General Relativity and Gravitation

, Volume 39, Issue 10, pp 1583–1624 | Cite as

Gravimagnetism, causality, and aberration of gravity in the gravitational light-ray deflection experiments

  • Sergei M. KopeikinEmail author
  • Edward B. Fomalont
Research Article


Experimental verification of the existence of gravimagnetic fields generated by currents of matter is important for a complete understanding and formulation of gravitational physics. Although the rotational (intrinsic) gravimagnetic field has been extensively studied and is now being measured by the Gravity Probe B, the extrinsic gravimagnetic field generated by the translational current of matter is less well studied. The present paper uses the post-Newtonian parametrized Einstein and light geodesics equations to show that the extrinsic gravimagnetic field generated by the translational current of matter can be measured by observing the relativistic time delay and/or light deflection caused by the moving mass. We prove that the extrinsic gravimagnetic field is generated by the relativistic effect of the aberration of the gravity force caused by the Lorentz transformation of the metric tensor and the Levi–Civita connection. We show that the Lorentz transformation of the gravity field variables is equivalent to the technique of the retarded Lienard–Wiechert gravitational potentials predicting that a light particle is deflected by gravitational field of a moving body from its retarded position so that both general-relativistic phenomena—the aberration and the retardation of gravity—are tightly connected and observing the aberration of gravity proves that gravity has a causal nature. We explain in this framework the 2002 deflection experiment of a quasar by Jupiter where the aberration of gravity from its orbital motion was measured with accuracy 20%. We describe a theory of VLBI experiment to measure the gravitational deflection of radio waves from a quasar by the Sun, as viewed by a moving observer from the geocentric frame, to improve the measurement accuracy of the aberration of gravity to a few percent.


Relativity Gravitation Gravitational deflection of light Speed of light Speed of gravity Very long baseline interferometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ashby N. (2004). General relativity: Frame-dragging confirmed. Nature 31: 918 CrossRefADSGoogle Scholar
  2. 2.
    Ashby N. and Shahid-Saless B. (1990). Geodetic precession or dragging of inertial frames?. Phys. Rev. D 42: 1118 CrossRefADSGoogle Scholar
  3. 3.
    Bekaert X, Boulanger N. and Vázquez-Poritz J.F. (2002). Gravitational Lorentz Violations from M-Theory. J. High Energy Phys. 10: 53 CrossRefADSGoogle Scholar
  4. 4.
    Bertotti B., Iess L. and Tortora P. (2003). A test of general relativity using radio links with the Cassini spacecraft. Nature 425: 374 CrossRefADSGoogle Scholar
  5. 5.
    Bini D., Jantzen R.T. and Mashhoon B. (2001). Gravitomagnetism and relative observer clock effects. Class. Quant. Grav. 18: 653 zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Bini D., Cherubini C., Jantzen R.T. and Mashhoon B. (2003). Gravitomagnetism in the Kerr-Newman-Taub-NUT spacetime. Class. Quant. Grav. 20: 457 zbMATHCrossRefADSGoogle Scholar
  7. 7.
    Bonilla M.Á.G. and Senovilla J.M.M. (1997). Very Simple Proof of the Causal Propagation of Gravity in Vacuum. Phys. Rev. Lett. 78: 783 CrossRefADSGoogle Scholar
  8. 8.
    Braginsky V.B., Caves C.M. and Thorne K.S. (1977). Laboratory experiments to test relativistic gravity. Phys. Rev D 15: 2047 CrossRefADSGoogle Scholar
  9. 9.
    Brumberg V.A. and Kopeikin S.M. (1990). Relativistic time scales in the solar system. Cel. Mech. Dyn. Astron. 48: 23 CrossRefADSGoogle Scholar
  10. 10.
    Camacho A. (2002). Coupling gravitomagnetism-spin and Berry’s phase. Gen. Rel. Grav. 34: 1963 zbMATHCrossRefGoogle Scholar
  11. 11.
