Advertisement

Covariant equations of motion beyond the spin-dipole particle approximation

  • Sergei M. KopeikinEmail author
Regular Article
  • 18 Downloads
Part of the following topical collections:
  1. Focus Point on Tests of General Relativity and Alternative Gravity Theories

Abstract.

The present paper studies the post-Newtonian dynamics of N-body problem in general relativity. We derive covariant equations of translational and rotational motion of N extended bodies having arbitrary distribution of mass and velocity of matter by employing the set of global and local coordinate charts on curved spacetime manifold M of N-body system along with the mathematical apparatus of the Cartesian symmetric trace-free tensors and Blanchet-Damour multipole formalism. We separate the self-field effects of the bodies from the external gravitational environment and construct the effective background spacetime manifold by making use of the asymptotic matching technique. We make worldline of the center of mass of each body identical with that of the origin of the body-adapted local coordinates by the appropriate choice of the dipole moments. The covariant equations of motion are obtained on the background manifold \(\bar{M}\) by applying the Einstein principle of equivalence and the Fermi-Walker law of transportation of the linear momentum and spin of each body. Our approach significantly extends the Mathisson-Papapetrou-Dixon covariant equations of motion beyond the spin-dipole particle approximation by accounting for the entire infinite set of the internal multipoles of the bodies which are gravitationally coupled with the curvature tensor of the background manifold \( \bar{M}\) and its covariant derivatives. The results of our study can be used for much more accurate prediction of orbital dynamics of extended bodies in inspiraling binary systems and construction of templates of gravitational waves at the merger stage when the strong gravitational interaction between the higher-order multipoles of the bodies play a dominant role. The covariant theory of the post-Newtonian equations of motion beyond the spin-dipole approximation is a solid foundation for future improvements in long-term accuracy of relativistic celestial ephemerides of the solar system bodies.

References

  1. 1.
    V.A. Brumberg, Relativistic Celestial Mechanics (Nauka, Moscow, 1972) (in Russian)Google Scholar
  2. 2.
    V.A. Brumberg, Essential Relativistic Celestial Mechanics (Adam Hilger, New York, 1991)Google Scholar
  3. 3.
    G.F.R. Ellis, J.-P. Uzan, Am. J. Phys. 73, 240 (2005)CrossRefGoogle Scholar
  4. 4.
    S.M. Kopeikin, Class. Quantum Grav. 21, 3251 (2004)CrossRefGoogle Scholar
  5. 5.
    E.B. Fomalont, S.M. Kopeikin, Astrophys. J. 598, 704 (2003)CrossRefGoogle Scholar
  6. 6.
    S.M. Kopeikin, Astrophys. J. Lett. 556, L1 (2001)CrossRefGoogle Scholar
  7. 7.
    S.M. Kopeikin, E.B. Fomalont, Found. Phys. 36, 1244 (2006)CrossRefGoogle Scholar
  8. 8.
    S.M. Kopeikin, E.B. Fomalont, Gen. Relativ. Gravit. 39, 1583 (2007)CrossRefGoogle Scholar
  9. 9.
    S.M. Kopeikin, Class. Quantum Grav. 22, 5181 (2005)CrossRefGoogle Scholar
  10. 10.
    S.M. Kopeikin, Int. J. Mod. Phys. D 15, 305 (2006)CrossRefGoogle Scholar
  11. 11.
    J. Ehlers, Ann. New York Acad. Sci. 336, 279 (1980)CrossRefGoogle Scholar
  12. 12.
    J. Frauendiener, Liv. Rev. Relativ. 7, 1 (2004)CrossRefGoogle Scholar
  13. 13.
    A.N. Petrov, S.M. Kopeikin, R.R. Lompay, B. Tekin, Metric Theories of Gravity: Perturbations and Conservation Laws (De Gruyter, Berlin, 2017)Google Scholar
  14. 14.
    S.M. Kopeikin, A.N. Petrov, Phys. Rev. D 87, 044029 (2013)CrossRefGoogle Scholar
  15. 15.
