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Relativity in the real world : R elativistic effects in GPS

Relativity in the real world : R elativistic effects in GPS. 20183284 Kiyoung Sun. How Global Positioning System (GPS) works?. GPS is a space-based radio-navigation system broadcasting synchronized timing signals to provide estimates of position, velocity, and time.

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Relativity in the real world : R elativistic effects in GPS

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  1. Relativityintherealworld: Relativisticeffectsin GPS 20183284 Kiyoung Sun

  2. How Global Positioning System (GPS) works? • GPS is a space-based radio-navigation system broadcasting synchronized timing signals to provide estimates of position, velocity, and time. • The GPS user measures, • The received times in receiver clock • The arrival times from satellite clock time-tagged information • “Simultaneity is a crucial concept in the GPS.” Source: Tim Gunther, National Geographic

  3. Clock corrections in GPS UTC ΩE Earth-Centered, Earth-Fixed (ECEF) Frame GPS Time Earth-Centered Inertial (ECI) Frame

  4. Clock corrections in GPS • Clocks in the GPS satellites are synchronized with GPS Time, in the Earth-Centered Inertial (ECI) frame. • Correction for each satellite clock to GPS Time • Receiver clock is based on Earth-Centered, Earth-Fixed (ECEF) rotating reference frame. • - Correction for GPS Time to UTC Time (clock on Earth) Source: Jan Van Sickle, The Pennsylvania State University

  5. Relativistic effects in GPS VSat • Reasons • GPS satellites have a large velocity • User-satellite gravitational potential difference • Earth rotation effect Gravitational potential ΩE

  6. Relativistic Effects in GPS • 1. Sagnac effect • Due to rotation of the Earth • 2. Einstein Gravitational blue shift • Due to gravitational potential difference • Equivalence principle of general relativity • 3. Doppler red shift of second order • Time dilation due to motion of the satellite • Special relativity

  7. Relativistic Effects in GPS1. Sagnac Effects • Sagnac delay is caused by the Earth’s rotation during the time of transit of the satellite signal to the ground. • The transformation from an Minkowski space to rotating frame with angular rate, , (1) (2) • Light travels along a null worldline, , (3) Where (4) • At Earth’s equator,

  8. Relativistic Effects in GPS2. Einstein Gravitational Blue Shift • Using a Schwarzschild coordinates, (5) • Let’s consider circular motion (), (6) • Assume the satellite clock and clock on the Earth surface are fixed in the ECI (, • The clock in the satellite runs faster about 45.7 μs/day (= 13.7 km/day)!

  9. Relativistic Effects in GPS3. Doppler Red Shift of Second Order • Using a Schwarzschild coordinates, (5) • Let’s consider circular motion (), (6) • With moving clocks based on ECI frame; , • The net result including blue shift and red shift is about 39 μs/day (=11.7 km/day)!

  10. Relativistic Effects in GPSEffects on Ground-based Clocks • An approximate solution of Einstein’s field equations in isotropic coordinates: (7) (8) • Transformation of the coordinates into a rotating ECEF coordinate system, ECI to ECEF, (9)

  11. Relativistic Effects in GPSEffects on Ground-based Clocks • For a clock at rest on Earth, Equation (9) reduces to, (10) • Clocks on the rotating geoid run slow compared to clocks at rest at infinity (11) • At the equator when and , the numerical value of reference potential gives, Attributed to Earth’s mass Attributed to Earth’s quadrapole moment Second-order Doppler shift due to Earth’s spin 1) 2) 3) • The resulting correction should be about 60 μs/day (=18 km/day)! 10,000 time larger than Cesium clock frequency stability

  12. Relativistic Effects in GPSEffects on Satellite Clocks • From Schwarzschild coordinates, with a spherically symmetric gravitational field, (12) • at a satellite orbit plane, , (13) (14) • At the GPS orbit radius , the clock rate offset is be set as,

  13. Example of Relativistic CorrectionFractional Frequency Shift in Satellites Source: Ashby, N., Living Rev. Relativity

  14. Example of Relativistic CorrectionMeasuring Relativistic Effects on Galileo satellites Source: Delva, P., ESA

  15. Example of Relativistic CorrectionMeasuring Relativistic Effects on Galileo satellites Source: Delva, P., ESA

  16. References Ashby, N. and Spilker, J. J., Introduction to relativistic effects on the Global Positioning System. In: Parkinson, B. W. iSpilker J. J. (eds.) Global Positioning System: Theory and application, Vol. I, Ch. 18. 1996. Ashby, N., Relativity in the Global Positioning System, Living Rev. Relativity, 6, 2003 Delva, P., (2018, December, 4), Galileo satellites prove Einstein's Relativity Theory to highest accuracy yet, ESA Delva, P. et al., Gravitational Redshift Test Using Eccentric Galileo Satellites, Physical Review Letters, 121, 231101, 2018 Pascual-Sanchez, J. F., Introducing Relativity in Global Navigation Satellite Systems, Ann. Phys. (Leipzig) 16, No. 4, 258-273, 2007 Weiss, M. and Ashby, N., GPS Receivers and Relativity, 29th Annual Precise Time and Time Interval (PTTI) Meeting, 1997

  17. Backup slides…

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