of weak gravitational field, by introducing.

êoo = (1+ f ) c 2 (5.2.0)

Then we take Boo in reciprocal relation with B\ 1 , implying that:

Il I f) (5 2 1)

and

... (5.2.2)

Thus we verify that;

... (5.2.3)

This is a solution of the Einstein field equation.

Recalling the Schwarchild's metric we have:

... (5.2.4)

It could be written as,

... (5.2.5)

It may be noted that in flat space time m = 0, with ?·>&·> Φ as constant

i.e ?φ = 0 = ($ and = 0.

Thus equation (5.2.5) becomes,

... (5.2.6)

Then equation (5.2.6) turns out to be:

... (5.2.7)

... (5.2.8)

It follows that the velocity of light signals in the radial direction is given by:

... (5.2.9)

And the co-ordinate time is given by:

... (5.3.0)

The time of the GPS signal for a single trip of signal is given as:

... (5.3.1)

But,

...

It therefore follows that,

...

Where ...

(5.2.6)

Then equation (5.3.2) becomes:

...

For a static observer at a radius r outside a gravitating body the proper time dt will have a time dilatation and so since the metric is inhomogeneous and static the coordinate clocks showing the time t must flow at equal pace compared to standard clocks at infinity r^ oo. As we descend deeper and deeper into the gravitational field, the standard clocks showing proper time tick slower and slower compared to the coordinate time clocks.

The Karl Schwarzchild derived above is what we use for the computation of the dilated time which we regard as our Time Error.

Where:

dt = Proper time, dt = Coordinate time, c = Speed of Light,

G = Newton's gravitational constant, M = Mass of the earth,

r= Distance between the receiving station and GPS satellite.

6.0 Materials and Method

We used a real time GPS data for the month of February 2012 of the monitoring GPS station of National Space Research and Development Agency; Centre for Geodesy and Geodynamics, Toro, Bauchi State, Nigeria. We also use Karl Schwarzchild time dilation equation stated above to estimate the time errors. MATLAB software was used as a technical computing tool for coordinate time (dt) and distance(r) between coordinate point and the receiving station computation. We focus mostly on estimating the geometric range delay time which we regarded as the Time Error. We then use these measurements along with information in the navigation messages to solve for four unknowns: Wavelengths, Distance, Coordinate time, and Time Difference. The signals from the satellites can be thought of effectively as continuous timing signals arriving at the receiver from the satellite clocks. The receiver has its own local clock for comparison, so we use the arrival times on the local clock of a specific time tag in the received timing signals. The difference between the reception time (according to the local clock) and the transmission time at one satellite (according to GPS time), multiplied by the speed of light, which we regard as the pseudorange i.e. pseudorange is c (dt - dt).The stationary GPS receiver used for this research work acquire and record data at regular, specified intervals (every 5 seconds, as configured by the receiver user). Practically, the pseudorange and the exact time at which the signal was sent from the satellite coordinate were not in the GPS navigation message tag.

We have the following parameters in the received GPS data: Frequency (LI & L2), Time of data reception and GPS receiver coordinate. The total counts of wavelengths from SV to receivers on approximation is 100,000,000 (French, 1996).

The parameters we need to compute are: GPS Signal Arrival Time conversion into Total Seconds, Distance and the Time error.

For the purpose of this research work, we make use of one of the GPS standard formats; RINEX format.

The observation file contains in its header information that describes the files contents such as the station name, antenna information, the approximate station coordinates, number and types of observation, observation interval in seconds, time of first observation record, and other information. The RINEX format is in standard ASCII format (i.e., readable text) and the data (Observational Data) file consists of two sections: a header and data. The header section is a 22-line section and the data section of the observational data file in RINEX format starts at observational line 23, which contains the data and time of the first record (epoch). In the header, the navigation message contains information such as the date of file creation, the agency name, and other relevant information. The fig. 1 below is the Screenshot of the observational data in ASCII format.

It should be noted that we make use only LI frequency and only considered G01 satellite because of its constancy amongst the satellites that are in range.

MATLAB script was written and used to compute the parameters needed stated below:

The parameters needed are: Speed of light (c = 3.0 χ 108m/sec ), an approximate total wavelengths count (Awc) from satellite to receivers (100,000,000), Earth mass (ME =5.9T4xl024kg), Gravitational constant (G= 6.67 χ 10~nN.m2/kg2). The script perform the following task: Convert the GPS observation time in Hours: Mins: Sec to Total Seconds in a regular 5 seconds interval, compute the corresponding GPS signal wavelength( λ )of frequency (LI), distance and the time difference (Time Error).

The difference between the proper time and coordinate time is the time dilated which we refer to as Time Error caused by gravity (Curve concept).

7.0 Result and Discussion

Having known that due to the fact that satellites are constantly moving relative to observers on the Earth, effects predicted by Special and General Theories of Relativity must be taken into account to achieve the desired GPS navigation necessary accuracy of 20 - 30 nanoseconds which must be ensured (Richard, 2009). On the contrary, the estimated time error values obtained in nanoseconds in the computation are shown on the table 2 below:

From the above estimated time error values, we can see that it's far higher than the GPS 20-30 (ns) appropriate accuracy range. Through the time error values obtained during computation, we can see that time is actually dilated which is in perfect agreement with Karl Schwarzchild idea.

Conclusion

Time Error in stationary DGPS signals has been computed, from gathered DGPS data-set at real time in RINEX format using schwartzchild time dilation equation. We looked into GPS navigation code, system architecture and segments and we also discussed different sources of GPS inaccuracy /error but now dwelt much on Gravitational Time dilation Effect as one of the general relativistic error sources. We used the Karl Schwarzchild Time Dilation equation to estimate and compute the dilated time in GPS signals and we found a difference between elapsed coordinate time and the elapsed proper time as measured at the orbit of the satellite which we regarded as Time Error.

Also, we discussed the fact that orbiting clocks ticking rate depends gravitational field that surrounds it and that the greater the acceleration of the satellite the slower the clock runs. We focus on estimating the geometric range delay as a conceptual basis, founded on general relativity using Karl Schwarzschild time dilation equation and that signals exchanged by atomic clocks at different altitudes are subject to general relativistic effects described using the Schwarzschild metric. General relativity (GR) explains this effect quantitatively and that the presence of matter distorts space-time, it becomes curved. The way it is curved is determined by the metric of space-time, where metric is the way how distances - the elapsed proper time between events - in space-time are measured. The field equations of Einstein describe the relationship between the presence of matter (and other sources of curvature), the metric and curvature of space-time.

It must be noted that GPS navigation time accuracy is within 20 - 30 nanoseconds. On computation, the estimated time error values did not fall within the GPS navigation accuracy time range. We considered the time errors obtained are the dilated time and vice-versa which is in agreement with the theoretical concept.

It is interesting to say that the time dilated values obtained were due to the fact that the signals pass through the lines of space (curves) and therefore takes a longer time, thereby giving a time delay as said by Albert Einstein.

References

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