    S.M. Kopeikin, A.N. Petrov, Ann. Phys. 350, 379 (2014)CrossRefGoogle Scholar
  16. 16.
    S.M. Kopeikin, Phys. Rev. D 86, 064004 (2012)CrossRefGoogle Scholar
  17. 17.
    A. Galiautdinov, S.M. Kopeikin, Phys. Rev. D 94, 044015 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    L. Blanchet, T. Damour, Ann. Inst. H. Poincaré 50, 337 (1989)Google Scholar
  19. 19.
    V.A. Fock, The Theory of Space, Time and Gravitation (Pergamon Press, New York, 1959)Google Scholar
  20. 20.
    N. Spyrou, Astrophys. J. 197, 725 (1975)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Arminjon, Phys. Rev. D 72, 084002 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    É Racine, Class. Quantum Grav. 23, 373 (2006)CrossRefGoogle Scholar
  23. 23.
    P. Havas, J.N. Goldberg, Phys. Rev. 128, 398 (1962)MathSciNetCrossRefGoogle Scholar
  24. 24.
    K.S. Thorne, J.B. Hartle, Phys. Rev. D 31, 1815 (1985)MathSciNetCrossRefGoogle Scholar
  25. 25.
    X.-H. Zhang, Phys. Rev. D 31, 3130 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    R.-M. Memmesheimer, G. Schäfer, Phys. Rev. D 71, 044021 (2005)MathSciNetCrossRefGoogle Scholar
  27. 27.
    S. Hergt, G. Schäfer, Phys. Rev. D 77, 104001 (2008)CrossRefGoogle Scholar
  28. 28.
    G. Schäfer, Post-Newtonian Methods: Analytic Results on the Binary Problem, in Mass and Motion in General Relativity. Fundamental Theories of Physics, edited by L. Blanchet, A. Spallicci, B. Whiting, Vol. 162 (Springer, Berlin, 2011) pp. 167--210  https://doi.org/10.1007/978-90-481-3015-3_6
  29. 29.
    S.M. Kopeikin, G. Schäfer, C.R. Gwinn, T.M. Eubanks, Phys. Rev. D 59, 084023 (1999)MathSciNetCrossRefGoogle Scholar
  30. 30.
    M. Mathisson, Gen. Relativ. Gravit. 42, 989 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    M. Mathisson, Gen. Relativ. Gravit. 42, 1011 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    A. Papapetrou, Proc. R. Soc. London Ser. A 209, 248 (1951)MathSciNetCrossRefGoogle Scholar
  33. 33.
    A. Papapetrou, Proc. Phys. Soc. A 64, 57 (1951)CrossRefGoogle Scholar
  34. 34.
    W.G. Dixon, R. Soc. London Proc. Ser. A 314, 499 (1970)CrossRefGoogle Scholar
  35. 35.
    W.G. Dixon, R. Soc. London Proc. Ser. A 319, 509 (1970)CrossRefGoogle Scholar
  36. 36.
    W.G. Dixon, R. Soc. London Philos. Trans. Ser. A 277, 59 (1974)CrossRefGoogle Scholar
  37. 37.
    W.G. Dixon, Gen. Relativ. Gravit. 4, 199 (1973)CrossRefGoogle Scholar
  38. 38.
    W.G. Dixon, Extended bodies in general relativity: their description and motion, in Isolated Gravitating Systems in General Relativity, edited by J. Ehlers, (North-Holland, Amsterdam, 1979) pp. 156--219Google Scholar
  39. 39.
    W.G. Dixon, Acta Phys. Pol. B Proc. Suppl. 1, 27 (2008)Google Scholar
  40. 40.
    W.G. Dixon, The New Mechanics of Myron Mathisson and Its Subsequent Development, in Equations of Motion in Relativistic Gravity, edited by D. Puetzfeld, C. Lämmerzahl, B. Schutz (Springer, Cham, 2015) pp. 1--66Google Scholar
  41. 41.