    Camacho A. (2002). Quantum Zeno effect and the detection of gravitomagnetism. In: Cianci, R., Collina, R., Francaviglia, M. and Fré, P. (eds) Recent developments in general relativity. 14th SIGRAV Conference on Gen. Rel. and Grav. Phys., pp 347–351. Springer, Milano Google Scholar
  12. 12.
    Carlip S. (2004). Model-dependence of Shapiro time delay and the ‘speed of gravity/speed of light’ controversy. Class. Quant. Grav. 21: 3803 zbMATHCrossRefADSGoogle Scholar
  13. 13.
    Ciufolini I. (1986). Measurement of the Lense-Thirring drag on high-altitude, laser-ranged artificial satellites. Phys. Rev. Lett. 56: 278 CrossRefADSGoogle Scholar
  14. 14.
    Ciufolini, I.: Gravitomagnetism, Lense-Thirring Effect and De Sitter Precession. In: Pascual-Sanchez, J.F., Flori, L., San Miguel, A., Vicente, F. (eds.) Reference Frames and Gravitomagnetism. In: Proc. XXIII Spanish Relativity Meeting. World Scientific, Singapore, pp. 25–34 (2001)Google Scholar
  15. 15.
    Ciufolini I. and Pavlis E.C. (2004). A confirmation of the general relativistic prediction of the Lense-Thirring effect. Nature 431: 958 CrossRefADSGoogle Scholar
  16. 16.
    Ciufolini I. and Pavlis E. (2005). On the measurement of the Lense-Thirring effect using the nodes of the LAGEOS satellites, in reply to “On the reliability of the so-far performed tests for measuring the Lense-Thirring effect with the LAGEOS satellites” by L. Iorio. New Astron. 10: 636 CrossRefADSGoogle Scholar
  17. 17.
    Ciufolini I. and Ricci F. (2002). Time delay due to spin and gravitational lensing. Class. Quant. Grav. 19: 3863 zbMATHCrossRefADSGoogle Scholar
  18. 18.
    Ciufolini I. and Ricci F. (2002). Time delay due to spin inside a rotating shell. Class. Quant. Grav. 19: 3875 zbMATHCrossRefADSGoogle Scholar
  19. 19.
    Ciufolini I. and Wheeler J.A. (1995). Gravitation and Inertia. Princeton University Press, Princeton zbMATHGoogle Scholar
  20. 20.
    Ciufolini I., Kopeikin S., Mashhoon B. and Ricci F. (2003). On the gravitomagnetic time delay. Phys. Lett. A 308: 101 CrossRefADSGoogle Scholar
  21. 21.
    Damour T. and Nordtvedt K. (1993). General relativity as a cosmological attractor of tensor-scalar theories. Phys. Rev. Lett. 70: 2217 CrossRefADSGoogle Scholar
  22. 22.
    Dittus H.L., Lämmerzahl C. and Turyshev S.G. (2007). Lasers, Clocks, and Drag-Free: Exploration of Relativistic Gravity in Space. Astrophysics and Space Science Library, vol. 349. Springer, Berlin Google Scholar
  23. 23.
    Einstein A. (1911). Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes. Ann. Phys. 35: 898 CrossRefGoogle Scholar
  24. 24.
    Ellis G.F.R. and Uzan J.-P. (2005). c is the speed of light, isn’t it?. Am. J. Phys. 73: 240 CrossRefADSGoogle Scholar
  25. 25.
    Fock V. (1957). Three Lectures on Relativity Theory. Rev. Mod. Phys. 29: 325 zbMATHCrossRefADSGoogle Scholar
  26. 26.
    Fock V. (1964). Theory of Space, Time and Gravitation 2nd edition. Pergaman Press, Macmillan Company Google Scholar
  27. 27.
    Fomalont E.B. and Kopeikin S.M. (2003). The Measurement of the Light Deflection from Jupiter: Experimental Results. Astrophys. J. 598: 704 CrossRefADSGoogle Scholar
  28. 28.
