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Special and General Theories of Relativity

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## Special and General Theories of Relativity

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**Practical Relativistic Timing Effectsin GPS and**GalileoRobert A. NelsonSatellite Engineering Research CorporationBethesda, MD301-657-9641CGSIC Timing Subcommittee Meeting Thursday, March 20, 2003**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.**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**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**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**Around the world atomic clock experiment(J.C. Hafele and**R.E. Keating (1971)**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**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**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**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)**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**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**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**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**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**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**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.**Measurement of pseudorange**(Coordinate time) (“Hardware” proper time)**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**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.