Local gravitational physics of the Hubble expansion

Einstein’s equivalence principle in cosmology
  • Sergei M. KopeikinEmail author
Regular Article


We study physical consequences of the Hubble expansion of Friedmann-Lemaıtre-Robertson-Walker (FLRW) manifold on measurement of space, time and light propagation in the local inertial frame. We use the results of this study to analyse the Solar System radar ranging and Doppler tracking experiments and time synchronization. FLRW manifold is covered by the coordinates (t, y i ), where t is the cosmic time coinciding with the proper time of the Hubble observers and identified with the barycentric coordinate time (TCB) used in ephemeris astronomy. We introduce the local inertial coordinates x α = (x 0, x i ) in the vicinity of a world line of a Hubble observer with the help of a special conformal transformation that respects the local equivalence between the tangent and FLRW manifold. The local inertial metric is Minkowski flat and is materialized by the congruence of time-like geodesics of static observers being at rest with respect to the local spatial coordinates x i . The static observers are equipped with the ideal clocks measuring their own proper time which is synchronized with the cosmic time t measured by the Hubble observer. We consider the geodesic motion of test particles and notice that the local coordinate time x 0 = x 0(t) taken as a parameter along the world line of the particle, is a function of Hubble’s observer time t. This function changes smoothly from x 0 = t for a particle at rest (observer’s clock), to x 0 = t + (1/2)Ht 2 for photons, where H is the Hubble constant. Thus, the motion of a test particle is non-uniform when its world line is parametrized by the cosmic time t. NASA JPL Orbit Determination Program operates under the assumption that the spacetime is asymptotically flat which presumes that the motion of light (after the Shapiro delay is excluded) is uniform with respect to the time t but it does not comply with the non-uniform motion of light on cosmological manifold. For this reason, the motion of light in the Solar System analysed with the Orbit Determination Program appears as having a systematic blue shift of frequency, of radio waves circulating in the Earth-spacecraft radio link. The magnitude of the anomalous blue shift of frequency is proportional to the Hubble constant H that may open an access to the measurement of this fundamental cosmological parameter in the Solar System radiowave experiments.


