Measuring the Gravitational Field in General Relativity: From Deviation Equations and the Gravitational Compass to Relativistic Clock Gradiometry

  • Yuri N. Obukhov
  • Dirk PuetzfeldEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 196)


How does one measure the gravitational field? We give explicit answers to this fundamental question and show how all components of the curvature tensor, which represents the gravitational field in Einstein’s theory of General Relativity, can be obtained by means of two different methods. The first method relies on the measuring the accelerations of a suitably prepared set of test bodies relative to the observer. The second method utilizes a set of suitably prepared clocks. The methods discussed here form the basis of relativistic (clock) gradiometry and are of direct operational relevance for applications in geodesy.



This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant PU 461/1-1 (D.P.). The work of Y.N.O. was partially supported by PIER (“Partnership for Innovation, Education and Research” between DESY and Universität Hamburg) and by the Russian Foundation for Basic Research (Grant No. 16-02-00844-A).


  1. 1.
    F.A.E. Pirani, On the physical significance of the Riemann tensor. Acta Phys. Pol. 15, 389 (1956)ADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    J.L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1960)zbMATHGoogle Scholar
  3. 3.
    E. Poisson, A. Pound, I. Vega, The motion of point particles in curved spacetime. Living Rev. Relativ. 14(1), 7 (2011)ADSCrossRefGoogle Scholar
  4. 4.
    T. Levi-Civita, Sur l’écart géodésique. Math. Ann. 97, 291 (1927)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J.L. Synge, The first and second variations of the length integral in Riemannian space. Proc. Lond. Math. Soc. 25, 247 (1926)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J.L. Synge, On the geometry of dynamics. Phil. Trans. R. Soc. Lond. A 226, 31 (1927)ADSCrossRefGoogle Scholar
  7. 7.
    D. Puetzfeld, Y.N. Obukhov, Generalized deviation equation and determination of the curvature in general relativity. Phys. Rev. D 93, 044073 (2016)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    P. Szekeres, The gravitational compass. J. Math. Phys. 6, 1387 (1965)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Puetzfeld, Y.N. Obukhov, C. Lämmerzahl, Gravitational clock compass in general relativity. Phys. Rev. D 98, 024032 (2018)ADSCrossRefGoogle Scholar
  10. 10.
    B.S. DeWitt, R.W. Brehme, Radiation damping in a gravitational field. Ann. Phys. (N.Y.) 9, 220 (1960)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Puetzfeld, Y.N. Obukhov, Covariant equations of motion for test bodies in gravitational theories with general nonminimal coupling. Phys. Rev. D 87, 044045 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    D. Puetzfeld, Y.N. Obukhov, Equations of motion in metric-affine gravity: a covariant unified framework. Phys. Rev. D 90, 084034 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    A.C. Ottewill, B. Wardell, Transport equation approach to calculations of Hadamard Green functions and non-coincident DeWitt coefficients. Phys. Rev. D 84, 104039 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    D.E. Hodgkinson, A modified theory of geodesic deviation. Gen. Relativ. Gravit. 3, 351 (1972)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    S.L. Bażański, Kinematics of relative motion of test particles in general relativity. Ann. H. Poin. A 27, 115 (1977)MathSciNetzbMATHGoogle Scholar
  16. 16.
    A.N. Aleksandrov, K.A. Piragas, Geodesic structure: I. Relative dynamics of geodesics. Theor. Math. Phys. 38, 48 (1978)CrossRefGoogle Scholar
  17. 17.
    B. Schutz, On generalized equations of geodesic deviation, in Galaxies, Axisymmetric Systems, and Relativity, vol 17, ed. by M.A.H. MacCallum (Cambridge University Press, Cambridge, 1985), p. 237Google Scholar
  18. 18.
    C. Chicone, B. Mashhoon, The generalized Jacobi equation. Class. Quantum Gravity 19, 4231 (2002)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    T. Mullari, R. Tammelo, On the relativistic tidal effects in the second approximation. Class. Quantum Gravity 23, 4047 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Vines, Geodesic deviation at higher orders via covariant bitensors. Gen. Relativ. Gravit. 47, 59 (2015)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    E. Fermi, Sopra i fenomeni che avvengono in vicinanza di una linea oraria. Atti. Accad. Naz. Lincei Cl, Sci. Fis. Mat. Nat. Rend. 31, 21, 51, 101 (1922)Google Scholar