    A.H. Taub, The motion of multipoles in general relativity, in IV Centenario Della Nascita di Galileo Galilei, 1564–1964, edited by G. Barbèra, (Pubblicazioni del Comitato Nazionale per le Manifestazioni Celebrative, Firenze, 1965) pp. 100--118Google Scholar
  42. 42.
    J. Madore, Ann. l'I.H.P. Phys. Théor. 11, 221 (1969)Google Scholar
  43. 43.
    J. Ehlers, E. Rudolph, Gen. Relativ. Gravit. 8, 197 (1977)CrossRefGoogle Scholar
  44. 44.
    R. Schattner, Gen. Relativ. Gravit. 10, 377 (1979)MathSciNetCrossRefGoogle Scholar
  45. 45.
    A. Ohashi, Phys. Rev. D 68, 044009 (2003)MathSciNetCrossRefGoogle Scholar
  46. 46.
    J. Steinhoff, D. Puetzfeld, Phys. Rev. D 81, 044019 (2010)CrossRefGoogle Scholar
  47. 47.
    D. Puetzfeld, Y.N. Obukhov, Phys. Lett. A 377, 2447 (2013)CrossRefGoogle Scholar
  48. 48.
    Y.N. Obukhov, D. Puetzfeld, Phys. Rev. D 90, 104041 (2014)CrossRefGoogle Scholar
  49. 49.
    D. Puetzfeld, Y.N. Obukhov, Phys. Rev. D 90, 084034 (2014)CrossRefGoogle Scholar
  50. 50.
    T. Damour, The problem of motion in Newtonian and Einsteinian gravity, in Three Hundred Years of Gravitation, edited by S.W. Hawking, W. Israel (Cambridge University Press, Cambridge, 1987) pp. 128--198Google Scholar
  51. 51.
    T. Mädler, J. Winicour, Scholarpedia 11, 33528 (2016)CrossRefGoogle Scholar
  52. 52.
    B. Schmidt, M. Walker, P. Sommers, Gen. Relativ. Gravit. 6, 489 (1975)CrossRefGoogle Scholar
  53. 53.
    S.M. Kopejkin, Celest. Mech. 44, 87 (1988)MathSciNetCrossRefGoogle Scholar
  54. 54.
    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, Astron. J. 126, 2687 (2003)CrossRefGoogle Scholar
  55. 55.
    E. Battista, G. Esposito, S. Dell'Agnello, Int. J. Mod. Phys. A 32, 1730022 (2017)CrossRefGoogle Scholar
  56. 56.
    B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry - Methods and Applications (Springer, New York, 1984)Google Scholar
  57. 57.
    F.A.E. Pirani, Introduction to Gravitational Radiation Theory, in Lectures on General Relativity, edited by A. Trautman, F.A.E. Pirani, H. Bondi, Vol. 1 (Prentice Hall, 1965) pp. 249--373Google Scholar
  58. 58.
    K.S. Thorne, Rev. Mod. Phys. 52, 299 (1980)CrossRefGoogle Scholar
  59. 59.
    L. Blanchet, T. Damour, R. Soc. London Philos. Trans. Ser. A 320, 379 (1986)CrossRefGoogle Scholar
  60. 60.
    S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (J. Wiley & Sons, New York, 1972)Google Scholar
  61. 61.
    W. Tulczyjew, Acta Phys. Pol. 18, 393 (1959)MathSciNetGoogle Scholar
  62. 62.
    B. Tulczyjew, W. Tulczyjew, On multipole formalism in general relativity, in Recent Developments in General Relativity. A collection of papers dedicated to Leopold Infeld (Pergamon Press, New York, 1962) pp. 465--472Google Scholar
  63. 63.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W.H. Freeman, San Francisco, 1973)Google Scholar
  64. 64.
    J.L. Synge, Relativity: The general theory, in Series in Physics (North-Holland, Amsterdam, 1964)Google Scholar
  65. 65.
    I.M. Gel'fand, G.E. Shilov, Generalized functions, Vol. I, Properties and operations (Academic Press, New York, 1964) translated by E. SaletanGoogle Scholar
  66. 66.