    Fomalont E. and Reid M. (2004). Microarcsecond astrometry using the SKA. New Astron. Rev. 48: 1473 CrossRefADSGoogle Scholar
  29. 29.
    Fomalont E.B. and Sramek R.A. (1976). Measurements of the solar gravitational deflection of radio waves in agreement with general relativity. Phys. Rev. Lett. 36: 1475 CrossRefADSGoogle Scholar
  30. 30.
    Frittelli S. (2003). Aberration by gravitational lenses in motion. Mon. Not. R. Astron. Soc. 344: L85 CrossRefADSGoogle Scholar
  31. 31.
    Futamase T. and Schutz B.F. (1983). Newtonian and post-Newtonian approximations are asymptotic to general relativity. Phys. Rev. D 28: 2363 CrossRefADSGoogle Scholar
  32. 32.
    Hannay J.H. (1985). Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian. J. Phys. A Math. Gen. 18: 221 CrossRefADSGoogle Scholar
  33. 33.
    Harada W. and Fukushima T. (2003). Harmonic Decomposition of Time Ephemeris TE405. Astron. J. (USA) 126: 2557 ADSGoogle Scholar
  34. 34.
    Hawking S.W. and Ellis G.F.R. (1975). The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge Google Scholar
  35. 35.
    Hellings, R.W.: Relativistic effects in astronomical timing measurements. Astron. J. (USA) 91, 650; Erratum in: Astron. J. (USA) 92, 1446 (1986)Google Scholar
  36. 36.
    IERS Conventions: Dennis D. McCarthy and Gérard Petit. (IERS Technical Note 32) Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie, 2004. Chap. 11. (2003)Google Scholar
  37. 37.
    Iorio L. (2003). On Some Gravitational Spin-Spin Effect for Astronomical Bodies. Int. J. Mod. Phys. D 12: 35 CrossRefADSGoogle Scholar
  38. 38.
    Iorio L. (2005). On the reliability of the so-far performed tests for measuring the Lense Thirring effect with the LAGEOS satellites. New Astron. 10: 603 CrossRefADSGoogle Scholar
  39. 39.
    Iorio L., Ciufolini I., Pavlis E.C., Schiller S., Dittus H. and Lämmerzahl C. (2004). On the possibility of measuring the Lense Thirring effect with a LAGEOS-LAGEOS II-OPTIS mission. Class. Quant. Grav. 21: 2139 zbMATHCrossRefADSGoogle Scholar
  40. 40.
    Jackson J.D. (1998). Classical Electrodynamics. Wiley, New York Google Scholar
  41. 41.
    Kaplan, G.H.: The IAU Resolutions on Astronomical Reference Systems, Time Scales, and Earth Rotation Models: Explanation and Implementation. USNO Circular 179 ( (2005)Google Scholar
  42. 42.
    Klioner S.A. (2003). Light propagation in the gravitational field of moving bodies by means of Lorentz transformation I. Mass monopoles moving with constant velocities. Astron. Astrophys. 404: 783 zbMATHCrossRefADSGoogle Scholar
  43. 43.
    Kopeikin S.M. (1990). Theory of Relativity in Observational Radio Astronomy. Sov. Astron. 34: 5 ADSGoogle Scholar
  44. 44.
    Kopeikin S.M. (2003). The post-Newtonian treatment of the VLBI experiment on September 8, 2002. Phys. Lett. A. 312: 147 CrossRefADSGoogle Scholar
  45. 45.
    Kopeikin S.M. (2004). The speed of gravity in general relativity and theoretical interpretation of the Jovian deflection experiment. Class. Quant. Grav. 21: 3251 zbMATHCrossRefADSGoogle Scholar
  46. 46.
    Kopeikin S.M. (2005). Comment on ‘Model-dependence of Shapiro time delay and the “speed of gravity/speed of light” controversy’. Class. Quant. Grav. 22: 5181 zbMATHCrossRefADSGoogle Scholar
  47. 47.