Solar System Proper Time Cosmic Time World Line Hubble Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Y. Baryshev, P. Teerikorpi, Fundamental Questions of Practical Cosmology, in Astrophysics and Space Science, Vol. 383 (Springer, Netherlands, 2012).Google Scholar
  2. 2.
    S. Kopeikin, M. Efroimsky, G. Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley-VCH, Weinheim, 2011).Google Scholar
  3. 3.
    A. Einstein, Relativity, the Special and the General Theory: A popular Exposition by Albert Einstein (Crown Publishers, 1961).Google Scholar
  4. 4.
    A. Einstein, Ann. Phys. 356, 639 (1916).CrossRefGoogle Scholar
  5. 5.
    J.D. Norton, Rep. Prog. Phys. 56, 791 (1993).ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    C.M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, Cambridge, 1993).Google Scholar
  7. 7.
    M. Soffel, S.A. Klioner, G. Petit, P. Wolf, S.M. Kopeikin, P. Bretagnon, V.A. Brumberg, N. Capitaine, T. Damour, T. Fukushima, B. Guinot, T.-Y. Huang, L. Lindegren, C. Ma, K. Nordtvedt, J.C. Ries, P.K. Seidelmann, D. Vokrouhlický, C.M. Will, C. Xu, Astron. J. 126, 2687 (2003).ADSCrossRefGoogle Scholar
  8. 8.
    V. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, Cambridge, 2005).Google Scholar
  9. 9.
    S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008).Google Scholar
  10. 10.
    M. Soffel, R. Langhans, Space-Time Reference Systems, in Astronomy and Astrophysics (Springer, Berlin, 2013).Google Scholar
  11. 11.
    N. Jarosik, C.L. Bennett, J. Dunkley, B. Gold, M.R. Greason, M. Halpern, R.S. Hill, G. Hinshaw, A. Kogut, E. Komatsu, D. Larson, M. Limon, S.S. Meyer, M.R. Nolta, N. Odegard, L. Page, K.M. Smith, D.N. Spergel, G.S. Tucker, J.L. Weiland, E. Wollack, E.L. Wright, Astrophys. J. Suppl. Ser. 192, 14 (2011).ADSCrossRefGoogle Scholar
  12. 12.
    M. Ibison, J. Math. Phys. 48, 122501 (2007).ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Carrera, D. Giulini, Rev. Mod. Phys. 82, 169 (2010).ADSCrossRefGoogle Scholar
  14. 14.
    S.M. Kopeikin, Phys. Rev. D 86, 064004 (2012).ADSCrossRefGoogle Scholar
  15. 15.
    F. Rohrlich, Ann. Phys. 22, 169 (1963).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    A.L. Harvey, Ann. Phys. 29, 383 (1964).ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    S. Kopeikin, Equivalence Principle in Cosmology, in CPT and Lorentz Symmetry - Proceedings of the Sixth Meeting, edited by A. Kostelecky (World Scientific, Singapore, 2014) p. 224--227.Google Scholar
  18. 18.
    L. Hongya, J. Math. Phys. 28, 1920 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    L. Hongya, J. Math. Phys. 28, 1924 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    S.A. Klioner, M.H. Soffel, Refining the Relativistic Model for Gaia: Cosmological Effects in the BCRS, in The Three-Dimensional Universe with Gaia, edited by C. Turon, K.S. O’Flaherty, M.A.C. Perryman, Vol. 576 (2005).Google Scholar
  21. 21.
    B. Mashhoon, N. Mobed, D. Singh, Class. Quantum Grav. 24, 5031 (2007).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    W.T. Ni, M. Zimmermann, Phys. Rev. D 17, 1473 (1978).ADSCrossRefGoogle Scholar
  23. 23.
    T.D. Moyer, Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation (John Wiley & Sons, Inc., Hoboken, New Jersey, 2003).Google Scholar
  24. 24.
    R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).Google Scholar
  25. 25.
    H.A. Kastrup, Phys. Rev. 150, 1183 (1966).ADSCrossRefGoogle Scholar
  26. 26.
    M. Schottenloher, A Mathematical Introduction to Conformal Field Theory, in Lecture Notes in Physics, Vol. 759 (Springer Verlag, Berlin, 2008).Google Scholar
  27. 27.
    H.A. Kastrup, Ann. Phys. 464, 388 (1962).CrossRefMathSciNetGoogle Scholar
  28. 28.
    David Klein, Evan Randles, Ann. Henri Poincaré 12, 303 (2011).ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    J. Ehlers, F.A.E. Pirani, A. Schild, Gen. Relativ. Gravit. 44, 1587 (2012).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    C.M. Will, Living Rev. Relativ. 9, 3 (2006).ADSCrossRefGoogle Scholar
  31. 31.
    E. Fischbach, C. Talmadge, Nature 356, 207 (1992).ADSCrossRefGoogle Scholar
  32. 32.
    S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley & Sons, Inc., New York, 1972).Google Scholar
  33. 33.
    J.G. Williams, S.G. Turyshev, D.H. Boggs, Class. Quantum Grav. 29, 184004 (2012).ADSCrossRefGoogle Scholar
  34. 34.
    L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1975).Google Scholar
  35. 35.
    W.M. Folkner, J.G. Williams, D.H. Boggs, Interplanet. Netw. Prog. Rep. 178, C1 (2009).Google Scholar
  36. 36.
    A. Fienga, J. Laskar, T. Morley, H. Manche, P. Kuchynka, C. Le Poncin-Lafitte, F. Budnik, M. Gastineau, L. Somenzi, Astron. Astrophys. 507, 1675 (2009).ADSCrossRefGoogle Scholar
  37. 37.
    T.W. Murphy, E.G. Adelberger, J.B.R. Battat, L.N. Carey, C.D. Hoyle, P. Leblanc, E.L. Michelsen, K. Nordtvedt, A.E. Orin, J.D. Strasburg, C.W. Stubbs, H.E. Swanson, E. Williams, Pub. Astron. Soc. Pacific 120, 20 (2008).ADSCrossRefGoogle Scholar
  38. 38.
    N. Capitaine, S. Klioner, D. McCarthy, The re-definition of the astronomical unit of length: reasons and consequences, in IAU Joint Discussion (IAU Joint Discussion, 2012).Google Scholar
  39. 39.
    M. Carrera, D. Giulini, Class. Quantum Grav. 23, 7483 (2006).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto, S.G. Turyshev, Phys. Rev. D 65, 082004 (2002).ADSCrossRefGoogle Scholar
  41. 41.
    S.G. Turyshev, M.M. Nieto, J.D. Anderson, Adv. Space Res. 39, 291 (2007).ADSCrossRefGoogle Scholar
  42. 42.
    B. Rievers, C. Lämmerzahl, Ann. Phys. 523, 439 (2011).CrossRefGoogle Scholar
  43. 43.
    S.G. Turyshev, V.T. Toth, G. Kinsella, S.-C. Lee, S.M. Lok, J. Ellis, Phys. Rev. Lett. 108, 241101 (2012).ADSCrossRefGoogle Scholar
  44. 44.
    D. Modenini, P. Tortora, Phys. Rev. D 90, 022004 (2014).ADSCrossRefGoogle Scholar
  45. 45.
    M. Alcubierre, Class. Quantum Grav. 11, L73 (1994).ADSCrossRefMathSciNetGoogle Scholar
  46. 46.
    J. Natário, Class. Quantum Grav. 19, 1157 (2002).ADSCrossRefzbMATHGoogle Scholar
  47. 47.
    M. Kramer, N. Wex, Class. Quantum Grav. 26, 073001 (2009).ADSCrossRefGoogle Scholar
  48. 48.
    T. Damour, J.H. Taylor, Astrophys. J. 366, 501 (1991).ADSCrossRefGoogle Scholar
  49. 49.
    J.D. Anderson, G. Schubert, Phys. Earth Planet. Inter. 178, 176 (2010).ADSCrossRefGoogle Scholar
  50. 50.
    A. Hees, B. Lamine, S. Reynaud, M.-T. Jaekel, C. Le Poncin-Lafitte, V. Lainey, A. Füzfa, J.-M. Courty, V. Dehant, P. Wolf, Class. Quantum Grav. 29, 235027 (2012).ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Physics & AstronomyUniversity of MissouriColumbiaUSA
  2. 2.Siberian State Geodetic AcademyNovosibirskRussia

Personalised recommendations