  22. 22.
    E. Fermi, Collected Papers, vol 1, ed. by E. Amaldi, E. Persico, F. Rasetti, E. Segrè (University of Chicago Press, Chicago, 1962)Google Scholar
  23. 23.
    O. Veblen, Normal coordinates for the geometry of paths. Proc. Natl. Acad. Sci. (USA) 8, 192 (1922)ADSCrossRefGoogle Scholar
  24. 24.
    O. Veblen, T.Y. Thomas, The geometry of paths. Trans. Am. Math. Soc. 25, 551 (1923)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J.L. Synge, A characteristic function in Riemannian space and its application to the solution of geodesic triangles. Proc. Lond. Math. Soc. 32, 241 (1931)MathSciNetCrossRefGoogle Scholar
  26. 26.
    A.G. Walker, Relative coordinates. Proc. R. Soc. Edinb. 52, 345 (1932)CrossRefGoogle Scholar
  27. 27.
    F.K. Manasse, C.W. Misner, Fermi normal coordinates and some basic concepts in differential geometry. J. Math. Phys. 4, 735 (1963)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973)Google Scholar
  29. 29.
    W.-T. Ni, On the proper reference frame and local coordinates of an accelerated observer in special relativity. Chin. J. Phys. 15, 51 (1977)Google Scholar
  30. 30.
    B. Mashhoon, Tidal radiation. Astrophys. J. 216, 591 (1977)ADSCrossRefGoogle Scholar
  31. 31.
    W.-T. Ni, M. Zimmermann, Inertial and gravitational effects in the proper reference frame of an accelerated, rotating observer. Phys. Rev. D 17, 1473 (1978)ADSCrossRefGoogle Scholar
  32. 32.
    W.-Q. Li, W.-T. Ni, On an accelerated observer with rotating tetrad in special relativity. Chin. J. Phys. 16, 214 (1978)Google Scholar
  33. 33.
    W.-T. Ni, Geodesic triangles and expansion of the metrics in normal coordinates. Chin. J. Phys. 16, 223 (1978)Google Scholar
  34. 34.
    W.-Q. Li, W.-T. Ni, Coupled inertial and gravitational effects in the proper reference frame of an accelerated, rotating observer. J. Math. Phys. 20, 1473 (1979)ADSCrossRefGoogle Scholar
  35. 35.
    W.-Q. Li, W.-T. Ni, Expansions of the affinity, metric and geodesic equations in Fermi normal coordinates about a geodesic. J. Math. Phys. 20, 1925 (1979)ADSCrossRefGoogle Scholar
  36. 36.
    N. Ashby, B. Bertotti, Relativistic effects in local inertial frames. Phys. Rev. D 34, 2246 (1986)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    A.M. Eisele, On the behaviour of an accelerated clock. Helv. Phys. Acta 60, 1024 (1987)Google Scholar
  38. 38.
    T. Fukushima, The Fermi coordinate system in the post-Newtonian framework. Celest. Mech. 44, 1024 (1988)CrossRefGoogle Scholar
  39. 39.
    O. Semerák, Stationary frames in the Kerr field. Gen. Relativ. Gravit. 25, 1041 (1993)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    K.-P. Marzlin, Fermi coordinates for weak gravitational fields. Phys. Rev. D 50, 888 (1994)ADSCrossRefGoogle Scholar
  41. 41.
    D. Bini, A. Geralico, R.T. Jantzen, Kerr metric, static observers and Fermi coordinates. J. Math. Phys. 22, 4729 (2005)MathSciNetzbMATHGoogle Scholar
  42. 42.
    C. Chicone, B. Mashhoon, Explicit Fermi coordinates and tidal dynamics in de Sitter and Gödel spacetime. Phys. Rev. D 74, 064019 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    D. Klein, P. Collas, General transformation formulas for Fermi-Walker coordinates. Class. Quantum Gravity 25, 145019 (2008)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    D. Klein, P. Collas, Exact Fermi coordinates for a class of space-times. J. Math. Phys. 51, 022501 (2010)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    P. Delva, M.-C. Angonin, Extended Fermi coordinates. Gen. Relativ. Gravit. 44, 1 (2012)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    S.G. Turyshev, O.L. Minazzoli, V.T. Toth, Accelerating relativistic reference frames in Minkowski space-time. J. Math. Phys. 53, 032501 (2012)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    F.W. Hehl, W.-T. Ni, Inertial effects of a Dirac particle. Phys. Rev. D 42, 2045 (1990)ADSCrossRefGoogle Scholar
  48. 48.
    I. Ciufolini, M. Demianski, How to measure the curvature of space-time. Phys. Rev. D 34, 1018 (1986)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    I. Ciufolini, Generalized geodesic deviation equation. Phys. Rev. D 34, 1014 (1986)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    R. Rodrigo, V. Dehant, L. Gurvits, M. Kramer, R. Park, P. Wolf, J. Zarnecki (eds.), High Performance Clocks with Special Emphasis on Geodesy and Geophysics and Applications to Other Bodies of the Solar System, vol. 63, Space Sciences Series of ISSI (Springer, Netherlands, 2018)Google Scholar