    I. Kolár, P.W. Michor, J. Slovák, Natural operations in differential geometry (Springer, Berlin, 1993)Google Scholar
  67. 67.
    O. Veblen, T.Y. Thomas, Trans. Am. Math. Soc. 25, 551 (1923)CrossRefGoogle Scholar
  68. 68.
    J.A. Schouten, Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications (Springer, Berlin, 1954) see review by K. Yano at https://doi.org/projecteuclid.org/download/pdf_1/euclid.bams/1183519893
  69. 69.
    E. Poisson, C.M. Will, Gravity (Cambridge University Press, Cambridge, 2014)Google Scholar
  70. 70.
    A.I. Nesterov, Class. Quantum Grav. 16, 465 (1999)CrossRefGoogle Scholar
  71. 71.
    W.-T. Ni, M. Zimmermann, Phys. Rev. D 17, 1473 (1978)CrossRefGoogle Scholar
  72. 72.
    W. Beiglböck, Commun. Math. Phys. 5, 106 (1967)CrossRefGoogle Scholar
  73. 73.
    I. Bailey, W. Israel, Ann. Phys. 130, 188 (1980)CrossRefGoogle Scholar
  74. 74.
    A. Pound, Motion of small objects in curved spacetimes: An introduction to gravitational self-force, in Equations of Motion in Relativistic Gravity, edited by D. Puetzfeld, C. Lämmerzahl, B. Schutz (Springer, Berlin, 2015) pp. 399--486Google Scholar
  75. 75.
    B.F. Schutz, Philos. Trans. R. Soc. London Ser. A 376, 20170279 (2018)CrossRefGoogle Scholar
  76. 76.
    S. Babak, J.R. Gair, R.H. Cole, Extreme mass ratio inspirals: Perspectives for their detection, in Equations of Motion in Relativistic Gravity, edited by D. Puetzfeld, C. Lämmerzahl, B. Schutz (Springer International Publishing, 2015) pp. 783--812Google Scholar
  77. 77.
    D. Bini, C. Cherubini, A. Geralico, A. Ortolan, Gen. Relativ. Gravit. 41, 105 (2009)CrossRefGoogle Scholar
  78. 78.
    V.A. Brumberg, S.M. Kopejkin, Relativistic theory of celestial reference frames, in Reference Frames in Astronomy and Geophysics, edited by J. Kovalevsky, I.I. Mueller, B. Kolaczek, Vol. 154 (Astrophysics and Space Science Library, Kluwer, 1989) pp. 115--141Google Scholar
  79. 79.
    T. Damour, M. Soffel, C. Xu, Phys. Rev. D 43, 3273 (1991)MathSciNetCrossRefGoogle Scholar
  80. 80.
    C.M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, Cambridge, 1993)Google Scholar
  81. 81.
    S.M. Kopejkin, Itogi Nauk. Tekhn. Ser. Astron. 41, 87 (1991)Google Scholar
  82. 82.
    V.A. Brumberg, S.M. Kopejkin, Nuovo Cimento B 103, 63 (1989)CrossRefGoogle Scholar
  83. 83.
    V.A. Brumberg, S.M. Kopeikin, Celest. Mech. Dyn. Astron. 48, 23 (1990)CrossRefGoogle Scholar
  84. 84.
    S. Kopeikin, I. Vlasov, Phys. Rep. 400, 209 (2004)MathSciNetCrossRefGoogle Scholar
  85. 85.
    S. Kopeikin, M. Efroimsky, G. Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley, Weinheim, 2011)Google Scholar
  86. 86.
    I. Ciufolini, J.A. Wheeler, Gravitation and Inertia (Princeton University Press, Princeton, 1995)Google Scholar
  87. 87.
    C.M. Will, Class. Quantum Grav. 32, 220301 (2015)CrossRefGoogle Scholar
  88. 88.
    I. Ciufolini, E.C. Pavlis, A. Paolozzi, J. Ries, R. Koenig, R. Matzner, G. Sindoni, K.H. Neumayer, New Astron. 17, 341 (2012)CrossRefGoogle Scholar
  89. 89.