    Kopeikin S.M. (2006). Gravitomagnetism and the Speed of Gravity. Int. J. Mod. Phys. D 15: 305 zbMATHCrossRefADSGoogle Scholar
  48. 48.
    Kopeikin S.M. (2001). Testing the Relativistic Effect of the Propagation of Gravity by Very Long Baseline Interferometry. Astrophys. J. Lett. 556: L1 CrossRefADSGoogle Scholar
  49. 49.
    Kopeikin S.M. and Fomalont E.B. (2006). Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment. Found. Phys. 36: 1244 zbMATHCrossRefADSGoogle Scholar
  50. 50.
    Kopeikin S.M. and Fomalont E.B. (2006). On the speed of gravity and relativistic v/c corrections to the Shapiro time delay. Phys. Lett. A. 355: 163 CrossRefADSGoogle Scholar
  51. 51.
    Kopeikin S.M. and Makarov V.V. (2007). Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers. Phys. Rev. D 75: 062002 CrossRefADSGoogle Scholar
  52. 52.
    Kopeikin S.M. and Mashhoon B. (2002). Gravitomagnetic effects in the propagation of electromagnetic waves in variable gravitational fields of arbitrary-moving and spinning bodies. Phys. Rev D 65: 064025 CrossRefADSGoogle Scholar
  53. 53.
    Kopeikin S.M. and Ozernoy L.M. (1999). Post-Newtonian Theory for Precision Doppler Measurements of Binary Star Orbits. Astrophys. J. 523: 771 CrossRefADSGoogle Scholar
  54. 54.
    Kopeikin S.M. and Schäfer G. (1999). Lorentz covariant theory of light propagation in gravitational fields of arbitrary-moving bodies. Phys. Rev D 60: 124002 CrossRefADSGoogle Scholar
  55. 55.
    Kopeikin S. and Vlasov I. (2004). Parametrized post-Newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem. Phys. Rep. 400: 209 CrossRefADSGoogle Scholar
  56. 56.
    Kopeikin S.M. and Wei-Tou Ni. (2007). Laser Ranging Delay in the Bi-Metric Theory of Gravity. In: Dittus, H.L., Lämmerzahl, C. and Turyshev, S.G. (eds) Lasers, Clocks, and Drag-Free: Exploration of Relativistic Gravity in Space. Astrophysics and Space Science Library, vol 249, pp 209–216. Springer, Berlin Google Scholar
  57. 57.
    Kovalevsky J. and Seidelmann P.K. (2004). Fundamentals of Astrometry. Cambridge University Press, Cambridge Google Scholar
  58. 58.
    Lämmerzahl, C., Neugebauer, G.: The Lense-Thirring Effect: From the Basic Notions to the Observed Effects; In: Gyros, Clocks, Interferometers... : Testing Relativistic Gravity in Space, Edited by C. Lämmerzahl, C.W.F. Everitt, F.W. Hehl, Lecture Notes in Physics, vol. 562, p. 31 Lecture Notes in Physics (2001)Google Scholar
  59. 59.
    Landau L.D. and Lifshitz E.M. (1971). The Classical Theory of Fields. Pergamon, Oxford Google Scholar
  60. 60.
    Lebach D.E., Corey B.E., Shapiro I.I., Ratner M.I., Webber J.C., Rogers A.E.E., Davis J.L. and Herring T.A. (1995). Measurement of the Solar Gravitational Deflection of Radio Waves Using Very-Long-Baseline Interferometry. Phys. Rev. Lett. 75: 1439 CrossRefADSGoogle Scholar
  61. 61.
    Mashhoon B. (1974). Can Einstein’s theory of gravitation be tested beyond the geometrical optics limit?. Nature 250: 316 CrossRefADSGoogle Scholar
  62. 62.
    Mashhoon B. (2001). Gravitoelectromagnetism. In: Pascual-Snchez, J.F., Floría, L., San Miguel, A. and Vicente, F. (eds) Reference Frames and Gravitomagnetism. Proc. XXIII Spanish Relativity Meeting, pp 121–132. World Scientific, Singapore Google Scholar
  63. 63.