  51. 51.
    F.W. Hehl, P. von der Heyde, G.D. Kerlick, J.M. Nester, General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393 (1976)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    M. Blagojević, F.W. Hehl, Gauge Theories of Gravitation: A Reader with Commentaries (Imperial College Press, London, 2013)CrossRefGoogle Scholar
  53. 53.
    V.N. Ponomarev, A.O. Barvinsky, Y.N. Obukhov, Gauge Approach and Quantization Methods in Gravity Theory (Nauka, Moscow, 2017)CrossRefGoogle Scholar
  54. 54.
    F.W. Hehl, Y.N. Obukhov, D. Puetzfeld, On Poincaré gauge theory of gravity, its equations of motion, and gravity probe B. Phys. Lett. A 377, 1775 (2013)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    Y.N. Obukhov, D. Puetzfeld, Multipolar test body equations of motion in generalized gravity theories, in Equations of Motion in Relativistic Gravity, vol. 179, Fundamental Theories of Physics, ed. by D. Puetzfeld, et al. (Springer, Cham, 2015), p. 67CrossRefGoogle Scholar
  56. 56.
    D. Puetzfeld, Y.N. Obukhov, Deviation equation in Riemann-Cartan spacetime. Phys. Rev. D 97, 104069 (2018)ADSCrossRefGoogle Scholar
  57. 57.
    A. Trautman, Einstein-Cartan theory, in Encyclopedia of Mathematical Physics, vol. 2, ed. by J.-P. Francoise, G.L. Naber, S.T. Tsou (Elsevier, Oxford, 2006), p. 189CrossRefGoogle Scholar
  58. 58.
    Y.N. Obukhov, Poincaré gauge gravity: selected topics. Int. J. Geom. Methods Mod. Phys. 03, 95 (2006)CrossRefGoogle Scholar
  59. 59.
    Y.N. Obukhov, Poincaré gauge gravity: an overview. Int. J. Geom. Methods Mod. Phys. 15, Supp. 1 (2018) 1840005ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    J.L. Synge, Geodesics in non-holonomic geometry. Math. Ann. 99, 738 (1928)MathSciNetCrossRefGoogle Scholar
  61. 61.
    W.H. Goldthorpe, Spectral geometry and \(SO(4)\) gravity in a Riemann-Cartan spacetime. Nucl. Phys. B 170, 307 (1980)ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    H.T. Nieh, M.L. Yan, Quantized Dirac field in curved Riemann-Cartan background: I. Symmetry properties, Green’s function. Ann. Phys. (N.Y.) 138, 237 (1982)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    N.H. Barth, Heat kernel expansion coefficient: I. An extension. J. Phys. A Math. Gen. 20, 857 (1987)ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    S. Yajima, Evaluation of the heat kernel in Riemann-Cartan space. Class. Quantum Gravity 13, 2423 (1996)ADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    S.S. Manoff, Auto-parallel equation as Euler-Lagrange’s equation in spaces with affine connections and metrics. Gen. Relativ. Gravit. 32, 1559 (2000)ADSCrossRefGoogle Scholar
  66. 66.
    S.S. Manoff, Deviation equations of Synge and Schild over spaces with affine connections and metrics. Int. J. Mod. Phys. A 16, 1109 (2001)ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    B.Z. Iliev. Deviation equations in spaces with a transport along paths. JINR Commun. E2-94-40, Dubna, 1994 (2003)Google Scholar
  68. 68.
    R.J. van den Hoogen, Towards a covariant smoothing procedure for gravitational theories. J. Math. Phys. 58, 122501 (2017)ADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    T.Y. Thomas, The Differential Invariants of Generalized Spaces (Cambridge University Press, Cambridge, 1934)zbMATHGoogle Scholar
  70. 70.
    J.A. Schouten, Ricci-Calculus. An Introduction to Tensor Analysis and its Geometric Applications, 2nd edn. (Springer, Berlin, 1954)zbMATHGoogle Scholar
  71. 71.
    I.G. Avramidi, A covariant technique for the calculation of the one-loop effective action. Nucl. Phys. B 355, 712 (1991)ADSMathSciNetCrossRefGoogle Scholar
  72. 72.
    I.G. Avramidi, Covariant methods for the calculation of the effective action in quantum field theory and investigation of higher-derivative quantum gravity. Ph.D. thesis, Moscow State University (1986), English version arXiv:hep-th/9510140
  73. 73.
    A.Z. Petrov, Einstein Spaces (Pergamon, Oxford, 1969)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Theoretical Physics LaboratoryNuclear Safety Institute, Russian Academy of SciencesMoscowRussia
  2. 2.Center of Applied Space Technology and Microgravity (ZARM), University of BremenBremenGermany

Personalised recommendations