    I. Ciufolini, A. Paolozzi, E.C. Pavlis, R. Koenig, J. Ries, V. Gurzadyan, R. Matzner, R. Penrose, G. Sindoni, C. Paris, H. Khachatryan, S. Mirzoyan, Eur. Phys. J. C 76, 120 (2016)CrossRefGoogle Scholar
  90. 90.
    T. Damour, M. Soffel, C. Xu, Phys. Rev. D 45, 1017 (1992)MathSciNetCrossRefGoogle Scholar
  91. 91.
    T. Damour, B.R. Iyer, Ann. I. H. P., sect. A 54, 115 (1991)Google Scholar
  92. 92.
    T. Damour, M. Soffel, C. Xu, Phys. Rev. D 47, 3124 (1993)MathSciNetCrossRefGoogle Scholar
  93. 93.
    N. Ashby, B. Bertotti, Phys. Rev. D 34, 2246 (1986)MathSciNetCrossRefGoogle Scholar
  94. 94.
    I.G. Fichtengoltz, JETP 20, 233 (1950)Google Scholar
  95. 95.
    S. Kopeikin, I. Vlasov, The Effacing Principle in the Post-Newtonian Celestial Mechanics, in The 11-th MG Meeting On Recent Developments in Theoretical and Experimental General Relativity, edited by H. Kleinert, R.T. Jantzen, R. Ruffini (World Scientific Publishing, 2008) pp. 2475--2477Google Scholar
  96. 96.
    W.-M. Suen, Phys. Rev. D 34, 3617 (1986)MathSciNetCrossRefGoogle Scholar
  97. 97.
    X.-H. Zhang, Phys. Rev. D 34, 991 (1986)MathSciNetCrossRefGoogle Scholar
  98. 98.
    E. Poisson, A. Pound, I. Vega, Living Rev. Relativ. 14, 7 (2011)CrossRefGoogle Scholar
  99. 99.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, 4th ed., edited by Yu.V. Geronimus, M.Yu. Tseytlin (Academic Press, New York, 1965). first appeared in 1942 as MT15 in the Mathematical tables series of the National Bureau of StandardsGoogle Scholar
  100. 100.
    G.E. Shilov, Generalized Functions and Partial Differential Equations: Mathematics and its Applications (Gordon & Breach, Philadelphia, 1968) translated by B. SecklerGoogle Scholar
  101. 101.
    L. Blanchet, G. Faye, J. Math. Phys. 42, 4391 (2001)MathSciNetCrossRefGoogle Scholar
  102. 102.
    J. Steinhoff, G. Schäfer, S. Hergt, Phys. Rev. D 77, 104018 (2008)MathSciNetCrossRefGoogle Scholar
  103. 103.
    L. Blanchet, Living Rev. Relativ. 5, 3 (2002)CrossRefGoogle Scholar
  104. 104.
    L. Blanchet, T. Damour, G. Esposito-Farè, Phys. Rev. D 69, 124007 (2004)MathSciNetCrossRefGoogle Scholar
  105. 105.
    L. Blanchet, B.R. Iyer, Phys. Rev. D 71, 024004 (2005)CrossRefGoogle Scholar
  106. 106.
    W.G. Dixon, Post-Newtonian approximation for isolated systems by matched asymptotic expansions I. General structure revisited, arXiv:1311.6028 [gr-qc] (2013)Google Scholar
  107. 107.
    A. Papapetrou, Proc. Phys. Soc. A 64, 302 (1951)CrossRefGoogle Scholar
  108. 108.
    D. Puetzfeld, Y.N. Obukhov, Phys. Rev. D 92, 081502 (2015)CrossRefGoogle Scholar
  109. 109.
    A.I. Harte, Motion in Classical Field Theories and the Foundations of the Self-force Problem, in Equations of Motion in Relativistic Gravity, edited by D. Puetzfeld, C. Lämmerzahl, B. Schutz (Springer International Publishing, 2015) pp. 327--398Google Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physics & AstronomyUniversity of MissouriColumbiaUSA

Personalised recommendations