Special and General Theories of Relativity

1. Practical Relativistic Timing Effectsin GPS and GalileoRobert A. NelsonSatellite Engineering Research CorporationBethesda, MD301-657-9641CGSIC Timing Subcommittee Meeting Thursday, March 20, 2003

2. Special and General Theories of Relativity • Special relativity • Created in 1905 • Concerns kinematics, mechanics, and electromagnetism • General relativity • Completed in 1916 • Concerns gravitation • Not a separate theory: includes special relativity • Today the general theory of relativity is not simply a subject of theoretical scientific speculation, but rather it has entered the realm of practical engineering necessity. • Relativistic effects must be considered in the transport of atomic clocks and the propagation of electromagnetic signals.

3. Proper Time vs. Coordinate Time • Proper time • The time provided by an ideal clock in its own rest frame • Different for clocks in different states of motion and in different gravitational potentials • “Hardware” proper time • The time provided by a real clock in its own rest frame corrupted by noise and environmental effects • Coordinate time • The time coordinate in the chosen space-time coordinate system • A global coordinate • Has same value everywhere for a given event

4. Relativistic Effects Three effects contribute to the net relativistic effect on a transported clock • Velocity (time dilation) • Makes transported clock run slow relative to a clock on the geoid • Function of speed only • Gravitational potential (red shift) • Makes transported clock run fast relative to a clock on the geoid • Function of altitude only • Sagnac effect • Makes transported clock run fast or slow relative to a clock on the geoid • Depends on direction and path traveled

5. Time dilation of muon lifetimeB. Rossi and D.B. Hall (1941); D.H. Frisch and J.H. Smith (1963) Muons observed in 1 h at top of Mt. Washington (elev. 1910 m) and at sea level. Number observed at elev. 1910 m is 568. Number observed at sea level is 412. Exponential law of decay with mean proper lifetime = 2.2 s Muons selected with velocity 0.9952 c Time of flight in laboratory frame = 6.4 s Time of flight in muon rest frame = 0.63 s

6. Around the world atomic clock experiment(J.C. Hafele and R.E. Keating (1971)

7. Around the world atomic clock experiment(Flying clock – Reference clock) predicted effectdirection East West Gravitational potential (redshift) + 144 ns + 179 ns Velocity (time dilation)  51 ns  47 ns Sagnac effect  133 ns + 143 ns Total  40  23 ns + 275  21 ns Measured  59  10 ns + 273  7 ns

8. Gravitational redshift of an atomic clockC.O. Alley, et al. (1975) Gravitational redshift 52.8 ns Time dilation 5.7 ns Net effect 47.1 ns

9. TWTT Flight Tests Tests conducted by Timing Solutions Corp., Zeta Associates, and AFRL Flight clock data collected on a C-135E aircraft to demonstrate TWTT in background of an active communications channel 6 flights in November 2002 from WPAFB L-Band Antenna

10. Relativistic Effects • Relativity effects on flight clock computed based on the position record over the flight interval • Gravitational (redshift) effect, velocity (time dilation) effect and Sagnac effect combine to a predicted net change in flight clock phase of 15 ns Relativistic Effects (Reference Clock – Flying Clock)

11. Processed TWTT Data • Averaging instantaneous data results in a sub-nanosecond, continuous record of the clock difference over the flight interval • Collected data agree well with predicted clock differences based on relativity calculations TWTT Data (60 s average) Approach/Landing

12. Sagnac effect (TWSTT)NIST to USNO via Telstar 5 at 97 WL Uplink 24.1 ns Downlink 57.7 ns Total Sagnac correction 81.1 ns

13. GPS • Gravitational redshift (blueshift) • Orbital altitude 20,183 km • Clock runs fast by 45.7 s per day • Time dilation • Satellite velocity 3.874 km/s • Clock runs slow by 7.1 s per day • Net secular effect (satellite clock runs fast) • Clock runs fast by 38.6 s per day • Residual periodic effect • Orbital eccentricity 0.02 • Amplitude of periodic effect 46 ns • Sagnac effect • Maximum value 133 ns for a stationary receiver on the geoid

14. GPS (Summary) • Net secular relativistic effect is 38.6 s per day • Nominal clock rate is 10.23 MHz • Satellite clocks are offset by – 4.464733 parts in 1010 to compensate effect • Resulting (proper) frequency in orbit is 10229999.9954326 Hz • Observed average rate of satellite clock is same as clock on the geoid • Residual periodic effect • Maximum amplitude 46 ns • Correction applied in receiver • Sagnac effect • Maximum value 133 ns • Correction applied in receiver

15. Galileo • Gravitational redshift (blueshift) • Orbital altitude 23,616 km • Clock runs fast by 47.3 s per day • Time dilation • Satellite velocity 3.645 km/s • Clock runs slow by 6.3 s per day • Net secular effect (satellite clock runs fast) • Clock runs fast by 47.3 s per day • Residual periodic effect • Orbital eccentricity 0.02 • Amplitude of periodic effect 49 ns • Sagnac effect • Maximum value 153 ns for a stationary receiver on the geoid

16. Molniya satellite

17. Molniya orbit ground trace Period = 11.967 h Apogee altitude = 39,362 km Perigee altitude = 1006 km Eccentricity = 0.722 Inclination = 63.4 Argument of perigee = 250

18. Eccentricity correction for Molniya orbit

19. GPS ICD-200 Must also consider effect of moving receiver on signal propagation time. Paragraph on “Geometric Range” in GPS ICD-200 revised in 1998. In the past, the ICD assumed the receiver was at rest on the rotating Earth. Paragraph is now completely general.

20. Measurement of pseudorange (Coordinate time) (“Hardware” proper time)

21. Additional relativistic effects • Contribution to gravitational redshift due to Earth oblateness • Amplitude of periodic effect for GPS is 24 ps • Tidal potentials of the Moon and Sun • Amplitude of periodic effect is on the order of 1 ps • Effect of gravitational potential on time of signal propagation • On the order of 3 ps • Intersatellite links (GPS III and beyond) • Eccentricity correction on the order of tens of nanoseconds

22. Conclusion • Relativity has become an important practical engineering consideration for modern precise timekeeping systems. • Far from being simply a textbook problem or merely of theoretical scientific interest, the analysis of relativistic effects is an essential practical engineering consideration. • These relativistic effects are well understood and have been applied successfully in the GPS. • Similar corrections will need to need to be applied in Galileo. • Common geodetic and time scale references will be needed for possible interoperability between GPS and Galileo. • Terrestrial reference system (WGS-84 and ITRF-2000) • Time (realization of common coordinate time by satellite clocks) • Of these two considerations, the measurement of time will be the most important.