    Mattingly, D.: Modern Tests of Lorentz Invariance. Living Rev. Relativity 8, (2005), 5. URL (cited on Sep 7, 2005): (2004)Google Scholar
  64. 64.
    Mashhoon B., Hehl F.W. and Theiss D.S. (1984). On the gravitational effects of rotating masses - The Thirring-Lense Papers. Gen. Rel. Grav. 16: 711 CrossRefADSGoogle Scholar
  65. 65.
    Mashhoon B., McClune J.C. and Quevedo H. (1999). On the gravitoelectromagnetic stress-energy tensor. Class. Quant. Grav. 16: 1137 zbMATHCrossRefADSGoogle Scholar
  66. 66.
    Mashhoon B., Iorio L. and Lichtenegger H. (2001). On the gravitomagnetic clock effect. Phys. Lett. A 292: 49 zbMATHCrossRefADSGoogle Scholar
  67. 67.
    Maartens R., Mashhoon B. and Matravers D.R. (2002). Holonomy and gravitomagnetism. Class. Quant. Grav. 19: 195 zbMATHCrossRefADSGoogle Scholar
  68. 68.
    Merloni A., Vietri M., Stella L. and Bini D. (1999). On gravitomagnetic precession around black holes. Mon. N. R. Astron. Soc. 304: 155 CrossRefADSGoogle Scholar
  69. 69.
    Misner C.W., Thorne K.S. and Wheeler J.A. (1973). Gravitation. Freeman, New York Google Scholar
  70. 70.
    Ni W.-T. (2005). Empirical Foundations of the Relativistic Gravity. Int. J. Mod. Phys. D 14: 901 zbMATHCrossRefADSGoogle Scholar
  71. 71.
    Nordtvedt K. (1988). Gravitomagnetic interaction and laser ranging to Earth satellites. Phys. Rev. Lett. 61: 2647 CrossRefADSGoogle Scholar
  72. 72.
    O’Connell R.F. (2004). Proposed New Test of Spin Effects in General Relativity. Phys. Rev. Lett. 93: 081103 CrossRefADSGoogle Scholar
  73. 73.
    Pascual-Sánchez J.-F. (2004). Speed of Gravity and Gravitomagnetism. Int. J. Mod. Phys. 13: 2345 zbMATHADSGoogle Scholar
  74. 74.
    Penrose R. (1968). Structure of Space-Time. In: De Witt, C.M. and Wheeler, J.A. (eds) Battelle Recontres, pp 121–235. Benjamin, New York Google Scholar
  75. 75.
    Petrov A.Z. (1969). Einstein Spaces. Pergamon, New York zbMATHGoogle Scholar
  76. 76.
    Pitjeva E.V. (2005). High-Precision Ephemerides of Planets?EPM and Determination of Some Astronomical Constants. Solar Syst. Res. 39: 176 CrossRefADSGoogle Scholar
  77. 77.
    Rafikov R.R. and Lai D. (2006). Effects of gravitational lensing and companion motion on the binary pulsar timing. Phys. Rev. D, vol. 73(6): 063003 CrossRefADSGoogle Scholar
  78. 78.
    Ruggiero M.L. and Tartaglia A. (2005). Post-Keplerian parameter to test gravitomagnetic effects in binary pulsar systems. Phys. Rev. D 72: 084030 CrossRefADSGoogle Scholar
  79. 79.
    Schilizzi, R.T.: The Square Kilometer Array. In: Proc. of the SPIE, vol. 5489, pp. 62–71Google Scholar
  80. 80.
    Schneider P., Ehlers J. and Falco E. (1992). Gravitational Lenses. Springer-Verlag, Berlin Google Scholar
  81. 81.
    Shapiro I.I. (1964). Fourth Test of General Relativity. Phys. Rev. Lett. 13: 789 CrossRefADSGoogle Scholar
  82. 82.
