Clock Synchronization and Navigation in the Vicinity of the Earth

Clock synchronization is the backbone of applications such as high-accuracy satellite navigation, geolocation, space-based interferometry, and cryptographic communication systems. The high accuracy of synchronization needed over satellite-to-ground and satellite-to-satellite distances requires the use of general relativistic concepts. The role of geometrical optics and antenna phase center approximations are discussed in high accuracy work. The clock synchronization problem is explored from a general relativistic point of view, with emphasis on the local measurement process and the use of the tetrad formalism as the correct model of relativistic measurements. The treatment makes use of J. L. Synge's world function of space-time as a basic coordinate independent geometric concept. A metric is used for space-time in the vicinity of the Earth, where coordinate time is proper time on the geoid. The problem of satellite clock syntonization is analyzed by numerically integrating the geodesic equations of motion for low-Earth orbit (LEO), geosynchronous orbit (GEO), and highly elliptical orbit (HEO) satellites. Proper time minus coordinate time is computed for satellites in these orbital regimes. The frequency shift as a function of time is computed for a signal observed on the Earth's geoid from a LEO, GEO, and HEO satellite. Finally, the problem of geolocation in curved space-time is briefly explored using the world function formalism.

[1]  J. C. Good,et al.  Aperture synthesis for microwave radiometers in space , 1983 .

[2]  RIEMANN NORMAL COORDINATES, FERMI REFERENCE SYSTEM AND THE GEODESIC DEVIATION EQUATION , 1999, gr-qc/0010096.

[3]  H. Steyskal,et al.  Pattern synthesis for TechSat21-a distributed spacebased radar system , 2001, 2001 IEEE Aerospace Conference Proceedings (Cat. No.01TH8542).

[4]  Desmond George King-Hele,et al.  The gravity field of the Earth , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[5]  Clifford M. Will,et al.  Theory and Experiment in Gravitational Physics , 1982 .

[6]  A. M. Volkov,et al.  The Propagation of Electromagnetic Waves in a Riemannian Space , 1971 .

[7]  Colin P. Williams,et al.  Quantum clock synchronization based on shared prior entanglement , 2000, Physical review letters.

[8]  B. Mashhoon.  Wave propagation in a gravitational field , 1987 .

[9]  Hong,et al.  Measurement of subpicosecond time intervals between two photons by interference. , 1987, Physical review letters.

[10]  Navigation in curved space-time , 2001, gr-qc/0101077.

[11]  Howard E. Brandt,et al.  Quantum computation and information : AMS Special Session Quantum Computation and Information, January 19-21, 2000, Washington, D.C. , 2002 .

[12]  B. J. Ramsey,et al.  Timekeeping and Time Dissemination in a Distributed Space-Based Clock Ensemble , 2002 .

[13]  John Preskill.  Quantum clock synchronization and quantum error correction , 2000 .

[15]  B Guinot.  Application of general relativity to metrology , 1997 .

[16]  Elliott D. Kaplan.  Understanding GPS : principles and applications , 1996 .

[17]  Robert A. Laskin,et al.  Separated-spacecraft interferometer concept for the New Millennium Program , 1996, Optics & Photonics.

[18]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[19]  Raymond J. Sedwick,et al.  Optimum aperture placement for a space-based radar system using separated spacecraft interferometry , 1999 .

[20]  D. Kirchner,et al.  Two-way time transfer via communication satellites , 1991, Proc. IEEE.

[21]  Dirk Bouwmeester,et al.  The physics of quantum information: quantum cryptography, quantum teleportation, quantum computation , 2010, Physics and astronomy online library.

[22]  K. C. Ho,et al.  Geolocation of a known altitude object from TDOA and FDOA measurements , 1997, IEEE Transactions on Aerospace and Electronic Systems.

[23]  Chuang.  Quantum algorithm for distributed clock synchronization , 2000, Physical review letters.

[24]  W. Szymanowski,et al.  BULLETIN DE L'ACADEMIE POLONAISE DES SCIENCES , 1953 .

[25]  Arthur Eddington.  The mathematical theory of relativity , 1923 .

[27]  David M. Le Vine,et al.  The sensitivity of synthetic aperture radiometers for remote sensing applications from space , 1990 .

[28]  C. Will.  The Confrontation between general relativity and experiment: A 1998 update , 1998, gr-qc/9811036.

[29]  H. S. Ruse.  Taylor's theorem in the tensor calculus , 1931 .

[30]  J. Synge.  A Characteristic Function in Riemannian Space and its Application to the Solution of Geodesic Triangles , 1931 .

[31]  Relativity of GPS measurement , 2003, gr-qc/0306076.

[32]  R. John.  Zur Berechnung des geodätischen Abstands und assoziierter Invarianten im relativistischen Gravitationsfeld , 1984 .

[33]  Bruce R. Schupler,et al.  Signal Characteristics of GPS User Antennas , 1994 .

[34]  G. Price.  On the relationship between the transmitting and receiving properties of an antenna , 1986 .

[35]  K. C. Ho,et al.  Solution and performance analysis of geolocation by TDOA , 1993 .

[36]  Christopher S. Jacobs,et al.  Astrometry and geodesy with radio interferometry: experiments, models, results , 1998 .

[37]  T. Bahder,et al.  Pyroelectric effect in semiconductor heterostructures , 1993 .

[38]  T. Fukushima.  The Fermi coordinate system in the post-Newtonian framework , 1988 .

[39]  J. A. Schouten,et al.  Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications , 1954 .

[40]  H. S. Ruse.  An absolute partial differential calculus , 1931 .

[41]  H. Buchdahl,et al.  On the world function of the Schwarzschild field , 1979 .

[42]  B. Hofmann-Wellenhof,et al.  Global Positioning System , 1992 .

[43]  S. I. Chou,et al.  Sensitivity analysis of dual-satellite geolocation , 2000, IEEE Trans. Aerosp. Electron. Syst..

[44]  B. Mashhoon.  Scattering of Electromagnetic Radiation from a Black Hole , 1973 .

[45]  T. Mo,et al.  Electromagnetic Field and Wave Propagation in Gravitation , 1971 .

[46]  Colin P. Williams,et al.  Jozsa et al. Reply , 2001 .

[47]  Victor Brumberg,et al.  Essential Relativistic Celestial Mechanics , 1991 .

[48]  Jonathan P. Dowling,et al.  Lorentz-invariant look at quantum clock-synchronization protocols based on distributed entanglement , 2000, quant-ph/0010097.

[49]  G. Sinclair,et al.  The Transmission and Reception of Elliptically Polarized Waves , 1950, Proceedings of the IRE.