    Skrotskii G.V. (1957). On the influence of gravity on the light propagation. Doklady Akad. Nauk SSSR 114: 73 Google Scholar
  83. 83.
    Soffel M., Klioner S.A., Petit G., Wolf P., Kopeikin S.M., Bretagnon P., Brumberg V.A., Capitaine N., Damour T., Fukushima T., Guinot B., Huang T.-Y., Lindegren L., Ma C., Nordtvedt K., Ries J.C., Seidelmann P.K., Vokrouhlick D., Will C.M. and Xu C. (2003). The IAU 2000 Resolutions for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory Supplement. Astron. J. (USA) 126: 2687 ADSGoogle Scholar
  84. 84.
    Spallicci A. (2004). Satellite measurement of the Hannay angle. Nuov. Cim. B. 119: 1215 ADSGoogle Scholar
  85. 85.
    Spallicci A., Morbidelli A. and Metris G. (2005). The three-body problem and the Hannay angle. Nonlinearity 18: 45 zbMATHCrossRefADSGoogle Scholar
  86. 86.
    Standish, E.M.: JPL planetary ephemeris DE410 Interoffice Memorandum 312.N-03-109 (2003)Google Scholar
  87. 87.
    Tartaglia A. (2000). Detection of the gravitomagnetic clock effect. Class. Quant. Grav. 17: 783 zbMATHCrossRefADSGoogle Scholar
  88. 88.
    Tartaglia A. and Ruggiero M.L. (2004). Gravito-electromagnetism versus electromagnetism. Eur. J. Phys. 25: 203 zbMATHCrossRefGoogle Scholar
  89. 89.
    Tartaglia A., Ruggiero M.L. and Nagar A. (2005). Time delay in binary systems. Phys. Rev. D 71: 023003 CrossRefADSGoogle Scholar
  90. 90.
    Urban, S.E., Bell, S., Kaplan, G.H., Hohenkerk, C.Y., Stewart, S.G., Bangert, J.A., Hilton, J.L.: The Astronomical Almanac 2006: Changes Resulting from IAU Resolutions. In: Bull. Am. Astron. Soc. Meeting 207, #25.05 (2005)Google Scholar
  91. 91.
    Van Flandern T. (1998). The speed of gravity - What the experiments say. Phys. Lett. A 250: 1 CrossRefADSGoogle Scholar
  92. 92.
    Van Flandern T. and Vigier J.P. (2002). Experimental Repeal of the Speed Limit for Gravitational, Electrodynamic and Quantum Field Interactions. Found. Phys 32: 1031 CrossRefGoogle Scholar
  93. 93.
    Wald R.M. (1984). General Relativity. The University of Chicago Press, Chicago zbMATHGoogle Scholar
  94. 94.
    Wex N. and Kopeikin S.M. (1999). Frame Dragging and Other Precessional Effects in Black Hole Pulsar Binaries. Astrophys. J. 514: 388 CrossRefADSGoogle Scholar
  95. 95.
    Will C.M. (1993). Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  96. 96.
    Will, C.M.: (2001) The Confrontation between General Relativity and Experiment. Living Rev. Relativity 4, 4. URL (cited on July 16, 2005): Scholar
  97. 97.
    Will C.M. (2003). Covariant calculation of general relativistic effects in an orbiting gyroscope experiment. Phys. Rev. D 67: 062003 CrossRefADSGoogle Scholar
  98. 98.
    Will C.M.: Has the speed of gravity been measured? (cited on July 14, 2007) (2005)Google Scholar
  99. 99.
    Wittman, D.: In: Courbin, F., Minniti, D. (eds.) Gravitational Lensing: An Astrophysical Tool. Springer-Verlag, Berlin, pp. 55–95 (2002)Google Scholar
  100. 100.
    Zakharov V.D. (1973). Gravitational Waves in Einstein’s Theory. Halsted Press, New York zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of MissouriColumbiaUSA
  2. 2.National Radio Astronomy ObservatoryCharlottesvilleUSA

Personalised recommendations